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Complex dynamics in two-dimensional endomorphisms
0362-546X/80/1101-3
Nonlineor Annlysir, Theory, Merhods & Applrcolionr, Vol. 4, No. 6, pp. 1167-l 187. @ Pergamon Press Ltd. 1980. Printed in Great Britain
COMPLEX
167 SOZ.oO/O
DYNAMICS IN TWO-DIMENSIONAL ENDOMORPHISMS C. MIRA
Groupe
Systemes Dynamiques Non Linkaires et Applications, Universitt Paul Sabatier, 31400 Toulouse, France (Received 24 July 1979)
Key words: Endomorphisms, tions, strange attractors.
recurrence
equations,
point mapping,
iteration,
chaotic
behaviour,
bifurca-
1. INTRODUCTION IN MANY practical contexts apparently stochastic processes are encountered.This apparent stochasticity cannot be attributed to some erratic fluctuations of the process parameters, nor to some faulty sampling or measurement technics. When a deterministic mathematical model of such a process is formulated, following a well established ‘standard procedure of the discipline concerned, then it is found that the model does not admit any stable constant or periodic stationary states, at least if the period of the latter is longer than the duration of observations. The models may have a wide range of forms, but one property is common to all of them: the presence of a strong non linearity. Such a complex behaviour of solutions appears in ordinary differential equations [l, 21. and in recurrence equations (equivalent names: recurrences, iterations, point mappings T). In the case of these latter equations of an order (dimension) m Q 2, Tbeing a mapping of real axis onto itself (m = l), or a mapping of a plane into itself, apparently stochastic processes are encountered when: T is a mapping with a unique inverse (diffeomorphism) sufficiently close to a conservative mapping [3] (order m = 2), or a conservative mapping; T is a mapping with a non-unique inverse T-’ (endomorphism), m d 2; T is close to a conservative mapping, but with a non-unique T- ’ [3,4], m = 2; T can be also a mapping with a constant Jacobian [5], or with a constant Jacobian perturbated by a function having a small coefficient. This paper considers only the second case, for the class of two-dimensional endomorphisms having a vanishing Jacobian, when T is defined by differentiable functions, or having the equivalent property for non-differentiable functions (cf. [28, Chapter 61): X n+
1 =
fb,,
Y,, 4,
Yn+ 1 = g(x,, yn, A),
II = n,, n, + 1,. . . .
(1)
In (l), x, Y are real variables, 1 a parameter,f, g, are single-valued functions either continuously differentiable, or piecewise differentiable, or piecewise continuous. The complex solutions are known as ‘completely invariant region’ in [6,7], ‘attractive limit set’ in [8,9], ‘bounded stochastic solutions’ in [3], and more recently as ‘chaotic solutions’ [lo], or ‘strange attractor’ [5]. Mathematically, such a regime corresponds to the property that a very simple point mapping, like (1) may define an extremely complex function xn = h, L-n,x(n,), Y(Q, 21,
Yn = h,Cn, x(n,), Y(Q, I167
4.
(2)
1168
C. MIRA
When (1) is considered as a dynamic system, then (2) constitutes its general solution, x(n,), y(n,), being the ‘integration’ constants, i.e., the insertion of (2) into (1) yields an identity. For one-dimensional endomorphisms, the first basic contribution to the problem of complex dynamics is that of Myrberg [ll]. Since the number of authors involved is large (see [lo]), only references pertinent to the specific context of this paper will be quoted. In one-dimensional endomorphisms, the chaotic behaviour of the solution appears as a manifestation of an infinite number of ‘basic’ bifurcations, and of the accumulations of the latter, forming ‘composite’ bifurcations. The complete bifurcation structure is orderly in the parameter space %, and it turns out to be of a box-within-a box type [12-161. This paper is devoted to the two-dimensional case: A first part considers the mechanism of the appearance of ‘chaos’ from orderly dynamics, in the case of a particular example, with differentiable data, first studied in [17]. The study of some fundamental bifurcations chains permits to conjecture the existence of a bifurcation structure of a box-within-a-box type. The second part concerns the characterization of the boundary (f’) of the ‘chaotic’ area (Ii) in the phase plane (x,, y,). This area is such that, for an initial condition (x(n,), y(n,)), sufficiently close to (6) (inside the domain (D) of attraction of(d)), after some iterations the sequence of points (xy, y,) enters into (d), and inside (6) these points have an apparently erratic movement without any possibility to escape. The ‘chaotic area’ is constituted by (d) and its boundary (f). Results are given when f and g of (1) are continuously differentiable functions, when f, g are piecewise differentiable functions, when J; g are piecewise continuous functions. The last part defines the bifurcation that destroys suddenly, for a small variation of the parameter k the ‘chaotic area’, a repulsive set taking up the place of(d). This bifurcation is a non-classical one, and corresponds to a contact of a certain order, depending on the differentiability off and g, between two singular curves of different nature. The aim of this paper is not a theory of two-dimensional endomorphisms, but only to give an introduction to these newly considered problems through some particular cases and ‘transparent’ examples.
2. TRANSITION
‘ORDER TO CHAOS’ DIFFERENTIABLE
IN
A RECURRENCE FUNCTIONS
DEFINED
BY
2.1. Purpose Following an idea of Poincart, the functions (2) can be characterized by the singularities of (l), similarly to the way a function of a complex variable is characterized by the enumeration of its zeros, poles and essentially singular points in the complex plane. These singularities are either of dimension zero-f%ed points and periodic stationary states called cycles (a cycle of order k is represented by k points verifying x, +k = x,, y, +k = y,, xn + j # x,, y, + j # y,, j -C k), or of dimension one-invariant curves crossing fixed points and cycles. The transition from orderly to complex dynamics is characterized by the appearance of an infinity of singularities in a finite domain of the (x,, y,) plane. These singularities are generated by typical mechanisms of bifurcations. As a first step towards the identification of the mechanism of dynamic complexity in two-dimensional endomorphisms, the following recurrence, or point mapping T: X
n+1
J’n+ 1
exp[L(l
=
xn
=
aXn[
- x,/K)
1 - exp( - ayJ]
- ayJ (3)
Complex
dynamics
in two-dimensional
1169
endomorphisms
model is considered [ 181, J, a and K being parameters. Recurrence (3) represents a predator-prey of two species with non-overlapping generations, first studied, from this point of view, in [17], and used as a starting point of the numerical study of some bifurcation sequencesgiving rise to the singularities of dimension zero. For a different recurrence a similar mechanism of dynamic complexity is described in [3, pp. 207-2141. 2.2 Bifurcation
of the stable constant stationary
state of (3)
Real roots of the algebraic system of equations x,,+ 1 - x,, = 0, y,, r - y, = 0 are called fixed points of (3). Fixed points correspond to a constant equilibrium between the two populations densities. When a > 0, the recurrence (3) admits three distinct fixed points. The first point is point qr(O, 0) is a saddle with eigenxn = yn = 0. In the phase plane .xn - yn the corresponding values s1 = exp I > 1, s2 = 0, and eigen directions coinciding with the xn and y, axes. The second and the third fixed points are given by the roots x, y, of: I.(1 - x/K) - ay = 0.
y = ax[l
- exp( - ay)].
(4)
One root of (4) is obvious, x = K, y = 0. The fned point q@, 0) is either a repulsive node, or a saddle depending on whether the eigenvalues s1 = 1 - 2, s, = aaK, satisfy Is11 > 1, s2 > 1 or Isi/ < 1, s2 > 1. The eigen directions of qz have slopes pr = 0, pZ = (1 - A - aaK)/a. The fixed points q1 and qz are continuations of the fixed points of the first equation of (3), when a = 0. The latter limits the behaviour of the prey on the x axis, which is an invariant curve of (3). The coordinates (x,, y,) of the third fixed point q3 cannot be given explicitly, because (4) is a transcendental one. They are however readily determined numerically. Several properties of q3 can be deduced directly from (4). Consider the auxiliary parameter y = x,/K, which is a measure of the extent to which the predator can depress the prey below its carrying capacity (cf. [17]). Using y, (4) can be rearranged into: a=v[l-exp(ny-A)]-‘. Inserting (4) and (5) into the characteristic equation of qJ, leads to an explicit expression sum G = s1 + s2, and the product A = slsZ of the eigenvalues sl, s2 of q3:
0=1-A+
1 - 41 expI(y
1 -
Y)- 1)’
A = A(1 - y)
of the
-
1 -y;lexp(yA exp(yl - 2)A)’
The expressions (6) are sufficient to define the (phase-plane) properties of the fixed point q3, except when neither 1.~~1= 1 nor lsZl = 1. The bifurcation of qj vs an increase of/z is now considered. Due to the non-availability of the required explicit expressions, the following analysis is carried out for the fixed parameter set K = 10, a = 1, y = 0.4 (cf. [17]), and 0 < 1 < Amax.The largest value iz,,, of biological interest is about five. For any given ,?, the value of a is fixed by (5) For the chosen values of K, a and y, the fixed point q3 is a focus because for 0 < 1 < 5, A > 0, 1 - (T + A < 0, 1 + c + A < 0, gz - 4A < 0. The eigenvalues of this focus are s1 z = A exp (+ icp), 9 = TC- arc tg J(4A/a2 - 1). The focus q3 is asymptotically stable when A < 1, i.e., when 2 < i, i, = 0.70695195. The critical value ACis a root of A = 1, for which, in linear approximation, the focus degenerates into a center. An analysis of the Liapunov critical case A = lc has shown that due to the non-linearity, q3 is actually an asymptotically stable composite focus of
1170
C. MIRA
multiplicity one [19]. For A = AC+ E,0 < E d 1, this composite focus bifurcates into an unstable (ordinary) focus and into a closed stable (attractive) invariant curve (C) [19]. The curve (C) encloses qJ and contracts into it as E --, 0. It has been found by numerical analysis [18] that the singly connected area DCenclosed by (C) increases simultaneously with ,? - AC. 2.3. Ttc,ofundamental bifurcations occurring in (3) When A varies and crosses a bifurcation value &, the cycles of order k appear or disappear by means of two fundamental bifurcations described below. (a) In the case of the first bifurcation, A = Ah - E, E being a small positive or negative value, corresponds to the absence of cycles of order k in a certain non connected region of the phase plane. When A = 1, + E, two cycles of order k exist in this region. One cycle is a node characterized by two eigenvalues si, s2, satisfying Isi1 < 1, lsZI < 1 (stable node) or Isi1 > 1, lsZl > 1 (unstable node). The other cycle is a saddle characterized by (si( < 1, jsZl > 1. For A = I,, E = 0, the node and the saddle merge into a double cycle having one of its eigenvalues equal to + 1 (cf. [20, 281, for details about this bifurcation). (b) In the case of the second bifurcation, a cycle of order k (either stable node or a saddle) exist for I = X,, - E with s2 -+ - 1 as E --f 0. For A = ;Z, + E, this cycle still exists with changed properties (a stable node becomes a saddle, or a saddle becomes an unstable node), and it gives rise to a cycle of order 2k, which merges into the cycle of order k (one of the s --f + 1) as E --+0 (cf. [20, 281 for details about this bifurcation). 2.4. Transition ‘order chaos’ When I increases beyond the bifurcation value Ah= 2.180 118 343, the invariant curve (C) enclosing q3 ceases to exist, and the bifurcation (a) (E > 0) takes place with the appearance of two cycles of order k = 5 (a stable node and saddle). Each of these cycles gives rise to two chains of bifurcations, when I - A,,increases. One chain is characterized by the appearance of stable nodes of order k = 5 x 2’, i = 1,2,3,. . (already
Fig. 1. 1 = 2.50200.
