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Once more on Hénon map: Analysis of bifurcations
Chaos, Solitons
& Froc&zls Vol. 7, No. 12, pp. 2215-2234, 19% Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. All rinhts resewed 09ao779p6 sis.00 + lO.cKl
PII: !3o%o-o779(%)ooo81-1
Once More on H&on Map: Analysis of Bifurcations MICHAEL
SONIS
Bar-Ban University, Israel
Abstract-Using the analysis of bifurcations approach the detailed description of bifurcation phenomena in the classical Henon map is presented. This description strongly supports the idea that the HCnon map contains all possible bifurcation phenomena known for two-dimensional discrete maps. It is interesting to note that the existence of two different equilibria in the Henon map generates additional - dual - appearance of bifurcation phenomena. The proposed analysis can serve as a prototype of the bifurcation analysis for finite-dimensional iterative processes with multiple equilibria. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
The well-studied Henon map (H&on [l]) x f+l
=
1
-
a3
+
Yt
t = 0, 1, 2, . . .)
Yr+1 = bxt,
presents a simple two-dimensional map with quadratic non-linearity. This map gave a first example of the strange attractor with a fractal structure. Because of its simplicity, the H&on map easily lends itself to numerical studies. Thus a large amount of computer investigations followed (see B-L. Hao [2]). Nevertheless, the complete picture of all possible bifurcations under the change of the parameters a and b is far from completion. The purpose of this study is to produce a more detailed and more complete description of the multiplicity of bifurcations in the H&on map on the basis of newly developed analytical/numerical procedure of the description of properties of equilibria of finite-dimensional iterative maps on the boundary of the stability of equilibria (see Sonis [3, 41). Next we present in brief an analytical description of the two-dimensional version of the analysis of bifurcations procedure for two-dimensional iteration maps of the type Xt+1
=
U(a,
b;
x,,
Yl)
(2) Yt+1 = V(4
b; x,3 YJ
where x,, yt represent the components of the iteration process at time t = 0, 1, 2, . . .; parameters a, b are the external bifurcation parameters and the functions U(a, b; x,, y,) and V(a, b; x,, y,) are differentiable (almost everywhere) functions in their components x,7 Yt. Analysis of bifurcations procedure includes first of all the calculation of possible equilibria x, y of the iteration process (2) given by the system of equations x = u(a, b; x, y),
Y = V(a, b; x, Y). 2215
(3:)
2216
M. SONIS
The algebraic equations (3) allow for a representation of two external bifurcation parameters a, b with the help of the components of the equilibrium x, y : h = g(x. y).
a = f(x. y);
(4)
As a result, the coordinates of the equilibrium X, y play a role of internal bifurcation parameters. Movement of equilibrium points can be placed on segments of straight lines. This allows for the complete computerized description of the appearance of different bifurcation phenomena in the space of orbits. Thus, the first basic element of the analysis of bifurcations procedure includes travels of equilibria in the space of orbits which reveal the qualitative features in the behavior of the orbits of the iteration process near the boundaries of domain of stability of the equilibria. The next step of the analysis of bifurcations procedure is the construction of the matrix of a linear approximation of the iteration process, the Jacobi matrix: / --‘j 3x,
--3v,‘[
J(t, t + 1) = j
(5)
/ av j 3x, and its value J* at the fixed point X, ,Y. r-3u * J” -; I z-r / j SV”
av 3v,
au* 3y
/ /
avz? / 31:
j 3x L..
(6)
/ -1
where U* = U(x, y), V” = V(x, y). The eigenvalues ,B,. k of the Jacobi matrix J* are the solutions of the characteristic equation ,l? - Tr J*p ?- det J* = 0.
(7)
where
(8)
By the well-known Von Neumann theorem. the equilibrium (x, y) is asymptotically stable if and only if, for all its eigenvalues p, ) k%,the following conditions hold: I& / i: 1.
:b / -I 1.
(91
The outcome of the general Routh-Hurvitz stability conditions is that the polynomial !I2 - Tr J*,l+ det J” has roots less than 1 in absolute value if and only if .-I
_S Tr J*
-; &t
J* CC 1
(see Samuelson ]S] p. 436. or Dendrinos and Sonis [6] p. 79).
(1Q)
Analysis of bifurcations
i!217
Presenting Tr J* and det J* through the coordinates X, y of the equilibrium one obtains in the space of orbits the domain of stability of equilibria; boundaries of this domain are the following curves: (1) the divergence boundary under the equation TrJ* = detJ* + 1;
(11)
(2) the flip boundary under the equation TrJ* = -(det J* + 1);
WY
(3) the flutter boundary under the equation detJ* = 1.