Complex
dynamics
in two-dimensional
endomorphisms
1171
Fig. 2. 1 = 2.50440.
described in [17] via the bifurcation (b), (E > 0)), as A crosses the value &. The other chain is characterized by the appearance of saddles of order k = 5 x 2’, i = 1,2,3,. . . , by the same type of bifurcation, as A crosses, the value Xg. When i --, 00, Xhi -+ AC, = 2.50101, & -+ A:, # 2.56. Each chain of cycles of order k = 5 x 2’ gives rise to points which are located on 20 segments of invariant curves, the slope of the latter at the cycle points being equal to the eigen direction associated with the eigenvalue s < 0, which was equal to -1 at the bifurcation. There exist thus two sets of invariant curve segments, say (Ci) and (C). When 1 increases beyond ;Icl, A:,, cascades of cycles of order k = lOk,, k, = 5,6,7,. . . , are produced on the segments (Ci), (Cl) by a mechanism similar to the mechanism of a first order recurrence, as long as 1 < AC2for the first chain, and
Fig. 3. 1 = 2.50445
1172
C. MIRA
w
4 4
8
12
16
X”
Fig. 4. 1 = 2.53000. A < A:2 for the second one. Ac2,AL, are critical values such as when Acz = 2.504 43, ;i > AL2 > Ac2, cycles of any order appear, without being located on the invariant curve segments (C,), (Ci). In the case of the first chain, invariant curve segments are shown in Figs. 1 and 2. These segments were traced by means of computer generated iterates of a point (x,, y,) chosen at random when
Table 1 I
k
2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8
1 4 4 8 8 9 9 11 11 11 11 12 12 12 12 13 13 13 13 14 14 15 15 15 15
X 4 3.312430 17.03128 3.180551 7.917645 3.829778 10.32207 0.626571 0.482222 0.567141 0.381109 3.485239 4.831342 3.201826 15.189918 4.318845 3-791522 4.621741 3.766182 4.363598 4.526641 2.104981 3.007712 1.047865 2.099552
Y
s, OrP
s,Or (Pra.3
3.254504 5.517332 0.844316 4.247864 3.515393 1.808717 1.597601 2.901126 2.622240 2.673334 2.136189 3.567009 4.948113 6.305706 1.135507 3.46004 1 4.430569 3.473271 4.842982 3.478267 3.492576 2.892633 2.902543 1578465 0.382328
p
cp 1.46702 - 2.2505 - 2.3289 - 6.1749 - 8.8692 - 4.6008 cp 1.8852 -6.1536 - 5.3358 9 0.76271 0.0352 - 13.7541 - 20.4779 7.9791 - 1.1860 12.2642 u, I.89920 rp 1.49976 -8.1386 - 14.9376 - 14.9965 -16.1313 rp I.48191 - 39.2925 - 20.4340
1.27821 2.1280 - 0.2800 3.3152 - 0.7794 4.9127 p 4.0586 6.8988 - 2.3767 p 6.50627 7.2258 7.8712 - 2.4775 13.3090 28.4793 13.6358 p 7.29776 p 12.1489 7.6521 10.9485 - 8.2747 19.7328 p 17.0089 - 0.2540 8.7335
Pl
p2
- 0.55700 -0.09633 - 0.05278 -1.71531 0.26724
- 0.18422 -4.09873 -0.47665 -2.01417
1.80342 2.68556
18.5582 1.84582
72.9319 0.19017 - 12.7961 24.6873 - 0.43033 -0.39927 - 12.7961 24.6873 - 2.72469 0.09972 0.08747 0.55963
4.87704 - 1.89137 - 0.42644 - 0.53797 -0.10295 1.41824
0.82067 - 1.37150 - 0.79053 - 1.75215
65.6279 1.08212
0.01395 0.00785
99ac&l
1173
Complex dynamics in two-dimensional endomorphisms
k, is large. The first thousand iterates were omitted in order not to show the transient leading to this ‘strange attractor’. Figure 3 concerns the case 1 = ACz + E, E > 0. When i > AC2,the cycles of any order k appear by the bifurcation process (a). If a pair: stable node-saddle appears, each of the two cycles of order k produces a chain of bifurcations of cycles of order k x 2’, i = 1,2,3,. . . already described for k = 5. If an unstable node-saddle pair
Table 2
2.18018343 2.20 2.24 2.32 2.36 2.40 2.418520
5 5 5 5 5 5 5
10.732994 11.104020 11.231903 11.098448 10.932706 10.718746 10.604265
1.260072 1.348086 1.416665 1.526232 1.582591 1.643820 1.674610
0.99969 0.40846 -0.01646 -0.53173 - 0.73400 -0.68764 - 0.60946
-0.81264 - 0.89921 -@95478 -0.90879 - 0.82000 -0.91829 -0.99998
021126 0.15856 0.11754 0.07421 O-06189 -0.13706 -0.19109
2.42 2.5044
5 5
10594690 9.917704
1.677154 1.857251
-0.60271 - 0.05269
- 1.0064 - 1.3577
-0.19443 - 0,30404
2.418526 2.42 2.46 2.48
10 10 10 10
10.607628 10.746561 11.528822 11.799213
1.674785 1.681601 1.651036 1.624379
0.99999 0.97388 0.30663 0.21995
0.37141 0.37488 - 0.02898 -0.83705
0.04828 0.07106 -0.19055 -0.17863
-0.19120 -0.19456 -0.23926 -0.29372
2.48387 2.5044
10 10
11.843415 1.620041 12.03 1549 1.602742
0.19289 @01628
- lGOO3 - 1.8624
-0.17728 -0.17304
-0.30108 -0.32963
2.48387 2,488 2.498
20 20 20
11.822445 1.626356 11.233094 1.817841 10074583 2.231221
0.99871 0.36889 0.16852
0.03700 0.02069 -0.77398
0.30127 - 0.30797 -0.36612
2.498982 2.5044
20 20
9.968058 2.272971 9.631132 2.417525
008125 - 0.2507
- 1.0001 -2.8165
- 0.37746 -0.44517
0.18373 Stable node - 0.28843 -0.33490 A cycle k = 40 appears when S = - 1 - 0.34258 -0.36966 Saddle
2.498982 2.5 2.501
40 40 40
9.973295 2.271177 10.370219 2.139227 lo.565799 2.076044
000658 -0.23465 - 0.996 11
-0.34268 -0.33867 -0.32990
-0.37683 -0.34391 -0.33895
2.501005 2.5044
40 40
10.566667 2.075765 11,008963 I.936289
- 0.32986 -0.30792
- 0.33893 - 0.33170
@99977 0.19930 0.03432
0~03400 - 1GOo4 - 0.0260 -4.8189
-0.27951 -0.33333 -0.40158 -0.94189 0.28019 0.05192 0.04902
Stable node
A cycle k = 10 appears whenS = -1
OQ4882 0.4655 1 Saddle Stable node A cycle k = 20 appears whenS = -1 Saddle
Stable mode A cycle k = 80 appears when S = - 1 Saddle
appears, only the saddle can give a chain of bifurcations of cycles of order k x 2’. Moreover, some cycles of order k = 5k,, k, = 1,2,3,. _. , generated in the parameter interval 1, < 1 < jlC2, disappear gradually by means of the bifurcation process (a) with E < 0. When A increases beyond AC2,the bifurcations processes referred to above (cycles of any order) give rise to a larger and larger number of cycles chains, each consisting of an infinite number of unstable cycles of any order. The points of these cycles are located inside a domain (0) of the x,-y,, phase plane, limited by five critical curves (see below). From any initial point (x,, y,), x0 > 0, y, > 0, the sequence (x,, y,) enters into (D) after some iterations, n > N, and wanders ‘erratically’ inside (D) without showing any tendency to settle down (see Fig. 4 showing the computer generated iterates points in (D)). For 1 = 2.8, Table 1 shows the first unstable cycles of any order inside (D) (a cycle is defined in
1174
C. MIRA Table 3 X
Y
5 5 5 5 5 5 5
10.732538 lo.287144 9.985692 9.807633 9.682230 9.681110 9.269490
1.259975 1.158971 1.068495 0.997228 0.934500 0.933971 0.682669
2.3196 2.36 2.40 2.44 2.48 2.50 2.5085 2.5090
10 10 10 10 10 10 10 10
9.718980 lo.597680 lo.937161 11.203921 11.437201 11.545597 11.590304 11.592911
0.927480 0.730962 0.655392 0.603801 0.565388 0.549551 0.543379 0.543015
2.5090 2.513 2.528 2.538 2.548 2.54809 2.5482
20 20 20 20 20 20 20
11.635391 11.791709 12.013580 12.109117 12.188625 12.189289 12.190099
0.5403 19 0.528800 0.509491 0.500358 0.492462 O-492395 0.492313
A
k
2.18018343 2.20 2.24 2-28 2.31955 2.32 2.5044
Sl
%
PI
P2
1.0003 1.5625 1.9174 2.1113 2.2233 2.2245 1.9297
-0.81258 - 0.79076 -0.84209 -0.91620 - 0.99991 - 10010 - 1.4523
0.21131 0.24693 0.25646 0.25380 0.24640 0.24629 0.18960
- 0.27947 -0.24399 - 0.22243 - 0.20746 -0.19435 - 0.19420 -0.13144
4.49448 5.3361 5.6827 5.9966 6.2690 6.3830 6.4256 6.3112
0.99957 0.62181 0.21550 -0.21508 -0.66528 -0.89681 -0.99641 - 1.0023
0.24364 0.19212 0.18256 0.17998 0.18179 0.18419 0.18552 0.18561
-0.19159 -0.11580 Saddle -0.09115 -0.05676 - 0.06764 - 0.06437 - 0.06318 A cycle k = 20 appears - 0.063 11 whenS, = -1
O-99082 0.80080 0.05925 -0.45923 -0.99508 -0.99997 - l+IO60
0.18170 0.16942 0.15633 0.15244 0.15020 0.15018 0.15017
- 0.06380 Saddle -0.06566 - 0.06759 - 0.06860 - 0.06977 - 0.06978 A cycle k = 40 appears - 0.06980 when S, = - 1
41.293 40.990 39.668 38.618 37,430 37.418 37.404
Saddle
A cycle k = 10 appears when S, = - 1 Unstable mode
Table 4 I
k
X
Y
s,
%
PI
2.7289833 2.7489839 2.7689839 2.82 2.82896 2.82898
4 4 4 4 4 4
2.315053 1.864128 1.701973 1.460036 1.429755 1.429690
5.918932 5.996512 6.009993 6.0 18542 6.019566 6.019157
0.99904 0.44321 0.16200 - 0.67724 -0.99914 - 1.0003
- 2.9055 -2.9019 -2.7334 - 1-9124 - 1.5748 - 1.5736
- 0.23454 0.13528 0.36153 1.0914 1.4022 1.4045
2.7289833 2.73 2.79
4 4 4
2.316809 2.427663 3.236830
5.918520 5.890867 5.609357
- 2.9048 - 2.8535 -2.3031
-0.23556 -0.29549 - 0.54579
30.407 50,206 Unstable node - 109.31
PI
P2
-0.41215 -0.41533 -0.70613 - 1.6686
-0.31945 -0.34287 -0.41026 - 0.42072
Stable node A cycle k = 60 appears when S = -1
-0.41206 - 0.41028 -0.39050 -0.38444
-0.31862 - 0.29927 0.09565 - 1.0761
Saddle
1QOlO 1.1208 2.0325
PI
30,200 8.2984 5.6266 Saddle 2.6027 2.0733 A cycle k = 8 appears 2.0717 when S = - 1
Table 5 1
k
X
Y
S,
2.49599439 2.496 2.497 2.5044
30 30 30 30
8.410206 8.405362 8.357578 8.317967
2.814921 2.816941 2.840472 2.881648
0.99322 -0.62768 0.60535 -0.68241 - 0.52492 - 6.6241 - -0.43887 -28,336
2.49599439 2.495998 2.4974 2.5044
30 30 30 30
8.410378 8.414381 8.519261 8.729455
2.814850 2.813216 2.777840 2.758054
1.0068 1.3171 1.1566 8.6853
S2
-0.62618 - 0.59660 -0.27635 -0.01316
Complex
dynamics
in two-dimensional
1175
endomorphisms
this table and in the following tables, by the coordinates of one point of the cycle, sl, s2, and the eigendirections pr, pz). When si, s2 are complex the columns sl, s2 give respectively the modulus p and the argument cpof the eigenvalues (the cycles is a focus). Tables 2 and 3 give the two chains of cycles of order k = 5 x 2’ from 2 = A,,.Table 4 shows the evolution of a pair of cycles of order k = 4. In Table 5 appears the possibility coexistence of two stable cycles for some sets of the A values. It seems that for 2 > ;lc2,all the cycles of (3), the number of which is infinite, are repulsive. Thus the presence of a chaotic area can be conjectured for these values. The study of some other fundamental bifurcation chains permits us also to conjecture the existence of a ‘box-within-a box’ type bifurcation structure. 3. DETERMINATION
OF
THE
CHAOTIC
AREA
3.1. Critical lines and absorptive area It was shown in [15] and [16] that, in the case of one-dimensional endomorphism ?: the domain of complex dynamics on the real axis is bounded by critical points (in the sense of JuliaFatou) of a certain entire power of ?: This notion can be extended in the case of a m-dimensional endomorph&m ?; m > 1. Then, a ‘critical variety’ (VC) (dimension rn - 1) of rank 1 is defined as the geometrical locus, in the phase space, of points having two or more coincident antecedents of the first rank (first iterates of T-l). In the general case, such a variety separates the phase space into two regions, the points of each of them having a different number of antecedents of first rank. The existence of real ‘critical varieties’ constitutes a typical property of the considered class of endomorphisms. In the two-dimensional case, m = 2, a critical curve, or ‘critical line’ (LC) of rank 1, is defined as the locus of points having two or more, coincident antecedents in the phase plane. A critical line (LCI) of rank r + 1 is the consequent curve of rank r from (LC). An ‘absorptive area’* (d’) is defined as a domain bounded by a closed curve made up of segments of critical lines (LX), (LC,), r = 1,2,. . . , such that, for an initial point (xc, y,) sufficiently close to (d’), the sequence of the points (x,, y”) enters into (d’) after some iterations, and inside (d’) these points cannot escape. This is an extension of the notion of ‘non-invariant attractive set’, or ‘absorptive segment’ E, in the one-dimensional endomorphism [15, 16,283. Two different situations are possible for an ‘absorptive area’ (d’). In a first case (d’) contains one attractive limit set (sometimes more) such as fixed point, cycle, invariant closed curve, invariant curve segments of the type appearing in Figs. 1 and 2. In a second situation, the absorptive area (d’) contains a chaotic area (d), in the absence of an attractive limit set of the above type inside (d’). When this situation occurs, whether the chaotic area (d) tills (d’), or (d) fills (d’) with the exception of some holes containing points of repulsive cycles. However, frequently the complete structure of bifurcations of (1) being not known, the absence of such cyclical limit set is not sure. From a practical point of view, the transient to such sets, with a cyclical order greater than the number of observed iterations, can also be considered as a chaotic area. At the present time, nothing can be said about the structure of the numerically obtained ‘strange attractors’. 3.2. Endomorphism defined by continuously differentiable functions When the functions J g of (1) are continuously differentiable with respect to x, y, (LC) is the * Terminology
proposed
by Prof. R. Thorn (private
communication).
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locus of the first rank consequents of the curve J(x, y) = 0, J(x, y) being the Jacobian of the righthand side of (1). Since the curve J = 0 constitutes an antecedent of (LC), it is designated by (LC _ 1). The critical lines of power k, k = integer, of T, (Tk) are the set of curves (LC), (,X,), i = 1,2,... , (k - l), (LC,) being the consequent curve of (LC,_ i). A curve (LC,) can also be considered as a critical line of rank (i + 1). To simplify the exposition, the following hypotheses are first made: The curve (LC _ 1) has only one branch. Then, (LC) and (LC,), i = 1,2,3, . . . , are also made up of one branch. Moreover it is supposed that (LC) and (LC_ i) have a single point of intersection, and that (LC,) and (LC) have no double point. Let h(x, y) = 0 be the equation of (LC). The curve (LC) separates the phase plane into two regions: h(x, y) > 0, and h(x, y) < 0. A point of the former possesses two antecedents of first rank, and a point of the latter none. Then, for the recurrence (l), the following lemmas can be shown to hold: (A) Let a, (x,, y,) be the intersection of the curve (LC_ i) and (LC). The consequent a, (x,, y,) of a, is a point of first-order contact between (tC) and (LC,), or exceptionally a cusp point of (LC,) on (LC). (B) The sequence of consequents an (x,, y,) of a, (x,, y,), n = 1,2,3,. . , constitutes a sequence of first order contact points (or exceptionally of cusp points) between (LC,_ i) and (LC,). (C) The consequent of the intersection of (LC_,) and (LC,) is a first order contact between WC) and WC,, i) (D) The curve (LC,) separates the region h(x, y) > 0 into three subregions, where the number of antecedents of second rank (second iterate of T - ‘) of (x,, ya is none, two, or four. Similarly for the (LC)‘s of a higher rank. . . . For the proof of Lemma A, it is noticed that the point a, belongs to (LC) and (Xi), since it is the consequent of the intersection of (LC_,) and (LC). The curve (LC,) cannot cross through (LC), otherwise (LC,) would have a branch in the region h(x, y) < 0, which is impossible because a point has no antecedents. Then a, is a point of first order contact between (LC) and (LC,), or a cusp point (exceptional case) of (LC,) located on (LC), in the region h(x, y) > 0. From the Lemma A, the Lemmas B, C, are deduced The Lemma D is the consequence of the separation by (LC) of the plane (x, y) into two regions, with zero and two antecedents of first rank, and the equivalent properties for the successive powers of T, T i , i = 2,3,4, . . . For m > 2, equivalent lemmas occur, the equivalent of points a, being varieties of dimension m - 2. From these lemmas, the following proposition results: PROPOSITION
1. Let the recurrence
(1) be such that T has an unique
branch
(LC_ r) crossing
(LC)
inan unique point a,. Let e i, i = 1,2,. . . , m, be a sequence of critical line segments of Tmel such that a, is a first order contact point, or exceptionally a cusp point of (LC,) on (LC,_ 1). Let b, be the first intersection between (LC_ 1) and a segment uzl of (LC,), b, the consequent of b,; therefore b, is a first order contact between (LC) and (LC,+l). In the simplest case, an absorptive
area (d’) is bounded
by the closed curve (9) constituted
by the segment G
of (LC),
the sequence a, and the segment 7 a,,,, 1 1 of (LC,+ i). Then the rank m, defining b, is the least rank such that the sequence uj, j > m + 1, cannot be outside of the domain bounded by (Z), as also the parts of the segments of (LCj), (LC,_ 1) located between uj and the first point of contact with (9). In this simplest case, when a chaotic area (d) exists, (d) c (d’), (9) E (d) and (9) is an ‘external’ boundary of the closed set (d).
Complexdynamicsin two-dimensional endomorphisms
1177
This proposition is a direct consequence of the definition of (d’) and of the above lemmas. (2’) is called an ‘external’ boundary because generally (d) does not fill all the domain bounded by this closed curve, taking into account some holes containing points of repulsive cycles. In more complex cases, parts of the segments of (LX,), (LC,,_ J i > m + 1, located between uj and the first point of contact with (LZ), can be outside of the domain bounded by (_Y), aj being inside, or outside of this domain (cf. [27] and [28]). It is worthy of note that, for a range of values of the parameter 1, the absorptive area (d’) contains one attractive limit set (sometimes more), such as fixed point, cycle, invariant closed curve, and invariant curve segments of the type appearing in Figs. 1 and 2. For another range of 1, when a ‘chaotic area’ exists, which implies the absence of an attractive limit set of the abovementioned type, the absorptive area (d’) turns into a chaotic area (d). In this case (d’) contains an enumererable infinity of repulsive cycles, and (d) c (d’), (9)~ (d). It is added that a segment
G
of (9) can be constituted
by two parts of (LC,), that are joined
is a point of non-differentiability
in a double point. This point
(cf. Figs. 5 and 7 for i = 1,2,3).
for a=,
.
C 4
8
12
16
X” Fig. 5. I = 2.80000.
For one-dimensional endomorphisms, xn + 1 = f(x,, A), f continuously differentiable function, this last situation is only possible for a nowhere dense set of the values R, denoted A: in [12, 15, 161, and giving rise to ‘cyclical stochastic segments’. With A = A:, the infinite chains of the antecedents of a critical point are everywhere dense on these segments. For 1 # A:, and inside the range giving an attractive set at finite distance, there exists at least an attractive cycle, because when a bifurcation occurs, in the one-dimensional case, there always appears at least such a cycle associated with one or more repulsive cycles. Iq two-dimensional endomorphisms, repulsive cycles can appear without being associated with an attractive one (see Section 2.3a). On the contrary of the one-dimensional case, the set of values A corresponding to a ‘chaotic area’ thus can be continuous. The analogy (due to the absence of attractive cycles) between a ‘chaotic area’ and the ‘stochastic segments’, obtained for 1 = AZ with one-dimensional endomorphisms, as numerical experiments, suggest that the infinite chains of antecedents of almost every point of the boundary (9) are dense on (d).
1178
C. MIRA
X” Fig. 6. 1 = 2.53.
Fig. 7. A = 2.80.
Complex dynamics in two-dimensional
1179
endomorphisms
The above results can be illustrated by the recurrence (3) and (5), with a = 1, K = 10, y = 0.4. The region x > 0, y > 0 is the domain of attraction of the ‘chaotic area’ considered above. The equation of (LC_,) and (LC) are respectively: x = K,?.-’ exp(ay); h(x, y) = X’K exp[;l 1 - A~/K] - x = 0. For Figs. 6 and 7, the critical lines of T5 are drawn with 1 = 2.53 and ,J = 2.80. (LC_ J intersects (LC,) in a point d,. However the segments a,d, cannot close (d’), otherwise a4 and d, would be outside (d’). The point b, consequent of b,, intersecz bzeenJLCA) and (IL,) gives the correct area (d’). This area is bounded by the segments b,a,, a,~,, a2aj, a3a4, n a,b,, the points aj, j = 5,6,. . . , belong to (d’). Figures 6 and 7 can be compared with the displays of numerical iterates of randomly chosen initial points (Figs. 4 and 5). It is worthy of note that some segments of critical lines appear as places of concentration of iterate points in these figures. Figures 6 and 7, suggest that, in the interval 2.53 < 2 < 2.80, there exists a value X of A, such that a point a, is a cusp of (LCJ located on (LC, _ 1). In order to determine this value, the parametric equation of (LC,) is considered: x = K,/I exp A exp[l - exp A - at], i y = K/1 [l - exp( -at)]
A = 1 - 1 - At,lK
exp A,
X is defined by equalling to zero the radius of curvature of (LC,) in a, (t = (A - 1) K,/(A + aK)). The cusp is obtained for # 2.624. If (LC) has two branches, separating a region where a point (x, y) has three antecedents, and a domain where a point has a unique antecedent, (d’) is again defined by segments of (LC), (LC,) having the same above properties. Examples of ‘chaotic area’, with (LC) having two branches, can be encountered in [3, pp. 207-218,4, 15,281. 3.3 Endomorphism defined by piecewise differentiable functions One of the two functions f, g, of(l), or the two functions are piece-wise differentiable such that the curves of the one parameter family, f(x, y) = GI(a = parameter), have a zero-order contact with the curves of the one parameter family g(x, y) = /I (y = parameter). The geometrical locus of these points of contact is designated by (LC_ 1), since in the differentiable case, the property of first order contact between these two families is expressed by J(x, y) = 0. The arguments of Section 3.2 give the same statement of the Lemmas A, B, C and D and of the Proposition 1, by taking the place of ‘first order contact points’ by ‘zero order contact points’, i.e., an angular point of (LC,) is located on (Xi_ i) on the same side of (LC,_ 1). A very simple example of such a situation is discussed in Section 4 (Figs 12 and 13). Another simple case corresponds to the recurrence T: Y” - 0,2&, X “+l
= (1 -
4Xn
+ Y,,
Y”+l =
Y” -
215
IX”1 d 5 +
yn-2Ax,-9&
91,
xn > 5 x,,<
(7)
-5.
For (7), WC_,) (x = 5), (U-J (x = - 5), are the equivalents of the curve J = 0 in the differentiable case. The equation of (LC) is y - x = 4il - 5, and that of (LC’) y - x = 5 - 41. In the region located between (LC) and (LC’) a point has 3 antecedents of first rank. Outside of this region a point has only 1 antecedent.
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Fig. 8
Fig. 9. 1 = 2.4, I div. = I.
Complex
dynamics
in two-dimensional
endomorphisms
1181
In T a cyclical chaotic area of order 2 appears of 1 = A, + E, A, = 1617.2 = 2.222.. . , E > 0 sufficiently small, i.e., there exist two chaotic areas of T2. When 1 31 2.4 these two regions have a common part, giving a simple chaotic area (for details, see [3]). When A > 2.532 this attractive limit set disappears, and it does not exist attractive set at finite distance. Using the above results, it is not difficult to construct the boundary (2) of the ‘chaotic area’ which corresponds to the ‘absorptive area’. The sequences (LC,) (LC), i = 1,2,3,. . . are drawn of the equion Fig. 8 for 1 = 2,4, the points a, bi, ci, d,, ei, gi, i = 1,2,. . . being the consequents valent points, with the index zero, located on (LC_ i). and (LC_ i). Numerical iterations give Fig 9, which can be compared with the boundary (2) drawn with thicker lines in Fig. 8. It is worthy of note that inside (d), some segments of (LC,), drawn in Figure 8, appear, in Fig. 9, as separations of regions with different densities of iterates. 3.4. Endomorphism defined by piecewise continuous functions One of the two functions f; g of (l), or the two functions, are piecewise continuous, such that the curves of the one-parameter family, f(x, y) = a, have a zero order contact with the curves of the one parameter family, g(x, y) = B, (a, p, parameters) in a point where the continuity ceases to exist. The geometrical locus of such points constitutes segments of (LC_ i). Then the segments of (LC), consequent of (K-i), separate in the phase plane regions where the number, or the nature, of antecedents of first rank is different. When a ‘chaotic area’ (d) exists, (6) is bounded by segments of (LC), (ZC,), i = 1,2,. . . , as for the previous cases. The following example illustrates this property. Let be the recurrence T: (8)
Fig. 10. 1 = 1.12, p = 11
1182
C. MIRE
Y”
C.,
Fig. 11.
with I = 1.12, p = 11. (LC_ r) is the straight line x = 6. (LC) and (LC’) verify respectively y = x - 62, y = x - 61 + I-(-(LC) and (LC’) limit a region where a point has two antecedents of first rank, the complementary regions being such that a point has a unique antecedent of first rank. Let a,, and e, be the respective intersections of (LC_ r) with (LC) and (LC’). By application of T, a, has two different consequents, a, defined by (8) for x < 6, and cr defined by (8) for x b 6. (LC,)is constituted by two half straight lines, a,, cr being one of the extremities. By the same way, e, has two consequents e,, h,, and (LC',) is constituted by two half straight lines. Numerical iterations give the chaotic area of Fig. 10 with regions of different density of iterated points. The sequences (LC), (LC'), (LC,), (LC),i= 1,2,3,4, are drawn on Fig. 11, the points ai, b, ei, hi being the consequents of a,, b,, e,, h,; di, gi, li, m, the consequents of d,, go, 1,, m, located on (LC_,). As in Section 3.3, regions with different densities of iterates appear, and segments of (LC,) and (LC)separate these regions. The two-dimensional recurrence (or point mapping T), associated to the Lorentz-Sal&man [I] differential equation of order 3, is piecewise continuous [2]. The chaos in the 3-dimensional phase space can be defined from the construction of the chaotic area of the two-dimensional associated point mapping. 4. BIFURCATION
DESTROYING
A CHAOTIC
AREA
Numerical experiments show that a ‘chaotic area’ (d) disappears suddenly, when the parameter I crosses a bifurcation value A*. Thus, if J. > 2.532 in (7), the unique attractive limit set is located at infinity.