(13)
On the divergence boundary at least one of the eigenvalues is equal to 1. Crossing of this boundary allows for orbits to approach infinity. Such divergence starts from within the domain of stability; this domain of divergence is the infinity-locking domain. On the flip boundary at least one of the eigenvalues is equal to - 1. Each point on the flip boundary corresponds to a two-periodic cycle, and movement outside the domain of stability generates the Feigenbaum type period doubling sequence, leading to chaos (Feigenbaum [7]). On the flutter boundary 1,~~ 1= 1~121 = 1. It is easy to describe the type of bifurcations in all points on the flutter boundary. The condition Ij.+( = 1~121 = 1 means that p1 = &R, OGQGl ~(2= eei2Y and therefore TrJ* = ,u~+ j,~ = 2cos27rQ.
(14)
If 52 is a rational fraction: Q = p/q, then we have q-periodic (resonance) fixed points; between them there are fixed points of strong resonance with Q = l/3, 52= l/4. Other rational fractions 52= p/q represent points of weak resonance. The same periodic behavior is also observed in a small domain of s2 near p/q. This domain, the mode-locking domain, is the image of the Arnold tongue from the corresponding domain of change in eigenvalues in the complex plane (Arnold [S]). For strong resonance, the mode-locking domain starts within the domain of stability (Kogan [9]). Thus, the movement of the equilibria - travels of the equilibria - in the space of orbits along the segments of straight lines and the crossing the boundaries of the stability domain reveal a plethora of possible ways from stability, periodicity, Arnold horns and quasiperiodicity to chaos. A numerical procedure of the description of such phenomena includes the construction of the spatial bifurcation diagrams in which the bifurcation parameter is the equilibrium itself. Organizing the movement of equilibria in the space of orbits along segments of straight lines can be achieved in the following way: it is possible to parametrize each segment of the straight line between the equilibria (x1, yl) and (x2, y2), as x(j) = 1 - f X1 + $x2 ( I j = 0, 1, . * *, T,
r(i)=(1- fI YI+
(15)
f~2
where j is a bifurcation parameter and T is the number of bifurcation steps. The usual bifurcation diagram can be obtained from eqn (15) by fixing x(j) or y(j).
2218
M. SONlS
Next, the ebmeMs of analysis of bifurcations procedure will be used for the study of the bifurcations of H&on map (1) with the purpose to show that under different parameter specifications this map can produce any preset dynamic behavior including stability, periodic motion, quasi-periodicity and various forms of chaotic movement. Our findings are in good agreement with previous results of numerical calculations and theoretical studies (see Simo [lo], Huang [lilt Alligood and Sauer [12], Devaney [13]. Part 2, Peitgen et al. 1141Ch. 12).
2.1.
Domain 0.f stability of equilibrium
for the Heizon map
The fixed points (x, y) of the HCnon map xr+.i = 1 .- ax; + yt .V*-7 i
=
lx,.
1 =
0.
1.
2,
.,
a, b f 0
satisfy the following quadratic system: .s =- 1 .- ax2 + )’ I
J’ = Iw
(16)
Thus, there are two equilibria satisfying a\.: .+ (1 - b).u -- 1 LT(I. The conditions
(16) imply that x f 0 and -9,,/.rz_ Ll - 1I -. f -+ _,
b -= y !.r .
(IX)
Therefore, instead of change of the external bifurcation parameters a and b one can change the location of equilibrium (x, y) in the space of orbits: in such a way the coordinates x and v can play a role of internal bifurcation parameters. The Jacobi matrix J(t. i i 11 of the H&on map is
and its value J” at the fixed point X. v ic
such that TrJ*
= -2ax
= -2(1 - .r + y)/.x (21)
detJ* = -h = -,y/:w.
Therefore. (10) implies that the domain of stability of equilibria (x. y) is given by the inequalities: -1 2 2(1 - x -!- y)/.u c: -v/x
<‘: 1,
x + 0
(22)
or. in a simpler form: 213
(.. .Y - ?; 5::2.
x + .\: ‘> 0.