Complex
dynamics
in two-dimensional
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1183
The properties of the boundary (F) of the domain of influence (D) of an attractive limit set (fixed point, cycle, chaotic area) play a prominent part in the mechanism of this bifurcation. Such a domain (D) is the zone of initial conditions (x,, y,), which will not prevent the sequences (x,, ya from returning to the considered attractive limit set. The boundary (F) of the domain (D) is invariant by application of T and T-‘. Generally (F) is defined from the invariant curve, crossing a saddle cycle, or a fixed point, which is associated to the saddle eigenvalue of modulus smaller than one, Properties of(F) are given in [7,28], and the plotting of an invariant curve crossing through a saddle fixed point is made by the means of mixed analytical-numerical methods [22, 23, 281. It is worthy of note that (F) can possess a complex nature. Indeed (F) can be a curve the tangent of which being indeterminate in each point. (F) can be also such that the domain (0) is not simply connected [9,24], (D) being constituted by an ‘immediate convergence domain’ (D,), containing the attractive limit set, and a certain number, sometimes infinite, of unconnected zones (D,). A more complicated situation occurs when (F) is a ‘fuzzy boundary’ or a ‘stochastic boundary’ [7, 251. In that case, the ‘fuzzy boundary’ is a strip of the phase plane, containing an infinity of repulsive cycles and invariant curves crossing through the points of these cycles. This strip separates the domain of influence of two attractive limit sets, and constitutes a region of uncertitude, for the evolution of the sequence (x,, y,) towards one of the two attractive sets, when the initial condition (x,, y,) is located insides the strip [28]. The following proposition can be shown to hold. 2. Given the recurrence (1) having a critical line (K) with one branch (situation of Proposition 1) or two branches (situation described at the end of Section 3.2 and with Example 7). Given an interval of the ,l values such that (1) has a chaotic area (d) in the (xn - y,J plane. This ‘chaotic area’ (6) disappears when i crosses through a bifurcation value I = il* defined in the two following cases: If the functions f, g of (1) are continuously differentiable, ;1* is the value such that a segment of a critical line (belonging to the boundary (2) of(d)) h as a first-order contact point with a segment of invariant curve, crossing through a saddle fixed point, or cycle, and belonging to the boundary (F) of the influence domain (D) of (4. If the functions f,g of (1) are piecewise differentiable, the same bifurcation occurs when a segment of critical line of (2) has a zero order contact point with a segment of an invariant curve of(F). PROPOSITION
Let be i = A* - E,E > 0 sufficiently small, a value giving a ‘chaotic area’ (d), and 2 = ;1* + E a value for which such an attractive area does not exist, and suppose that (F) is not a ‘fuzzy boundary’. This theorem is the consequence of the Proposition 1 and the properties of the boundary (F). Indeed, let be, for 1 = A*, a contact of first, or zero order, in a point PO,between (F) and the segment of (JX) belonging to (T), boundary of(d), such that (d) c (D). The consequents of PO are points of contact between (F) and the segments of (LC,), i = 1,2,. , . , belonging to (9). In this situation, the boundary (F) of(D) consists of the limit of the boundary of(D), with i = A* - E, E --f 0, and the infinite number of chains of antecedents j?_i of PO, i = 1,2,. . . , dense into (4. Each chain tends towards one point of a node repulsive cycle, which appears for il < I*. When i = A* + E, around each point /?_i antecedent of p, for E = 0, it appears a region belonging to the influence domain of an attractive set, located outside of(D). The area of these regions tend to zero when E + 0. The infinite chains of these regions cover the place, filled by (d) when ,l = ;1* - E,
1184
C. MIRA
except the set of measure zero consisting of the repulsive cycles points, that appear for L < A*, and certain invariant curves crossing through these points. Thus, for ,?. = A* + E, (F) no longer exists and, in the place of(d), there is a ‘repulsive’ region containing the repulsive cycles that have appeared for i < A*. A very simple generic example, showing this phenomena, corresponds to the point mapping, with piecewise differentiable data: XIIt1 = yn; i Ynil
y,+i
= y, - Ax,,
=yn+2x,,-6(2+1J
ifx,, < 6;
(9)
ifx,36.
Indeed, for (9) the boundaries (2) and (F) are made up of segments of straight lines, and A* can be calculated without approximations. Here the futed point 0 (0; 0) is a focus of type 1 [7], which is repulsive for 2 > 1, II = 1 being the value giving rise to a chaotic area. The boundary (8) is determined from the second fixed point A (X = y = 3(2 + A)), and the segment BC of the eigen direction (slope = - l), associated to the multiplicator (eigen value) S, = - 1, the second multiplicator of A being S, = 2 (slope of the eigen direction = 2). A is a degenerated saddle, and BC is a segment of repulsive cycles of order 2, with xB = y, = 6, y, = y, = 6(1 + 2). Then (F) is made up of BC and its antecedents B_iB_i+l,i = O,l,..., ~,B,,zB,B’_~B’_~+~~=~ ,..., 4,B’_, 3 C. (F) is closed by the segments B_ 4a,,, B’_ JZ~, x0 being the antecedent of the point ai, intersection between B _ 3B_4 and the critical line (LC), xjrO = 6. When I > 1, all the cycles are repulsive, and thus if 1 < ,J d i*, the absorptive zone (d’) merges with a chaotic area. For 1 < 13.< ;1,, 2, N 1.1, the chaotic area is an attractive limit set made up of 6 regions, the points of which exchanging themselves by application of T. For &, < 1 < A*, these 6 regions join, forming a ring shaped attractive zone, constituting the chaotic area. (LC_,) is the straight line x = 6. The equation of (Lc) is y = x - 62, and (LC) separates the phase plane into a region where a point has two antecedents of rank one, and a region where a point has no antecedent. The point C is located on (ZC). The sequence of the points ai(xi, y,), defined in Section 3 is such that: i =O,l,..., ‘i+l
=
yi,
x0 = 6,
y, = 6(1 - 4il + 3A2),
y, = 6(1 - J.),
y, = 6( 1 - 2L),
y, = 6(1 - 51 + 6A2 - L3),
y1 = 6( 1 - 3;1 + A2),
ys = 6(1 - 62 + 101’ - 4L3).
For this example up is a zero order contact point between (LC,_ i) and (LC,). When the chaotic area (d) exists, 2 < ;1*, a, is inside the region where a point has no antecedent. The bifurcation destroying (d) is defined for J. = A* such that CI~is located on (IX), thus merged with a,. Then the point a, is located on B_ 3B _4, which defines A* as the solution A = 1.2 of 52’ - 161 + 12 = 0. Numerical iterations give Fig. 12 which can be compared with the construction of the boundary (2) of(d) appearing in Fig. 13 (here 6,, d, located on (LC) are the antecedents of b,, d, respective “*, besides the indicated contour, the intersections of (LC_,) with uT5 and (IL,). For A = A boundary F is made up of all the sequences of antecedents of the point x,, E a,. When A = A* + E, E small positive value, c(~ is located into a region where a point has two antecedents of first rank. Around a, and its 2’ chains of antecedents, appear regions, the area of which tending towards zero if E + 0, that are part of the influence domain of the attractive point at the infinity. With ,I = I* + E, the sequence generated by (9) can remain into a domain reproducing the form of (61
Complex
dynamics
in two-dimensional
endomorphisms
1185
&
-10
Fig. 12. A = 1.199999.
for L = I*, during a large number of iterations, if (x0, y,) is located into a region containing an antecedent of a sufficiently large rank of ccO. In the following example (differentiable data): x n-l =y
Y
= y” - Jxn + x.”
(10)
(F) has the form indicated in [26] for an ~clui~z&t equation. A contact of first order between (9) and (8’) takes place, when ;1* # 155. For 1 < II < 1.4 there exists an attractive close curve inside
Fig. 13. 3, = i* = 1.2.
C. MIRA
1186
(D) bounded by (8’). This curve surrounds O(0; 0), a repulsive focus. For I N 1.4 a pair of cycles of order 7 appears. One is an attractive node, the other a saddle. The bifurcations of some cycles give rise to a chaotic area. 5. CONCLUDING
REMARKS
When a chaotic area exists in the differentiable case, numerical experiments have shown that segments of critical lines are places of concentration of iterate points. This property is explained by the fact that the consequent of a large zone, located on each side of the curve (LC _ ,), J(x, y) = 0, is a narrow strip bounded by (LC). In the piecewise differentiable case, it was seen that segments of critical lines appear as separations of regions with different densities of iterates. Remarking that the critical lines (LX,) can be considered as lines of folding of a Riemann surface with several sheets, (each sheet corresponding to a determination of the inverse mapping T-‘), the projection of these sheets on the x-y plane gives rise to regions with different densities of iterates. This paper does not claim to give a complete, or a strict theory of chaotic areas in two-dimensional endomorphisms. In particular, the conjecture related to the density on(d) of the antecedents of almost every point of (2) would require a rigorous mathematical proof (this conjecture is based on numerical experiments and analogies with the one-dimensional case), and its limits are not known. Nevertheless, the knowledge of the bifurcations giving rise to a chaotic area, and destroying it, the properties of the boundary of such an area here limited to particular cases, can help the understanding of more general situations.
Acknowledgements-The problem of ‘absorptive herein is supported by Direction des Recherches,
area’ was discussed with Professor R. Thorn. The research Etudes et Techniques under the Grant 77/l 170.
reported
REFERENCES 1. LORENZ E. N., Deterministic non periodic flows, J. Atmos. Sci. 20, 13s-141 (1963). 2. AFRAIMOVITCHV. S., BYKOV V. V. & CHILNIKOV L. P., Formation and structure of a Lorenz attractor, Dokl. Akad. Nauk. SSSR, T. 234, No. 2, pp. 336339 (1977). 3. BERNUWXJ J., Hsu Lru & MIRA C., Quelques exemples de solutions stochastiques born&es dans les rkcurrences du 2e ordre, Actes du Colloque Transformations PonctueNes et Applications (Toulouse, 10-14 Septembre 1973). EdItIons du C.R.N.S., Paris, 1976, pp. 194220. 4. GUMOWSKI I. & MIRA C., Point sequences generated by two-dimensional recurrences, Information Processing 74 (1.F.I.P. Congress), Stockholm, July 1974, pp. 851-855. North Holland Publishing Company, Amsterdam (1974). 5. HENON M., A two-dimensional mapping with a strange attractor, Commun. Math. Phys. 50, 69-77 (1976). 6. PIJLKIN S. P., Oscillating iterative sequences, Dokl. Akad. Nuuk SSSR, T. 73, NO. 6, pp. 1129-l 132. 7. MIRA C., Etude de la front&e de stabilitk d’un point double d’une rCcurrence non IinCaire du 2e ordre, Internutional Pulse Symposium, Budapest, 9-11 April 1968, D 43-7/11, pp. 1-128. 8. STEIN P. R. & ULAM S., Lectures in non linear algebraic transformations, Lectures Nato Adti. Studies Institutes, Istambul, Aug. 1970. D. Reidel Publishing Company, Dordrecht (D&Z. 1973). 9. GUMOWSKI I. & MIRA C., Sensitivity problems related to certain bifurcations in non linear recurrence relations, Automatica 5, 303-3 17 (1969). 10. MAY R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467 (1976). Il. MYRBERG P. J., Iteration der reellen polynome zweiten grades, Annales Acad. SC. Fennicue, math., 256 (1958), 268 (1959), 336 (1963). 12. GUMOWSKI I. & MIRA C., Accumulations de bifurcations dans une rtcurrence, C.R. Acad. Sci. Paris, t. 281, strie A, pp. 41-48 (1975). 13. MIRA C., Structure de bifurcations ‘boites-emboitkes’ dans les r¤ces du premier ordre dont la fonction prtsente un seul extrimum, C.r. Acud. Sci. Paris, t. 282, %rie A, pp. 219-222 (Jan. 1976). 14. MIRA C., Etude d’un modCle de croissance d’une population biologique, C.r. Acad. Sci. Paris, t. 282, sCrie A, pp. 1441-1444 (1976).