(23)
This system of inequalities geometrically represents the infinite half-strip (see Fie. 1)
2219
Analysis of bifurcations
Divergence boundary
Fig. 1. A half-strip of stability of equilibrium for the Htnon map.
such that if the fixed point (x, y) lies within the half-strip (23) then its become an attractor. The three sides of this half-strip represent the flip boundary with the equation x - y = 2/3; the divergence boundary with the equation x - y = 2; and the flutter boundary with the equation x + y = 0. If, starting within the half-strip of stability, the equilibrium (x, y) crosses the divergence line x - y = 2 then it is transformed from an attractor to a repeller and the orbits became divergent. The most interesting bifurcation behavior appears when equilibrium approach the flutter line x + y = 0. Only a segment of this straight line with the end points (l/3, -l/3) and (1, - 1) belongs to the boundary of the domain of stability of equilibrium. The points on this segment generate a multitude of different bifurcation phenomena. It is easy to calculate the coordinates of equilibria generating a q-periodic orbit. The conditions (14) and (21), together with condition x = - y imply that TrJ* = -2(1 - x + y)/x = 2cos- 23i-P 9
or
Table 1 represents the coordinates of equilibria generating a q-periodic orbits for q c 12; this table can be easily extended. The important feature of this periodic behavior is the existence on the flutter segment of mode-locking domains - the Arnold tongues - interwoven within each other (Arnold [8]). For strong resonance, the mode-locking domain starts within the domain of stability
(Wan PI).
Figure 2 represents the planar bifurcation diagram describing a travel of equilibria
M. SONIS Table 1. Coordinates of equilibria eerating Type of periodic&y 4 ..- -...--.--_.____
Arguments of eigenvahes Q=2np 4
periodicity of H&non map
Values of trace TrJ * = 2cos.9 4 -----.._-.
Coordinates of equilibria .t =: .- \ ___.. --.-__
2
i:
-2
0.333 33
3
2n 3 i; 1‘ 27i -.5 477 $
-1
0.4
4 5
0
0.5
0.61803
Cl.59137
1.61803
0.356
._--.
0.666 67 27 .7 4, hi;
f
::
10
4 2a 9 47 (4 x-I4 0 I:
4 .?o? .-.. 5
II
1.24698
0.72647
- 0.44504
0.44994
--1.x01 94
(1.34471
1.41421
0.773 46
I.53209
0.8104
0.347 30
0.547 54
--1.87939
1.h1803 -0.61803
0.341) 17 n.tw 64 0.43309 0.X6.7 0.63251 0.46678
towards the point of three periodic bifurcation (strong resonance p/q = l/3) along the segment with the end points: (0.46, -0.34) -+ (0.4, -0.4). This planar bifurcation diagram includes 500 orbits, each starting from the initial position: x(O) = y(0) = 1; each orbit includes 1000 points, 200 of them hidden. The three-periodic mode-locking domain is started within the domain of stability; moreover, its interlacement with the nine-periodic mode-locking domain is clearly visible. It is easy with the help of Table 1 to produce analogical planar bifurcation diagrams for other periodic bifurcation phenomena.
Analysis of bifurcations
:!221
1.5
x, = 0.46 y, = -0.34 x2 = 0.4
y2
= -0.4
x0 = I ( Yg= 1 t= 1000 q=200 Points = 500 I=
1.5
-1.
Fig. 2. Planar bifurcation diagram representing three-periodic flutter.
Further, for the amplitudes 52 of eigenvalues which are not rational one obtains the quasi-periodic orbits and different ways to chaos. The mode-locking domains of different quasi-periodicities, chaos and divergence also appeared. Figure 3 represent the bifurcations appearing when equilibrium is moving along the flutter boundary segment. It is interesting to note that on the flutter segment there are the mode-locking domains of divergence; after their appearance the periodicity, quasi-periodicity and chaos appeared once more. Figure 4 represents such an appearance of 5-period hyper-cycle of periodic and quasi-periodic curves. 2.2
Second iterate of the H&non map
If the equilibrium (x, y) crosses the flip line x - y = 2/3 then it enters the domain of stability of two-periodic cycle. The complete description of behavior of two-periodic bifurcations can be offered with the help of analysis of the second iterate of the H&on map. The second iterate of the Henon map is: x,+2
= 1 - ax:+1 + yt+l = 1 - a(1 - ax: + yt)2 + bx, (25)
Yt+2 = bxt+, Let
x, = u,,
x,+2 =
ut+1;
Yt
= w
-
a-d
+ Yth
then one obtains the following map:
= “t,
Yt+2 = Ut+1,
ut+1
= 1 - a(1 - auf + u,)~ + but
ut+l = b(1 - auf + u,).
(26)
2222
M. SONIS 3
I i
'r
/"
/
,//
x, = 0.4474
;,
i' J
i'
L
yt = -0.4474
*'
,, " _/'
./
/
,' ,,"
,I'
x2 = 0.55 y2 = -0.55 .x0 = I 0 Y* = 1 t = 1000 q = 200 Points = 20 r--3
-3 :J
-3
3
Fig. 3. Periodic and quasi-periodic flutter
1, = 0.562 y, = --0.562 x2 = 0.566 y2 = --0.566 x0=
?
p()= 1 I = 1000 g=200 Points = 10 r=3
-3
0
Fig. 4. Speriodic hyper-cycle.
3
Analysis of bifurcations
2223
The fixed points (u, u) of the map (26) are two-period cycles of the H&on map. The coordinates u, u satisfy the equations: u = 1 - a(1 - au2 + u)* + bu (27)
u = b(1 - au* + u). It is possible to prove that u satisfies the following equation: a3u4 - 2a2u2 + (1 - b)3~ + [a - (1 - b)*] = 0.
(28)
It is easy to check that u3u4 - 2a2u2 + (1 - b)% + [a - (1 - b)2] = ( a*u* - a(1 - b)u + [(l - b)* - a])[uu* + (1 - b)u - 11. (29)
The equation au* + (1 - b)u - 1 = 0 gives the fixed points of the H&on map (see eqn (17)). Therefore the equations a214- a*(1 - b)u + [(l - b)* - a] = 0,
u = 6(1 - au* + u)
give the coordinates of the proper two-periodic cycle of the H&on conditions (30) imply that
(30)
map. Note that
u*uu = b[(l - !7)2 - a].
(31) The discriminant of the quadratic equation (30) gives the conditions of the existence of a proper two-periodic cycle of the Henon map: a*(1 - b)* - 4a2[(1 - 6)2 - a] 2 0, or 4a a 3(1 - by.
(32)
Using the conditions (18) one obtains 4(1 - x + y) 3 3(x - y)2
or -2 c x - y < 213
(33)
This means that if the equilibrium (x, y) crosses the flip line x - y = 2/3 then the two-periodic cycle (u, u) appears. Next we define the domain of stability of two-periodic cycle (u, u) which is a part of the strip (33). The Jacobi matrix JJ(t, t + 1) of the second iterate of Henon map is
JJ(t, t + 1) =
b + 4a2u,u,+l/b -2abu,
--2au,+db b
1
(34)
and its value JJ* at the two-periodic cycle (u, u) JJ* =
such that one obtains, using (31)
b + 4a*uu/b -2abu
-2au/b b
1
(35)
TrJJ* = 4a*uu/b + 2b = 4[(1 - b)* - a] + 26 detJJ* = b*.
(36)
M. SONS
2224
Therefore, (10) implies that the domain of stabihty of two-periodic cycle (u, u) is given by the inequalities: -1 + (4[(1 - b)’ - (~1+ 26) < b’ e: 1.
(37)
or. using (18), -2
?I [4(x - y)Z - 4(1 - x f y)] + 2xy < y2 < x2.
(38)
Further, one obtains the inequalities 3(x - y)” - 4(1 - x + y) < 0 4(x - y)” - 4(1 - x c y) + (x + yy > 0 x? - p > 0. This system of inequalities is equivalent to two systems, defining a right strip of stability I) <. x - “y < 2/x
s + y ‘*’ 0
(39)
without the part inside the ellipse 4(x - yj2 - 4(1 - x -t- y) + (X +
yj2 > 0
(4)
and the left strip of stability of two-periodic cycles -2 -c .Y - y i 0.
x + ,y < 0
(41)
without the part inside the same ellipse (see Fig. 5). The domain of stability of two-periodic cycles includes the divergence boundary x y = 2/3 belonging to the right half-strip of stability and the divergence boundary x - y = -2 belonging to the left half-strip of stability: the flutter boundaries are x = y, x + y = 0 and the flip boundary is 4(x -- .\‘)2 - 4(1 - .Y + y) -t (x + y)’ = 0
Flutter bowday
Fig. 5. Left and right half-strips of stability of two-periodic cycle for the H&on map.