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endomorphisms
1187
15. MIRA C., Accumulations de bifurcations et structures boites-emboitkes dans les rkcurrences et transformations ponctuelles. Proceedings of the VIIe International Conference on Non Linear Oscillations. Berlin. Sept. 1975. Akademic Verlag, Berlin (1977). 16. MIRA C.. Systimes & dynamiques complexe et bifurcations boites-emboitCes, RAIRO Automatique, Vol. 12, No. 1 pp. 63-94, Vol. 12, No. 2, pp. 171-190 (1978). 17. BEDDINGTON J. R., FRLL C. A. & LAWTON J. H., Dynamics complexity in predator-prey models framed in difference equations, Nature 255, 58-60 (1975). 18. CLERC R., HARTMAN C. & MIRA C., Transition order to chaos in a predator-prey model in the form of a recurrence. Proceedings of Informatica 77, Bled (Yougoslavie), 3-l 16, pp. 1-4 (Oct. 1977). 19. GUMOWSKI I. & MIRA C., Bifurcation pour une rkcurrence d’ordre deux, par travershe d’un cas critique avec deux multiplicateurs complexes conjugks, C.r. Acad. Sci. Paris, t. 278, strie A, pp. 1591-1594 (1974). 20. MIRA C. & ROUBELLAT F., Etude de la traversCe d’un cas critique pour une recurrence d’ordre deux, sous l’effet d’une variation de paramktre, C.r. Acad. Sci. Paris, t. 267, s&rie A, pp. 969-972 (1968). 21. GUMOWSKI I. & MIRA C., Solutions chaotiques born&es d’une rbcurrence, ou transformation ponctuelle du 2e ordre A inverse non unique, C.R. Acad. Sci. Paris, t. 285, strie A, pp. 477-480 (1977). 22. GIRAUD A., Application des rCcurrences B 1’Ctude de certains systimes de commande, ThBse de Docteur-Ingenieur. Toulouse, 21 avril 1969. 23. GUMOWSKI I., Structure des solutions d’une rCcurrence conservative du 2e ordre avec non lin&aritC non bornbe. Actes du Colloque Transformations Ponctuelles et Applications, Toulouse, 10-14 Septembre 1973. Editions du C.N.R.S., Paris, pp. 79-92 (1976). 24. MIRA C. & ROUBELLAT F., Cas ou le domaine de stabilitC d’un ensemble limite attractif d’une rCcurrence du 2e ordre n’est pas simplement connexe, C.r. Acad. Sci. Paris, t. 268, sCrie A, pp. 1657-1660 (1969). 25. MIRA C., Sur la notion de frontihre floue de stabilitt, Proceedings of the Third Brazilian Congress of Mechanical Engineering, Rio de Janeiro. D-4, pp. 905-918 (Dec. 1975). 26. MIRA C., DCtermination pratique du domain de stabilitk d’un point fixe d’une rtcurrence du 2e ordre, C.r. Acud. Sci. Paris, t. 261, Groupe 2, pp. 53145317 (1965). 27. BARRUGOLA A., (to appear). 28. GUMOWSKI 1. & MIRA C., Dynamique Chaotique. ??ansformations Ponctuelles. Transition ordre-dtsordre. Editions CCpadues, Toulouse (March 1980).
Nonlineor Annlysir, Theory, Merhods & Applrcolionr, Vol. 4, No. 6, pp. 1167-l 187. @ Pergamon Press Ltd. 1980. Printed in Great Britain
COMPLEX
167 SOZ.oO/O
DYNAMICS IN TWO-DIMENSIONAL ENDOMORPHISMS C. MIRA
Groupe
Systemes Dynamiques Non Linkaires et Applications, Universitt Paul Sabatier, 31400 Toulouse, France (Received 24 July 1979)
Key words: Endomorphisms, tions, strange attractors.
recurrence
equations,
point mapping,
iteration,
chaotic
behaviour,
bifurca-
1. INTRODUCTION IN MANY practical contexts apparently stochastic processes are encountered.This apparent stochasticity cannot be attributed to some erratic fluctuations of the process parameters, nor to some faulty sampling or measurement technics. When a deterministic mathematical model of such a process is formulated, following a well established ‘standard procedure of the discipline concerned, then it is found that the model does not admit any stable constant or periodic stationary states, at least if the period of the latter is longer than the duration of observations. The models may have a wide range of forms, but one property is common to all of them: the presence of a strong non linearity. Such a complex behaviour of solutions appears in ordinary differential equations [l, 21. and in recurrence equations (equivalent names: recurrences, iterations, point mappings T). In the case of these latter equations of an order (dimension) m Q 2, Tbeing a mapping of real axis onto itself (m = l), or a mapping of a plane into itself, apparently stochastic processes are encountered when: T is a mapping with a unique inverse (diffeomorphism) sufficiently close to a conservative mapping [3] (order m = 2), or a conservative mapping; T is a mapping with a non-unique inverse T-’ (endomorphism), m d 2; T is close to a conservative mapping, but with a non-unique T- ’ [3,4], m = 2; T can be also a mapping with a constant Jacobian [5], or with a constant Jacobian perturbated by a function having a small coefficient. This paper considers only the second case, for the class of two-dimensional endomorphisms having a vanishing Jacobian, when T is defined by differentiable functions, or having the equivalent property for non-differentiable functions (cf. [28, Chapter 61): X n+
1 =
fb,,
Y,, 4,
Yn+ 1 = g(x,, yn, A),
II = n,, n, + 1,. . . .
(1)
In (l), x, Y are real variables, 1 a parameter,f, g, are single-valued functions either continuously differentiable, or piecewise differentiable, or piecewise continuous. The complex solutions are known as ‘completely invariant region’ in [6,7], ‘attractive limit set’ in [8,9], ‘bounded stochastic solutions’ in [3], and more recently as ‘chaotic solutions’ [lo], or ‘strange attractor’ [5]. Mathematically, such a regime corresponds to the property that a very simple point mapping, like (1) may define an extremely complex function xn = h, L-n,x(n,), Y(Q, 21,
Yn = h,Cn, x(n,), Y(Q, I167
4.
(2)
1168
C. MIRA
When (1) is considered as a dynamic system, then (2) constitutes its general solution, x(n,), y(n,), being the ‘integration’ constants, i.e., the insertion of (2) into (1) yields an identity. For one-dimensional endomorphisms, the first basic contribution to the problem of complex dynamics is that of Myrberg [ll]. Since the number of authors involved is large (see [lo]), only references pertinent to the specific context of this paper will be quoted. In one-dimensional endomorphisms, the chaotic behaviour of the solution appears as a manifestation of an infinite number of ‘basic’ bifurcations, and of the accumulations of the latter, forming ‘composite’ bifurcations. The complete bifurcation structure is orderly in the parameter space %, and it turns out to be of a box-within-a box type [12-161. This paper is devoted to the two-dimensional case: A first part considers the mechanism of the appearance of ‘chaos’ from orderly dynamics, in the case of a particular example, with differentiable data, first studied in [17]. The study of some fundamental bifurcations chains permits to conjecture the existence of a bifurcation structure of a box-within-a-box type. The second part concerns the characterization of the boundary (f’) of the ‘chaotic’ area (Ii) in the phase plane (x,, y,). This area is such that, for an initial condition (x(n,), y(n,)), sufficiently close to (6) (inside the domain (D) of attraction of(d)), after some iterations the sequence of points (xy, y,) enters into (d), and inside (6) these points have an apparently erratic movement without any possibility to escape. The ‘chaotic area’ is constituted by (d) and its boundary (f). Results are given when f and g of (1) are continuously differentiable functions, when f, g are piecewise differentiable functions, when J; g are piecewise continuous functions. The last part defines the bifurcation that destroys suddenly, for a small variation of the parameter k the ‘chaotic area’, a repulsive set taking up the place of(d). This bifurcation is a non-classical one, and corresponds to a contact of a certain order, depending on the differentiability off and g, between two singular curves of different nature. The aim of this paper is not a theory of two-dimensional endomorphisms, but only to give an introduction to these newly considered problems through some particular cases and ‘transparent’ examples.
2. TRANSITION
‘ORDER TO CHAOS’ DIFFERENTIABLE
IN
A RECURRENCE FUNCTIONS
DEFINED
BY
2.1. Purpose Following an idea of Poincart, the functions (2) can be characterized by the singularities of (l), similarly to the way a function of a complex variable is characterized by the enumeration of its zeros, poles and essentially singular points in the complex plane. These singularities are either of dimension zero-f%ed points and periodic stationary states called cycles (a cycle of order k is represented by k points verifying x, +k = x,, y, +k = y,, xn + j # x,, y, + j # y,, j -C k), or of dimension one-invariant curves crossing fixed points and cycles. The transition from orderly to complex dynamics is characterized by the appearance of an infinity of singularities in a finite domain of the (x,, y,) plane. These singularities are generated by typical mechanisms of bifurcations. As a first step towards the identification of the mechanism of dynamic complexity in two-dimensional endomorphisms, the following recurrence, or point mapping T: X
n+1
J’n+ 1
exp[L(l
=
xn
=
aXn[
- x,/K)
1 - exp( - ayJ]
- ayJ (3)
Complex
dynamics
in two-dimensional
1169
endomorphisms
model is considered [ 181, J, a and K being parameters. Recurrence (3) represents a predator-prey of two species with non-overlapping generations, first studied, from this point of view, in [17], and used as a starting point of the numerical study of some bifurcation sequencesgiving rise to the singularities of dimension zero. For a different recurrence a similar mechanism of dynamic complexity is described in [3, pp. 207-2141. 2.2 Bifurcation
of the stable constant stationary
state of (3)
Real roots of the algebraic system of equations x,,+ 1 - x,, = 0, y,, r - y, = 0 are called fixed points of (3). Fixed points correspond to a constant equilibrium between the two populations densities. When a > 0, the recurrence (3) admits three distinct fixed points. The first point is point qr(O, 0) is a saddle with eigenxn = yn = 0. In the phase plane .xn - yn the corresponding values s1 = exp I > 1, s2 = 0, and eigen directions coinciding with the xn and y, axes. The second and the third fixed points are given by the roots x, y, of: I.(1 - x/K) - ay = 0.
y = ax[l
- exp( - ay)].
(4)
One root of (4) is obvious, x = K, y = 0. The fned point q@, 0) is either a repulsive node, or a saddle depending on whether the eigenvalues s1 = 1 - 2, s, = aaK, satisfy Is11 > 1, s2 > 1 or Isi/ < 1, s2 > 1. The eigen directions of qz have slopes pr = 0, pZ = (1 - A - aaK)/a. The fixed points q1 and qz are continuations of the fixed points of the first equation of (3), when a = 0. The latter limits the behaviour of the prey on the x axis, which is an invariant curve of (3). The coordinates (x,, y,) of the third fixed point q3 cannot be given explicitly, because (4) is a transcendental one. They are however readily determined numerically. Several properties of q3 can be deduced directly from (4). Consider the auxiliary parameter y = x,/K, which is a measure of the extent to which the predator can depress the prey below its carrying capacity (cf. [17]). Using y, (4) can be rearranged into: a=v[l-exp(ny-A)]-‘. Inserting (4) and (5) into the characteristic equation of qJ, leads to an explicit expression sum G = s1 + s2, and the product A = slsZ of the eigenvalues sl, s2 of q3:
0=1-A+
1 - 41 expI(y
1 -
Y)- 1)’
A = A(1 - y)
of the
-
1 -y;lexp(yA exp(yl - 2)A)’
The expressions (6) are sufficient to define the (phase-plane) properties of the fixed point q3, except when neither 1.~~1= 1 nor lsZl = 1. The bifurcation of qj vs an increase of/z is now considered. Due to the non-availability of the required explicit expressions, the following analysis is carried out for the fixed parameter set K = 10, a = 1, y = 0.4 (cf. [17]), and 0 < 1 < Amax.The largest value iz,,, of biological interest is about five. For any given ,?, the value of a is fixed by (5) For the chosen values of K, a and y, the fixed point q3 is a focus because for 0 < 1 < 5, A > 0, 1 - (T + A < 0, 1 + c + A < 0, gz - 4A < 0. The eigenvalues of this focus are s1 z = A exp (+ icp), 9 = TC- arc tg J(4A/a2 - 1). The focus q3 is asymptotically stable when A < 1, i.e., when 2 < i, i, = 0.70695195. The critical value ACis a root of A = 1, for which, in linear approximation, the focus degenerates into a center. An analysis of the Liapunov critical case A = lc has shown that due to the non-linearity, q3 is actually an asymptotically stable composite focus of
1170
C. MIRA
multiplicity one [19]. For A = AC+ E,0 < E d 1, this composite focus bifurcates into an unstable (ordinary) focus and into a closed stable (attractive) invariant curve (C) [19]. The curve (C) encloses qJ and contracts into it as E --, 0. It has been found by numerical analysis [18] that the singly connected area DCenclosed by (C) increases simultaneously with ,? - AC. 2.3. Ttc,ofundamental bifurcations occurring in (3) When A varies and crosses a bifurcation value &, the cycles of order k appear or disappear by means of two fundamental bifurcations described below. (a) In the case of the first bifurcation, A = Ah - E, E being a small positive or negative value, corresponds to the absence of cycles of order k in a certain non connected region of the phase plane. When A = 1, + E, two cycles of order k exist in this region. One cycle is a node characterized by two eigenvalues si, s2, satisfying Isi1 < 1, lsZI < 1 (stable node) or Isi1 > 1, lsZl > 1 (unstable node). The other cycle is a saddle characterized by (si( < 1, jsZl > 1. For A = I,, E = 0, the node and the saddle merge into a double cycle having one of its eigenvalues equal to + 1 (cf. [20, 281, for details about this bifurcation). (b) In the case of the second bifurcation, a cycle of order k (either stable node or a saddle) exist for I = X,, - E with s2 -+ - 1 as E --f 0. For A = ;Z, + E, this cycle still exists with changed properties (a stable node becomes a saddle, or a saddle becomes an unstable node), and it gives rise to a cycle of order 2k, which merges into the cycle of order k (one of the s --f + 1) as E --+0 (cf. [20, 281 for details about this bifurcation). 2.4. Transition ‘order chaos’ When I increases beyond the bifurcation value Ah= 2.180 118 343, the invariant curve (C) enclosing q3 ceases to exist, and the bifurcation (a) (E > 0) takes place with the appearance of two cycles of order k = 5 (a stable node and saddle). Each of these cycles gives rise to two chains of bifurcations, when I - A,,increases. One chain is characterized by the appearance of stable nodes of order k = 5 x 2’, i = 1,2,3,. . (already
Fig. 1. 1 = 2.50200.