(421
2225
Analysis of bifurcations
which is an ellipse with center (-l/4, l/4) and main diameters x + y = 0 and x - y = -l/2. Let us start from the description of bifurcation phenomena appearing in the right half-strip. Figure 6 represents the movement of equilibria along the segment (5, 4.2) -+ (5, 5). This movement starts within the domain of stability of equilibrium in which the orbits attracted to the equilibrium. When the equilibrium crosses the flip boundary, the stable two-periodic cycle appeared; when equilibrium touches the flutter boundary for two-periodic cycles, the two-periodic hyper-cycle of quasi-periodic orbits appears. The analysis of the bifurcations in the form of the two-periodic hyper-cycle of quasi-periodic orbits can be done with the help of movement of equilibria along the flutter line x = y (see Fig. 7). As in the case of fixed point it is possible to calculate the coordinates of all bifurcation phenomena. It is interesting to note that when one moves the equilibria left along the flutter line x = y the two-periodic hyper-cycles of chaotic attractors appear (see Fig. 8). The movement to the right along the flutter line x = y smoothes out invariant curves incorporated in 2-hyper-cycles (see Fig. 9). When the equilibrium crosses the elliptic flip boundary of the left half-strip of stability of two-periodic cycles it enters the domain of stability of four-periodic cycles. This domain is also bounded by flip, flutter and divergence curves. This opens the periodic doubling way to chaos. Figure 10 represents the planar bifurcation diagram of the Feigenbaum way to chaotic Henon attractor by the movement of equilibria through the domains of stability of different double-periodic cycles along the segment (1, 0.3) + (0.62, 0.186). This pheno-
x, = 5
y, = 4.2 x* = 5 Y2= 5 x0= 1 a Yo= 1 t= 1000 q = 200 Points = 150 r= 10
-10
0
10
Fig. 6. Movement of equilibrium, starting within its domain of stability, crossing the domain of stability of two-periodic cycles and touching the flutter boundary of this domain.
M. SONIS
2226
X)
3
=
Y, =3 x2 = s Y, = 5 x0 = 1 Y”=
1
1= 1000 q=2w
Points = 20 r= to
IQ
0
Fig. 7. Movement of equilibrium along the flutter curve of two-periodic cycle.
x1 =
2.44
y, =
2.44
x7 =
2.44
yz =
2.44
x0=
I
y*=
I
f := 1000
q=o Points = 1
-6
Q
Fig. 8. Two-periodic hyper.cycle of chaotic flutter phenomena.
6
Analysis of bifurcations
2227
220
x, = 1
Y, = 1
x2 = 100
y* = 100
x0 = 1
0 Y()=
1
t=20oa q = 200 Points = 100 r=220
-22C Fig. 9. Two-hyper-cycles on the flutter boundary belonging to the right half-strip of the domain of stability of two-periodic cycles.
menon is much more visible if one considers the usual bifurcation diagram (see Fig. 11) representing the well-known Feigenbaum period-doubling cascade. We will start the description on bifurcations generated in the right half-strip by the example of movement (on a big distance from the point (-2.5, -2.5) to the point (-1000, -1000)) of equilibrium (x, y) along the flutter boundary x = y (see Fig. 12). The same phenomena (as on Fig. 9) of the 2-hyper-cycle of periodicity, quasi-periodicity and chaos appeared. The crossing of the left half-strip into direction to the flip boundary ellipse generates consequently two-periodic flutter, stable two-periodic cycles and two-hyper cycle of the period-doubling way to chaos. Figure 13 represents such a set of bifurcations starting from two-hyper-cycle of 6-periodic resonances, followed by two-hyper-cycle of periodic-doubling and finishing by the moving two-hyper-cycle of the H&on strange attractors. 3. BIFURCATIONS GENERATED BY THE MOVEMENT OF SECOND EQUILIBRIUM
As has been mentioned above, the H&on map has two equilibria: a primary equilibrium (x, y) and the second - dual - equilibrium (x’, y ‘) satisfying the quadratic system: x=1--&+y,
y = bx.
The Vietta conditions immediately imply that x + x’ = (b - 1)/u, XX’ = -l/a means that x’ = -l/ax
= -x/(1
-x + y),
y’ = -y/(1
- X + y).
which (43)
M. SONIS
2228 1.8
/ Right half-strip of &main of stability of two-periodic cycle
x, = I y, = 0.3
yz = 0.186
x0 = 1
Yo = I
1=1000
q=mo Points = 50 r=
1.8
Fig. 10. Planar bifurcation diagram of the period-doubling way to the chaotic H&non attractor. 1.8
y, = 0.3
y7 = 0.186
x0 = I (I, yo=
I
!=
1000
0.3)
(0.62. 0.186)
y = 200
Points
= 50
rz1.8
Fig. Il. Bifurcation diagram FOIthe Feigenbaum way to chaos when equilibrium crosses boundaries of domains of stability of various period-doubling cycles.
Analysis of bifurcations
2229
x, = -1000 y, = -1000 x2 = -2.5 y2 = -2.5 x0=-I y, = 0.5 I = 20.000 q=
1000
Points = 50 r = 2200
-22ot 10
2200
0
Fig. 12. Two-hyper-cycles generated by the travel of equilibria along the flutter boundary of left half-strip of stability.
x, =-2 y, = -2 x* = -1.1 Y, = 0
t = 1000 Movement
q = 500
of
Points = 100 r=3
-3 -3
0
Fig. 13. Two-hyper-cycle way to the moving two-hyper cycle of the H&on stranee attractors.
3
2230
M. SONlS
The movement of the equilibrium (x, y j implies the movement of the dual equilibrium (x’ . y ‘). This dual equilibrium has its own dual domain of stability defined by its own dual divergence, flip and flutter bountis. In the vicinity of these curves the different additional bifurcation phenomena oczm. The forn&as (43) &low us to represent the dual stability domain and dual bifurcations in the terms of coordinates of the primary equilibrium (X , y ) The Jacobi matrix J( t, r + 1) of the J%!Snonmap in the point of dual equilibrium (x’ . ,v‘) obtains a value
such that
TrJ ‘* = -2ffx’ = -2/x
(45)
det J’* = -b = -y/x.
Therefore, (10) implies that Ehe dom&n of. statiiity of dual equilibrium (x’, y ‘) is given in terms of the coordinates of the primary equifibrium (x. y) by the inequalities: -1 2 2/x < -y/s
cd 1.
f f 0
(46)
which are equivalent to the two systems of inequalities: y:>(), .-\‘-.- I“;2. I i- \’ -> 0: .K I-- y *- -2”
y *_ I)
(47)
.K + 1’ < 0.
(48)
These systems of inequalities geometrically represent two infinite straight angles (see Fig. 14) bounded by dual divergence straight line x’ - ; = 2, dual flip line x - ,y = -2 and dual flutter line x’ -+ y = 0. If, starting within the dual stability domain, the primary equilibrium (x. y) crosses the divergence line x - v = 2 then the corresponding dual equilibrium (x’. .v’> is transformed from an attractor to a repeller and orbits became divergent; if (x, y) crosses the flip boundary Y - y = -2 then the dual equilibrium is transformed into two-periodic cycle. When (x, y) approaches the flutter line .Y -t. y - 0 then the dual equilibrium generates a multitude of different bifurcation phenomena. It is easy to calculate the coordinates of equilibria generating a dual q-periodic orbit. The conditions (14) and (45), together with condition .Y ‘- -. \‘, imply that ‘Tr J”s
-‘T - 2 is
=
27ip
2 00s _- ._
or
,y II
.- 1’ -:- - 1 :cos
??!f.
LT -l/iTr
J’“,
(49)
ii
cl
Table 2 represents the coordmates of primary and dual equihbria generating a q-periodic orbits for q G 12; this table can be easily extended. Figures 15 and 16 give the examples of generating S-periodic and h-periodic phenomena while crossing the dual flutter boundary. The important feature of this periodic behavior is the existence on the flutter segment of mode-locking domains - the Arnold tongues. Next. the dual stability domain for 2-periodic cycle can be considered. As ascertained in Section 2.2 the domain of stability of two-periodic cycle (u. I>) is given by the inequalities: :, A (G((I -- !>\I -- [l] + 7))) -. /-I:
/
(SO)
Using the conditions (IX) and (43) one can see that the dual stability domain for 2-periodic cycle coincides with the stability domain for L&periodic cycle given by the inequalities (39)-(41). Therefore, it is possible to find the Feigenbaum period-doubling way
2231
Analysis of bifurcations
Dual stability
Fig. 14. Dual stability domain.
x, = 3.2341 y, = -3 x2 = 3.2341 y2 = -3.3 x0 = 0.5 y. = -0.5
t=
500
q = 200
Points = 100 r=2
--L
”
Fig. 15. Mode-locking of dual Speriodic cycles.
2232
M. SONIS
x, = 2 y, = -1.1 x* = 2 -y? = -2.01 X” = 0.5 0 1’” = -4.5 t = 500 q = 200 Points - 100
Fig. 16. Mode-locking of dual 6-periodic cycles.