Complex
dynamics
in two-dimensional
endomorphisms
1171
Fig. 2. 1 = 2.50440.
described in [17] via the bifurcation (b), (E > 0)), as A crosses the value &. The other chain is characterized by the appearance of saddles of order k = 5 x 2’, i = 1,2,3,. . . , by the same type of bifurcation, as A crosses, the value Xg. When i --, 00, Xhi -+ AC, = 2.50101, & -+ A:, # 2.56. Each chain of cycles of order k = 5 x 2’ gives rise to points which are located on 20 segments of invariant curves, the slope of the latter at the cycle points being equal to the eigen direction associated with the eigenvalue s < 0, which was equal to -1 at the bifurcation. There exist thus two sets of invariant curve segments, say (Ci) and (C). When 1 increases beyond ;Icl, A:,, cascades of cycles of order k = lOk,, k, = 5,6,7,. . . , are produced on the segments (Ci), (Cl) by a mechanism similar to the mechanism of a first order recurrence, as long as 1 < AC2for the first chain, and
Fig. 3. 1 = 2.50445
1172
C. MIRA
w
4 4
8
12
16
X”
Fig. 4. 1 = 2.53000. A < A:2 for the second one. Ac2,AL, are critical values such as when Acz = 2.504 43, ;i > AL2 > Ac2, cycles of any order appear, without being located on the invariant curve segments (C,), (Ci). In the case of the first chain, invariant curve segments are shown in Figs. 1 and 2. These segments were traced by means of computer generated iterates of a point (x,, y,) chosen at random when
Table 1 I
k
2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8
1 4 4 8 8 9 9 11 11 11 11 12 12 12 12 13 13 13 13 14 14 15 15 15 15
X 4 3.312430 17.03128 3.180551 7.917645 3.829778 10.32207 0.626571 0.482222 0.567141 0.381109 3.485239 4.831342 3.201826 15.189918 4.318845 3-791522 4.621741 3.766182 4.363598 4.526641 2.104981 3.007712 1.047865 2.099552
Y
s, OrP
s,Or (Pra.3
3.254504 5.517332 0.844316 4.247864 3.515393 1.808717 1.597601 2.901126 2.622240 2.673334 2.136189 3.567009 4.948113 6.305706 1.135507 3.46004 1 4.430569 3.473271 4.842982 3.478267 3.492576 2.892633 2.902543 1578465 0.382328
p
cp 1.46702 - 2.2505 - 2.3289 - 6.1749 - 8.8692 - 4.6008 cp 1.8852 -6.1536 - 5.3358 9 0.76271 0.0352 - 13.7541 - 20.4779 7.9791 - 1.1860 12.2642 u, I.89920 rp 1.49976 -8.1386 - 14.9376 - 14.9965 -16.1313 rp I.48191 - 39.2925 - 20.4340
1.27821 2.1280 - 0.2800 3.3152 - 0.7794 4.9127 p 4.0586 6.8988 - 2.3767 p 6.50627 7.2258 7.8712 - 2.4775 13.3090 28.4793 13.6358 p 7.29776 p 12.1489 7.6521 10.9485 - 8.2747 19.7328 p 17.0089 - 0.2540 8.7335
Pl
p2
- 0.55700 -0.09633 - 0.05278 -1.71531 0.26724
- 0.18422 -4.09873 -0.47665 -2.01417
1.80342 2.68556
18.5582 1.84582
72.9319 0.19017 - 12.7961 24.6873 - 0.43033 -0.39927 - 12.7961 24.6873 - 2.72469 0.09972 0.08747 0.55963
4.87704 - 1.89137 - 0.42644 - 0.53797 -0.10295 1.41824
0.82067 - 1.37150 - 0.79053 - 1.75215
65.6279 1.08212
0.01395 0.00785
99ac&l
1173
Complex dynamics in two-dimensional endomorphisms
k, is large. The first thousand iterates were omitted in order not to show the transient leading to this ‘strange attractor’. Figure 3 concerns the case 1 = ACz + E, E > 0. When i > AC2,the cycles of any order k appear by the bifurcation process (a). If a pair: stable node-saddle appears, each of the two cycles of order k produces a chain of bifurcations of cycles of order k x 2’, i = 1,2,3,. . . already described for k = 5. If an unstable node-saddle pair
Table 2
2.18018343 2.20 2.24 2.32 2.36 2.40 2.418520
5 5 5 5 5 5 5
10.732994 11.104020 11.231903 11.098448 10.932706 10.718746 10.604265
1.260072 1.348086 1.416665 1.526232 1.582591 1.643820 1.674610
0.99969 0.40846 -0.01646 -0.53173 - 0.73400 -0.68764 - 0.60946
-0.81264 - 0.89921 -@95478 -0.90879 - 0.82000 -0.91829 -0.99998
021126 0.15856 0.11754 0.07421 O-06189 -0.13706 -0.19109
2.42 2.5044
5 5
10594690 9.917704
1.677154 1.857251
-0.60271 - 0.05269
- 1.0064 - 1.3577
-0.19443 - 0,30404
2.418526 2.42 2.46 2.48
10 10 10 10
10.607628 10.746561 11.528822 11.799213
1.674785 1.681601 1.651036 1.624379
0.99999 0.97388 0.30663 0.21995
0.37141 0.37488 - 0.02898 -0.83705
0.04828 0.07106 -0.19055 -0.17863
-0.19120 -0.19456 -0.23926 -0.29372
2.48387 2.5044
10 10
11.843415 1.620041 12.03 1549 1.602742
0.19289 @01628
- lGOO3 - 1.8624
-0.17728 -0.17304
-0.30108 -0.32963
2.48387 2,488 2.498
20 20 20
11.822445 1.626356 11.233094 1.817841 10074583 2.231221
0.99871 0.36889 0.16852
0.03700 0.02069 -0.77398
0.30127 - 0.30797 -0.36612
2.498982 2.5044
20 20
9.968058 2.272971 9.631132 2.417525
008125 - 0.2507
- 1.0001 -2.8165
- 0.37746 -0.44517
0.18373 Stable node - 0.28843 -0.33490 A cycle k = 40 appears when S = - 1 - 0.34258 -0.36966 Saddle
2.498982 2.5 2.501
40 40 40
9.973295 2.271177 10.370219 2.139227 lo.565799 2.076044
000658 -0.23465 - 0.996 11
-0.34268 -0.33867 -0.32990
-0.37683 -0.34391 -0.33895
2.501005 2.5044
40 40
10.566667 2.075765 11,008963 I.936289
- 0.32986 -0.30792
- 0.33893 - 0.33170
@99977 0.19930 0.03432
0~03400 - 1GOo4 - 0.0260 -4.8189
-0.27951 -0.33333 -0.40158 -0.94189 0.28019 0.05192 0.04902
Stable node
A cycle k = 10 appears whenS = -1
OQ4882 0.4655 1 Saddle Stable node A cycle k = 20 appears whenS = -1 Saddle
Stable mode A cycle k = 80 appears when S = - 1 Saddle
appears, only the saddle can give a chain of bifurcations of cycles of order k x 2’. Moreover, some cycles of order k = 5k,, k, = 1,2,3,. _. , generated in the parameter interval 1, < 1 < jlC2, disappear gradually by means of the bifurcation process (a) with E < 0. When A increases beyond AC2,the bifurcations processes referred to above (cycles of any order) give rise to a larger and larger number of cycles chains, each consisting of an infinite number of unstable cycles of any order. The points of these cycles are located inside a domain (0) of the x,-y,, phase plane, limited by five critical curves (see below). From any initial point (x,, y,), x0 > 0, y, > 0, the sequence (x,, y,) enters into (D) after some iterations, n > N, and wanders ‘erratically’ inside (D) without showing any tendency to settle down (see Fig. 4 showing the computer generated iterates points in (D)). For 1 = 2.8, Table 1 shows the first unstable cycles of any order inside (D) (a cycle is defined in
1174
C. MIRA Table 3 X
Y
5 5 5 5 5 5 5
10.732538 lo.287144 9.985692 9.807633 9.682230 9.681110 9.269490
1.259975 1.158971 1.068495 0.997228 0.934500 0.933971 0.682669
2.3196 2.36 2.40 2.44 2.48 2.50 2.5085 2.5090
10 10 10 10 10 10 10 10
9.718980 lo.597680 lo.937161 11.203921 11.437201 11.545597 11.590304 11.592911
0.927480 0.730962 0.655392 0.603801 0.565388 0.549551 0.543379 0.543015
2.5090 2.513 2.528 2.538 2.548 2.54809 2.5482
20 20 20 20 20 20 20
11.635391 11.791709 12.013580 12.109117 12.188625 12.189289 12.190099
0.5403 19 0.528800 0.509491 0.500358 0.492462 O-492395 0.492313
A
k
2.18018343 2.20 2.24 2-28 2.31955 2.32 2.5044
Sl
%
PI
P2
1.0003 1.5625 1.9174 2.1113 2.2233 2.2245 1.9297
-0.81258 - 0.79076 -0.84209 -0.91620 - 0.99991 - 10010 - 1.4523
0.21131 0.24693 0.25646 0.25380 0.24640 0.24629 0.18960
- 0.27947 -0.24399 - 0.22243 - 0.20746 -0.19435 - 0.19420 -0.13144
4.49448 5.3361 5.6827 5.9966 6.2690 6.3830 6.4256 6.3112
0.99957 0.62181 0.21550 -0.21508 -0.66528 -0.89681 -0.99641 - 1.0023
0.24364 0.19212 0.18256 0.17998 0.18179 0.18419 0.18552 0.18561
-0.19159 -0.11580 Saddle -0.09115 -0.05676 - 0.06764 - 0.06437 - 0.06318 A cycle k = 20 appears - 0.063 11 whenS, = -1
O-99082 0.80080 0.05925 -0.45923 -0.99508 -0.99997 - l+IO60
0.18170 0.16942 0.15633 0.15244 0.15020 0.15018 0.15017
- 0.06380 Saddle -0.06566 - 0.06759 - 0.06860 - 0.06977 - 0.06978 A cycle k = 40 appears - 0.06980 when S, = - 1
41.293 40.990 39.668 38.618 37,430 37.418 37.404
Saddle
A cycle k = 10 appears when S, = - 1 Unstable mode
Table 4 I
k
X
Y
s,
%
PI
2.7289833 2.7489839 2.7689839 2.82 2.82896 2.82898
4 4 4 4 4 4
2.315053 1.864128 1.701973 1.460036 1.429755 1.429690
5.918932 5.996512 6.009993 6.0 18542 6.019566 6.019157
0.99904 0.44321 0.16200 - 0.67724 -0.99914 - 1.0003
- 2.9055 -2.9019 -2.7334 - 1-9124 - 1.5748 - 1.5736
- 0.23454 0.13528 0.36153 1.0914 1.4022 1.4045
2.7289833 2.73 2.79
4 4 4
2.316809 2.427663 3.236830
5.918520 5.890867 5.609357
- 2.9048 - 2.8535 -2.3031
-0.23556 -0.29549 - 0.54579
30.407 50,206 Unstable node - 109.31
PI
P2
-0.41215 -0.41533 -0.70613 - 1.6686
-0.31945 -0.34287 -0.41026 - 0.42072
Stable node A cycle k = 60 appears when S = -1
-0.41206 - 0.41028 -0.39050 -0.38444
-0.31862 - 0.29927 0.09565 - 1.0761
Saddle
1QOlO 1.1208 2.0325
PI
30,200 8.2984 5.6266 Saddle 2.6027 2.0733 A cycle k = 8 appears 2.0717 when S = - 1
Table 5 1
k
X
Y
S,
2.49599439 2.496 2.497 2.5044
30 30 30 30
8.410206 8.405362 8.357578 8.317967
2.814921 2.816941 2.840472 2.881648
0.99322 -0.62768 0.60535 -0.68241 - 0.52492 - 6.6241 - -0.43887 -28,336
2.49599439 2.495998 2.4974 2.5044
30 30 30 30
8.410378 8.414381 8.519261 8.729455
2.814850 2.813216 2.777840 2.758054
1.0068 1.3171 1.1566 8.6853
S2
-0.62618 - 0.59660 -0.27635 -0.01316
Complex
dynamics
in two-dimensional
1175
endomorphisms
this table and in the following tables, by the coordinates of one point of the cycle, sl, s2, and the eigendirections pr, pz). When si, s2 are complex the columns sl, s2 give respectively the modulus p and the argument cpof the eigenvalues (the cycles is a focus). Tables 2 and 3 give the two chains of cycles of order k = 5 x 2’ from 2 = A,,.Table 4 shows the evolution of a pair of cycles of order k = 4. In Table 5 appears the possibility coexistence of two stable cycles for some sets of the A values. It seems that for 2 > ;lc2,all the cycles of (3), the number of which is infinite, are repulsive. Thus the presence of a chaotic area can be conjectured for these values. The study of some other fundamental bifurcation chains permits us also to conjecture the existence of a ‘box-within-a box’ type bifurcation structure. 3. DETERMINATION
OF
THE
CHAOTIC
AREA
3.1. Critical lines and absorptive area It was shown in [15] and [16] that, in the case of one-dimensional endomorphism ?: the domain of complex dynamics on the real axis is bounded by critical points (in the sense of JuliaFatou) of a certain entire power of ?: This notion can be extended in the case of a m-dimensional endomorph&m ?; m > 1. Then, a ‘critical variety’ (VC) (dimension rn - 1) of rank 1 is defined as the geometrical locus, in the phase space, of points having two or more coincident antecedents of the first rank (first iterates of T-l). In the general case, such a variety separates the phase space into two regions, the points of each of them having a different number of antecedents of first rank. The existence of real ‘critical varieties’ constitutes a typical property of the considered class of endomorphisms. In the two-dimensional case, m = 2, a critical curve, or ‘critical line’ (LC) of rank 1, is defined as the locus of points having two or more, coincident antecedents in the phase plane. A critical line (LCI) of rank r + 1 is the consequent curve of rank r from (LC). An ‘absorptive area’* (d’) is defined as a domain bounded by a closed curve made up of segments of critical lines (LX), (LC,), r = 1,2,. . . , such that, for an initial point (xc, y,) sufficiently close to (d’), the sequence of the points (x,, y”) enters into (d’) after some iterations, and inside (d’) these points cannot escape. This is an extension of the notion of ‘non-invariant attractive set’, or ‘absorptive segment’ E, in the one-dimensional endomorphism [15, 16,283. Two different situations are possible for an ‘absorptive area’ (d’). In a first case (d’) contains one attractive limit set (sometimes more) such as fixed point, cycle, invariant closed curve, invariant curve segments of the type appearing in Figs. 1 and 2. In a second situation, the absorptive area (d’) contains a chaotic area (d), in the absence of an attractive limit set of the above type inside (d’). When this situation occurs, whether the chaotic area (d) tills (d’), or (d) fills (d’) with the exception of some holes containing points of repulsive cycles. However, frequently the complete structure of bifurcations of (1) being not known, the absence of such cyclical limit set is not sure. From a practical point of view, the transient to such sets, with a cyclical order greater than the number of observed iterations, can also be considered as a chaotic area. At the present time, nothing can be said about the structure of the numerically obtained ‘strange attractors’. 3.2. Endomorphism defined by continuously differentiable functions When the functions J g of (1) are continuously differentiable with respect to x, y, (LC) is the * Terminology
proposed
by Prof. R. Thorn (private
communication).