Table 2. Coordinates of primary and dual equilibria generating dual periodicity of Hhon map Type of periodicity 4
2
Arguments of eigenvalues Q = ??. Y
Values of trace TrJ ‘* = 2cos2nL (7
Coordinates of primary equilibria x”- v
Coordinates of dual equilibria x ’ = - ,2’
1
P
0.66667
0.61803 -1.61803
x
not exists
not exists
-3.23609
0.43309
1.23607
0.83964
7
0.4
I .24698
-- 1.60388
0.381 17
-0.445 04
4.49398
0.562 59
-- i .80194
1.10991
0.909 90
1.41421
--1.41421
0.36940
2233
Analysis of bifurcations Table 2. (continued) Type of periodicity
Arguments of eigenvalues 2s 9 4a 9
9
8a 9 IT 3
Coordinates of dual equilibria
1.53209
-1.30541
0.36153
0.347 30
-5.758 71
0.46006
1.064 18
0.943 12
-1.87939 1.61803
3a 5
-1.23607 3.23609
-0.61803
2a 11 4a ii 6n 11 ba 11 10T 11 12
Coordinates of primary equilibria
Values of trace
-1.18870
0.35196
0.830 83
-2.385 78
0.413 37
-0.284 63
7.026 67
0.53830
-1.30972
1.52704
0.74342
-1.91899
1.04221
0.96108
-1.15470
x, = -1.2357 y, = -1 x2 = -1.2357 Yz= 1 x0 = 0.5 y. = -0.5 r=500 p = 200 Points = 100 r=2
-2
0.59137
1.68251
1.73205
s 7
0.35600
”
Fig. 17. Feigenbaum’s way to chaos in the vicinity of dual equilibrium.
0.34892
2234
M. SONIS
to chaos in the vicinity of dual equilibrium via the movement of the primary equilibria. Figure 17 presents an example of such bifurcation events when the primary equilibrium crosses the dual flip boundary for two-periodic cycle. 4. CONCLUSION
In spite of the fact that we restricted ourselves by considering only the first and second iterate of the H&ton map, this gave a systematic and much more complete way for the analysis and control of bifurcations in Henon map. The study can be extended further by the consideration of other iterates and by more detailed travels of equilibria in the space of orbits. Three immediate conclusions can be outlined: first, the H&on map contains all possible bifurcation phenomena known for two-dimensional discrete maps; second, the existence of two equilibria generates the dual cascade of the bifurcation events in the vicinity of a second dual-equilibrium; and third, the analysis presented can serve as a prototype of the bifurcation analysis for finite-dimensional iterative processes with multiple equilibria. /\rknclM,IPdR~ments-The author is much indebted to Mr Eran Barzilai for his help with computer simulations
REFERENCES 1. M. H&ton, A two-dimensional mapping with a strange attractor. Commun Muth Whys 50, 69-77 (1078). I. Hao Bai-Lin, Chaos, Vol. II. World Scientific Publishing Co (1990). 3. M. Sonis, Behavior of iteration processes near the boundary of stability domain, with applications to the socio-spatial relative dynamics. in Functional-Differentia[ Equations. eds M. Drachlin and E. Litsyn. 198-227 (1993). 4. M. Sonis, Calculus of iterations and labor-capital core-periphery relative redistribution dynamics. Chaos, Solitons and Fractals U(4), 535-552 (1994). 5. P. A. Samuelson, Foundations of Economic Analysi.s. enlarged edition, Harvard Economic Studies. Vol. X0. Harvard University Press, Cambridge, MA (1983). h. D. S. Dendrinos and M. Sonis, Chaos and Sorio-Spntial Dpnamics. Applied Mathematical Sciences, Vol. 86. Springer, Berlin (1990). 7. M. Feigenbaum. Qualitative universality for a class of nonlinear transformations. J. Sfutivticul Whys. 19, 25-52 (1978). X. V. I. Arnold, Loss of stability of self-oscillations close to resonance and versa1 deformations of equivalent vector fields. Functional Anal. Appl. 11, X5-92 (1977). 9. V. Kogan, Loss of stability of auto-oscillations in the vicinity of resonances of order 3 and 4, in .Systems Dynamics. Gorky University Press, Gorky, Russia (1991). 10. C. Simo, On the H&cm-Pomeau attractor. .7. Stuti~l P&s. 21(4), 465-494 (1079). Il. Y-N. Huang, An raie ar@ytieal method for exploring periodic orbits of the Henon map. Scientiu Sinicu (series A) Z!#(lz), 1317(E&q. 12. K. T. ALligogd ad ‘P. Sauer, Rotation numbers of periodic orbits in the Henon’s map. C’ommun. Math Phys. 129, 105 (1988). 13. R. L. Devaney, An Introduction to Chaotic Dynamic Systems, 2nd edn, Part 2. Addison-Wesley, Reading, MA (1989). 14. H-O. Peitgen, H. Jurgens and D. Saupe. Chaos and Fractuls. Ch. 12. Springer, Berlin (1992).