1176
C. MIRA
locus of the first rank consequents of the curve J(x, y) = 0, J(x, y) being the Jacobian of the righthand side of (1). Since the curve J = 0 constitutes an antecedent of (LC), it is designated by (LC _ 1). The critical lines of power k, k = integer, of T, (Tk) are the set of curves (LC), (,X,), i = 1,2,... , (k - l), (LC,) being the consequent curve of (LC,_ i). A curve (LC,) can also be considered as a critical line of rank (i + 1). To simplify the exposition, the following hypotheses are first made: The curve (LC _ 1) has only one branch. Then, (LC) and (LC,), i = 1,2,3, . . . , are also made up of one branch. Moreover it is supposed that (LC) and (LC_ i) have a single point of intersection, and that (LC,) and (LC) have no double point. Let h(x, y) = 0 be the equation of (LC). The curve (LC) separates the phase plane into two regions: h(x, y) > 0, and h(x, y) < 0. A point of the former possesses two antecedents of first rank, and a point of the latter none. Then, for the recurrence (l), the following lemmas can be shown to hold: (A) Let a, (x,, y,) be the intersection of the curve (LC_ i) and (LC). The consequent a, (x,, y,) of a, is a point of first-order contact between (tC) and (LC,), or exceptionally a cusp point of (LC,) on (LC). (B) The sequence of consequents an (x,, y,) of a, (x,, y,), n = 1,2,3,. . , constitutes a sequence of first order contact points (or exceptionally of cusp points) between (LC,_ i) and (LC,). (C) The consequent of the intersection of (LC_,) and (LC,) is a first order contact between WC) and WC,, i) (D) The curve (LC,) separates the region h(x, y) > 0 into three subregions, where the number of antecedents of second rank (second iterate of T - ‘) of (x,, ya is none, two, or four. Similarly for the (LC)‘s of a higher rank. . . . For the proof of Lemma A, it is noticed that the point a, belongs to (LC) and (Xi), since it is the consequent of the intersection of (LC_,) and (LC). The curve (LC,) cannot cross through (LC), otherwise (LC,) would have a branch in the region h(x, y) < 0, which is impossible because a point has no antecedents. Then a, is a point of first order contact between (LC) and (LC,), or a cusp point (exceptional case) of (LC,) located on (LC), in the region h(x, y) > 0. From the Lemma A, the Lemmas B, C, are deduced The Lemma D is the consequence of the separation by (LC) of the plane (x, y) into two regions, with zero and two antecedents of first rank, and the equivalent properties for the successive powers of T, T i , i = 2,3,4, . . . For m > 2, equivalent lemmas occur, the equivalent of points a, being varieties of dimension m - 2. From these lemmas, the following proposition results: PROPOSITION
1. Let the recurrence
(1) be such that T has an unique
branch
(LC_ r) crossing
(LC)
inan unique point a,. Let e i, i = 1,2,. . . , m, be a sequence of critical line segments of Tmel such that a, is a first order contact point, or exceptionally a cusp point of (LC,) on (LC,_ 1). Let b, be the first intersection between (LC_ 1) and a segment uzl of (LC,), b, the consequent of b,; therefore b, is a first order contact between (LC) and (LC,+l). In the simplest case, an absorptive
area (d’) is bounded
by the closed curve (9) constituted
by the segment G
of (LC),
the sequence a, and the segment 7 a,,,, 1 1 of (LC,+ i). Then the rank m, defining b, is the least rank such that the sequence uj, j > m + 1, cannot be outside of the domain bounded by (Z), as also the parts of the segments of (LCj), (LC,_ 1) located between uj and the first point of contact with (9). In this simplest case, when a chaotic area (d) exists, (d) c (d’), (9) E (d) and (9) is an ‘external’ boundary of the closed set (d).
Complexdynamicsin two-dimensional endomorphisms
1177
This proposition is a direct consequence of the definition of (d’) and of the above lemmas. (2’) is called an ‘external’ boundary because generally (d) does not fill all the domain bounded by this closed curve, taking into account some holes containing points of repulsive cycles. In more complex cases, parts of the segments of (LX,), (LC,,_ J i > m + 1, located between uj and the first point of contact with (LZ), can be outside of the domain bounded by (_Y), aj being inside, or outside of this domain (cf. [27] and [28]). It is worthy of note that, for a range of values of the parameter 1, the absorptive area (d’) contains one attractive limit set (sometimes more), such as fixed point, cycle, invariant closed curve, and invariant curve segments of the type appearing in Figs. 1 and 2. For another range of 1, when a ‘chaotic area’ exists, which implies the absence of an attractive limit set of the abovementioned type, the absorptive area (d’) turns into a chaotic area (d). In this case (d’) contains an enumererable infinity of repulsive cycles, and (d) c (d’), (9)~ (d). It is added that a segment
G
of (9) can be constituted
by two parts of (LC,), that are joined
is a point of non-differentiability
in a double point. This point
(cf. Figs. 5 and 7 for i = 1,2,3).
for a=,
.
C 4
8
12
16
X” Fig. 5. I = 2.80000.
For one-dimensional endomorphisms, xn + 1 = f(x,, A), f continuously differentiable function, this last situation is only possible for a nowhere dense set of the values R, denoted A: in [12, 15, 161, and giving rise to ‘cyclical stochastic segments’. With A = A:, the infinite chains of the antecedents of a critical point are everywhere dense on these segments. For 1 # A:, and inside the range giving an attractive set at finite distance, there exists at least an attractive cycle, because when a bifurcation occurs, in the one-dimensional case, there always appears at least such a cycle associated with one or more repulsive cycles. Iq two-dimensional endomorphisms, repulsive cycles can appear without being associated with an attractive one (see Section 2.3a). On the contrary of the one-dimensional case, the set of values A corresponding to a ‘chaotic area’ thus can be continuous. The analogy (due to the absence of attractive cycles) between a ‘chaotic area’ and the ‘stochastic segments’, obtained for 1 = AZ with one-dimensional endomorphisms, as numerical experiments, suggest that the infinite chains of antecedents of almost every point of the boundary (9) are dense on (d).
1178
C. MIRA
X” Fig. 6. 1 = 2.53.
Fig. 7. A = 2.80.
Complex dynamics in two-dimensional
1179
endomorphisms
The above results can be illustrated by the recurrence (3) and (5), with a = 1, K = 10, y = 0.4. The region x > 0, y > 0 is the domain of attraction of the ‘chaotic area’ considered above. The equation of (LC_,) and (LC) are respectively: x = K,?.-’ exp(ay); h(x, y) = X’K exp[;l 1 - A~/K] - x = 0. For Figs. 6 and 7, the critical lines of T5 are drawn with 1 = 2.53 and ,J = 2.80. (LC_ J intersects (LC,) in a point d,. However the segments a,d, cannot close (d’), otherwise a4 and d, would be outside (d’). The point b, consequent of b,, intersecz bzeenJLCA) and (IL,) gives the correct area (d’). This area is bounded by the segments b,a,, a,~,, a2aj, a3a4, n a,b,, the points aj, j = 5,6,. . . , belong to (d’). Figures 6 and 7 can be compared with the displays of numerical iterates of randomly chosen initial points (Figs. 4 and 5). It is worthy of note that some segments of critical lines appear as places of concentration of iterate points in these figures. Figures 6 and 7, suggest that, in the interval 2.53 < 2 < 2.80, there exists a value X of A, such that a point a, is a cusp of (LCJ located on (LC, _ 1). In order to determine this value, the parametric equation of (LC,) is considered: x = K,/I exp A exp[l - exp A - at], i y = K/1 [l - exp( -at)]
A = 1 - 1 - At,lK
exp A,
X is defined by equalling to zero the radius of curvature of (LC,) in a, (t = (A - 1) K,/(A + aK)). The cusp is obtained for # 2.624. If (LC) has two branches, separating a region where a point (x, y) has three antecedents, and a domain where a point has a unique antecedent, (d’) is again defined by segments of (LC), (LC,) having the same above properties. Examples of ‘chaotic area’, with (LC) having two branches, can be encountered in [3, pp. 207-218,4, 15,281. 3.3 Endomorphism defined by piecewise differentiable functions One of the two functions f, g, of(l), or the two functions are piece-wise differentiable such that the curves of the one parameter family, f(x, y) = GI(a = parameter), have a zero-order contact with the curves of the one parameter family g(x, y) = /I (y = parameter). The geometrical locus of these points of contact is designated by (LC_ 1), since in the differentiable case, the property of first order contact between these two families is expressed by J(x, y) = 0. The arguments of Section 3.2 give the same statement of the Lemmas A, B, C and D and of the Proposition 1, by taking the place of ‘first order contact points’ by ‘zero order contact points’, i.e., an angular point of (LC,) is located on (Xi_ i) on the same side of (LC,_ 1). A very simple example of such a situation is discussed in Section 4 (Figs 12 and 13). Another simple case corresponds to the recurrence T: Y” - 0,2&, X “+l
= (1 -
4Xn
+ Y,,
Y”+l =
Y” -
215
IX”1 d 5 +
yn-2Ax,-9&
91,
xn > 5 x,,<
(7)
-5.
For (7), WC_,) (x = 5), (U-J (x = - 5), are the equivalents of the curve J = 0 in the differentiable case. The equation of (LC) is y - x = 4il - 5, and that of (LC’) y - x = 5 - 41. In the region located between (LC) and (LC’) a point has 3 antecedents of first rank. Outside of this region a point has only 1 antecedent.
1180
C. MIRA
Fig. 8
Fig. 9. 1 = 2.4, I div. = I.
Complex
dynamics
in two-dimensional
endomorphisms
1181
In T a cyclical chaotic area of order 2 appears of 1 = A, + E, A, = 1617.2 = 2.222.. . , E > 0 sufficiently small, i.e., there exist two chaotic areas of T2. When 1 31 2.4 these two regions have a common part, giving a simple chaotic area (for details, see [3]). When A > 2.532 this attractive limit set disappears, and it does not exist attractive set at finite distance. Using the above results, it is not difficult to construct the boundary (2) of the ‘chaotic area’ which corresponds to the ‘absorptive area’. The sequences (LC,) (LC), i = 1,2,3,. . . are drawn of the equion Fig. 8 for 1 = 2,4, the points a, bi, ci, d,, ei, gi, i = 1,2,. . . being the consequents valent points, with the index zero, located on (LC_ i). and (LC_ i). Numerical iterations give Fig 9, which can be compared with the boundary (2) drawn with thicker lines in Fig. 8. It is worthy of note that inside (d), some segments of (LC,), drawn in Figure 8, appear, in Fig. 9, as separations of regions with different densities of iterates. 3.4. Endomorphism defined by piecewise continuous functions One of the two functions f; g of (l), or the two functions, are piecewise continuous, such that the curves of the one-parameter family, f(x, y) = a, have a zero order contact with the curves of the one parameter family, g(x, y) = B, (a, p, parameters) in a point where the continuity ceases to exist. The geometrical locus of such points constitutes segments of (LC_ i). Then the segments of (LC), consequent of (K-i), separate in the phase plane regions where the number, or the nature, of antecedents of first rank is different. When a ‘chaotic area’ (d) exists, (6) is bounded by segments of (LC), (ZC,), i = 1,2,. . . , as for the previous cases. The following example illustrates this property. Let be the recurrence T: (8)
Fig. 10. 1 = 1.12, p = 11
1182
C. MIRE
Y”
C.,
Fig. 11.
with I = 1.12, p = 11. (LC_ r) is the straight line x = 6. (LC) and (LC’) verify respectively y = x - 62, y = x - 61 + I-(-(LC) and (LC’) limit a region where a point has two antecedents of first rank, the complementary regions being such that a point has a unique antecedent of first rank. Let a,, and e, be the respective intersections of (LC_ r) with (LC) and (LC’). By application of T, a, has two different consequents, a, defined by (8) for x < 6, and cr defined by (8) for x b 6. (LC,)is constituted by two half straight lines, a,, cr being one of the extremities. By the same way, e, has two consequents e,, h,, and (LC',) is constituted by two half straight lines. Numerical iterations give the chaotic area of Fig. 10 with regions of different density of iterated points. The sequences (LC), (LC'), (LC,), (LC),i= 1,2,3,4, are drawn on Fig. 11, the points ai, b, ei, hi being the consequents of a,, b,, e,, h,; di, gi, li, m, the consequents of d,, go, 1,, m, located on (LC_,). As in Section 3.3, regions with different densities of iterates appear, and segments of (LC,) and (LC)separate these regions. The two-dimensional recurrence (or point mapping T), associated to the Lorentz-Sal&man [I] differential equation of order 3, is piecewise continuous [2]. The chaos in the 3-dimensional phase space can be defined from the construction of the chaotic area of the two-dimensional associated point mapping. 4. BIFURCATION
DESTROYING
A CHAOTIC
AREA
Numerical experiments show that a ‘chaotic area’ (d) disappears suddenly, when the parameter I crosses a bifurcation value A*. Thus, if J. > 2.532 in (7), the unique attractive limit set is located at infinity.
Complex
dynamics
in two-dimensional
endomorphisms
1183
The properties of the boundary (F) of the domain of influence (D) of an attractive limit set (fixed point, cycle, chaotic area) play a prominent part in the mechanism of this bifurcation. Such a domain (D) is the zone of initial conditions (x,, y,), which will not prevent the sequences (x,, ya from returning to the considered attractive limit set. The boundary (F) of the domain (D) is invariant by application of T and T-‘. Generally (F) is defined from the invariant curve, crossing a saddle cycle, or a fixed point, which is associated to the saddle eigenvalue of modulus smaller than one, Properties of(F) are given in [7,28], and the plotting of an invariant curve crossing through a saddle fixed point is made by the means of mixed analytical-numerical methods [22, 23, 281. It is worthy of note that (F) can possess a complex nature. Indeed (F) can be a curve the tangent of which being indeterminate in each point. (F) can be also such that the domain (0) is not simply connected [9,24], (D) being constituted by an ‘immediate convergence domain’ (D,), containing the attractive limit set, and a certain number, sometimes infinite, of unconnected zones (D,). A more complicated situation occurs when (F) is a ‘fuzzy boundary’ or a ‘stochastic boundary’ [7, 251. In that case, the ‘fuzzy boundary’ is a strip of the phase plane, containing an infinity of repulsive cycles and invariant curves crossing through the points of these cycles. This strip separates the domain of influence of two attractive limit sets, and constitutes a region of uncertitude, for the evolution of the sequence (x,, y,) towards one of the two attractive sets, when the initial condition (x,, y,) is located insides the strip [28]. The following proposition can be shown to hold. 2. Given the recurrence (1) having a critical line (K) with one branch (situation of Proposition 1) or two branches (situation described at the end of Section 3.2 and with Example 7). Given an interval of the ,l values such that (1) has a chaotic area (d) in the (xn - y,J plane. This ‘chaotic area’ (6) disappears when i crosses through a bifurcation value I = il* defined in the two following cases: If the functions f, g of (1) are continuously differentiable, ;1* is the value such that a segment of a critical line (belonging to the boundary (2) of(d)) h as a first-order contact point with a segment of invariant curve, crossing through a saddle fixed point, or cycle, and belonging to the boundary (F) of the influence domain (D) of (4. If the functions f,g of (1) are piecewise differentiable, the same bifurcation occurs when a segment of critical line of (2) has a zero order contact point with a segment of an invariant curve of(F). PROPOSITION
Let be i = A* - E,E > 0 sufficiently small, a value giving a ‘chaotic area’ (d), and 2 = ;1* + E a value for which such an attractive area does not exist, and suppose that (F) is not a ‘fuzzy boundary’. This theorem is the consequence of the Proposition 1 and the properties of the boundary (F). Indeed, let be, for 1 = A*, a contact of first, or zero order, in a point PO,between (F) and the segment of (JX) belonging to (T), boundary of(d), such that (d) c (D). The consequents of PO are points of contact between (F) and the segments of (LC,), i = 1,2,. , . , belonging to (9). In this situation, the boundary (F) of(D) consists of the limit of the boundary of(D), with i = A* - E, E --f 0, and the infinite number of chains of antecedents j?_i of PO, i = 1,2,. . . , dense into (4. Each chain tends towards one point of a node repulsive cycle, which appears for il < I*. When i = A* + E, around each point /?_i antecedent of p, for E = 0, it appears a region belonging to the influence domain of an attractive set, located outside of(D). The area of these regions tend to zero when E + 0. The infinite chains of these regions cover the place, filled by (d) when ,l = ;1* - E,
1184
C. MIRA
except the set of measure zero consisting of the repulsive cycles points, that appear for L < A*, and certain invariant curves crossing through these points. Thus, for ,?. = A* + E, (F) no longer exists and, in the place of(d), there is a ‘repulsive’ region containing the repulsive cycles that have appeared for i < A*. A very simple generic example, showing this phenomena, corresponds to the point mapping, with piecewise differentiable data: XIIt1 = yn; i Ynil
y,+i
= y, - Ax,,
=yn+2x,,-6(2+1J
ifx,, < 6;
(9)
ifx,36.
Indeed, for (9) the boundaries (2) and (F) are made up of segments of straight lines, and A* can be calculated without approximations. Here the futed point 0 (0; 0) is a focus of type 1 [7], which is repulsive for 2 > 1, II = 1 being the value giving rise to a chaotic area. The boundary (8) is determined from the second fixed point A (X = y = 3(2 + A)), and the segment BC of the eigen direction (slope = - l), associated to the multiplicator (eigen value) S, = - 1, the second multiplicator of A being S, = 2 (slope of the eigen direction = 2). A is a degenerated saddle, and BC is a segment of repulsive cycles of order 2, with xB = y, = 6, y, = y, = 6(1 + 2). Then (F) is made up of BC and its antecedents B_iB_i+l,i = O,l,..., ~,B,,zB,B’_~B’_~+~~=~ ,..., 4,B’_, 3 C. (F) is closed by the segments B_ 4a,,, B’_ JZ~, x0 being the antecedent of the point ai, intersection between B _ 3B_4 and the critical line (LC), xjrO = 6. When I > 1, all the cycles are repulsive, and thus if 1 < ,J d i*, the absorptive zone (d’) merges with a chaotic area. For 1 < 13.< ;1,, 2, N 1.1, the chaotic area is an attractive limit set made up of 6 regions, the points of which exchanging themselves by application of T. For &, < 1 < A*, these 6 regions join, forming a ring shaped attractive zone, constituting the chaotic area. (LC_,) is the straight line x = 6. The equation of (Lc) is y = x - 62, and (LC) separates the phase plane into a region where a point has two antecedents of rank one, and a region where a point has no antecedent. The point C is located on (ZC). The sequence of the points ai(xi, y,), defined in Section 3 is such that: i =O,l,..., ‘i+l
=
yi,
x0 = 6,
y, = 6(1 - 4il + 3A2),
y, = 6(1 - J.),
y, = 6( 1 - 2L),
y, = 6(1 - 51 + 6A2 - L3),
y1 = 6( 1 - 3;1 + A2),
ys = 6(1 - 62 + 101’ - 4L3).
For this example up is a zero order contact point between (LC,_ i) and (LC,). When the chaotic area (d) exists, 2 < ;1*, a, is inside the region where a point has no antecedent. The bifurcation destroying (d) is defined for J. = A* such that CI~is located on (IX), thus merged with a,. Then the point a, is located on B_ 3B _4, which defines A* as the solution A = 1.2 of 52’ - 161 + 12 = 0. Numerical iterations give Fig. 12 which can be compared with the construction of the boundary (2) of(d) appearing in Fig. 13 (here 6,, d, located on (LC) are the antecedents of b,, d, respective “*, besides the indicated contour, the intersections of (LC_,) with uT5 and (IL,). For A = A boundary F is made up of all the sequences of antecedents of the point x,, E a,. When A = A* + E, E small positive value, c(~ is located into a region where a point has two antecedents of first rank. Around a, and its 2’ chains of antecedents, appear regions, the area of which tending towards zero if E + 0, that are part of the influence domain of the attractive point at the infinity. With ,I = I* + E, the sequence generated by (9) can remain into a domain reproducing the form of (61
Complex
dynamics
in two-dimensional
endomorphisms
1185
&
-10
Fig. 12. A = 1.199999.
for L = I*, during a large number of iterations, if (x0, y,) is located into a region containing an antecedent of a sufficiently large rank of ccO. In the following example (differentiable data): x n-l =y
Y
= y” - Jxn + x.”
(10)
(F) has the form indicated in [26] for an ~clui~z&t equation. A contact of first order between (9) and (8’) takes place, when ;1* # 155. For 1 < II < 1.4 there exists an attractive close curve inside
Fig. 13. 3, = i* = 1.2.
C. MIRA
1186
(D) bounded by (8’). This curve surrounds O(0; 0), a repulsive focus. For I N 1.4 a pair of cycles of order 7 appears. One is an attractive node, the other a saddle. The bifurcations of some cycles give rise to a chaotic area. 5. CONCLUDING
REMARKS
When a chaotic area exists in the differentiable case, numerical experiments have shown that segments of critical lines are places of concentration of iterate points. This property is explained by the fact that the consequent of a large zone, located on each side of the curve (LC _ ,), J(x, y) = 0, is a narrow strip bounded by (LC). In the piecewise differentiable case, it was seen that segments of critical lines appear as separations of regions with different densities of iterates. Remarking that the critical lines (LX,) can be considered as lines of folding of a Riemann surface with several sheets, (each sheet corresponding to a determination of the inverse mapping T-‘), the projection of these sheets on the x-y plane gives rise to regions with different densities of iterates. This paper does not claim to give a complete, or a strict theory of chaotic areas in two-dimensional endomorphisms. In particular, the conjecture related to the density on(d) of the antecedents of almost every point of (2) would require a rigorous mathematical proof (this conjecture is based on numerical experiments and analogies with the one-dimensional case), and its limits are not known. Nevertheless, the knowledge of the bifurcations giving rise to a chaotic area, and destroying it, the properties of the boundary of such an area here limited to particular cases, can help the understanding of more general situations.
Acknowledgements-The problem of ‘absorptive herein is supported by Direction des Recherches,
area’ was discussed with Professor R. Thorn. The research Etudes et Techniques under the Grant 77/l 170.
reported
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