& Froc&zls Vol. 7, No. 12, pp. 2215-2234, 19% Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. All rinhts resewed 09ao779p6 sis.00 + lO.cKl
PII: !3o%o-o779(%)ooo81-1
Once More on H&on Map: Analysis of Bifurcations MICHAEL
SONIS
Bar-Ban University, Israel
Abstract-Using the analysis of bifurcations approach the detailed description of bifurcation phenomena in the classical Henon map is presented. This description strongly supports the idea that the HCnon map contains all possible bifurcation phenomena known for two-dimensional discrete maps. It is interesting to note that the existence of two different equilibria in the Henon map generates additional - dual - appearance of bifurcation phenomena. The proposed analysis can serve as a prototype of the bifurcation analysis for finite-dimensional iterative processes with multiple equilibria. Copyright 0 1996 Elsevier Science Ltd.
1. INTRODUCTION
The well-studied Henon map (H&on [l]) x f+l
=
1
-
a3
+
Yt
t = 0, 1, 2, . . .)
Yr+1 = bxt,
presents a simple two-dimensional map with quadratic non-linearity. This map gave a first example of the strange attractor with a fractal structure. Because of its simplicity, the H&on map easily lends itself to numerical studies. Thus a large amount of computer investigations followed (see B-L. Hao [2]). Nevertheless, the complete picture of all possible bifurcations under the change of the parameters a and b is far from completion. The purpose of this study is to produce a more detailed and more complete description of the multiplicity of bifurcations in the H&on map on the basis of newly developed analytical/numerical procedure of the description of properties of equilibria of finite-dimensional iterative maps on the boundary of the stability of equilibria (see Sonis [3, 41). Next we present in brief an analytical description of the two-dimensional version of the analysis of bifurcations procedure for two-dimensional iteration maps of the type Xt+1
=
U(a,
b;
x,,
Yl)
(2) Yt+1 = V(4
b; x,3 YJ
where x,, yt represent the components of the iteration process at time t = 0, 1, 2, . . .; parameters a, b are the external bifurcation parameters and the functions U(a, b; x,, y,) and V(a, b; x,, y,) are differentiable (almost everywhere) functions in their components x,7 Yt. Analysis of bifurcations procedure includes first of all the calculation of possible equilibria x, y of the iteration process (2) given by the system of equations x = u(a, b; x, y),
Y = V(a, b; x, Y). 2215
(3:)
2216
M. SONIS
The algebraic equations (3) allow for a representation of two external bifurcation parameters a, b with the help of the components of the equilibrium x, y : h = g(x. y).
a = f(x. y);
(4)
As a result, the coordinates of the equilibrium X, y play a role of internal bifurcation parameters. Movement of equilibrium points can be placed on segments of straight lines. This allows for the complete computerized description of the appearance of different bifurcation phenomena in the space of orbits. Thus, the first basic element of the analysis of bifurcations procedure includes travels of equilibria in the space of orbits which reveal the qualitative features in the behavior of the orbits of the iteration process near the boundaries of domain of stability of the equilibria. The next step of the analysis of bifurcations procedure is the construction of the matrix of a linear approximation of the iteration process, the Jacobi matrix: / --‘j 3x,
--3v,‘[
J(t, t + 1) = j
(5)
/ av j 3x, and its value J* at the fixed point X, ,Y. r-3u * J” -; I z-r / j SV”
av 3v,
au* 3y
/ /
avz? / 31:
j 3x L..
(6)
/ -1
where U* = U(x, y), V” = V(x, y). The eigenvalues ,B,. k of the Jacobi matrix J* are the solutions of the characteristic equation ,l? - Tr J*p ?- det J* = 0.
(7)
where
(8)
By the well-known Von Neumann theorem. the equilibrium (x, y) is asymptotically stable if and only if, for all its eigenvalues p, ) k%,the following conditions hold: I& / i: 1.
:b / -I 1.
(91
The outcome of the general Routh-Hurvitz stability conditions is that the polynomial !I2 - Tr J*,l+ det J” has roots less than 1 in absolute value if and only if .-I
_S Tr J*
-; &t
J* CC 1
(see Samuelson ]S] p. 436. or Dendrinos and Sonis [6] p. 79).
(1Q)
Analysis of bifurcations
i!217
Presenting Tr J* and det J* through the coordinates X, y of the equilibrium one obtains in the space of orbits the domain of stability of equilibria; boundaries of this domain are the following curves: (1) the divergence boundary under the equation TrJ* = detJ* + 1;
(11)
(2) the flip boundary under the equation TrJ* = -(det J* + 1);
WY
(3) the flutter boundary under the equation detJ* = 1.
(13)
On the divergence boundary at least one of the eigenvalues is equal to 1. Crossing of this boundary allows for orbits to approach infinity. Such divergence starts from within the domain of stability; this domain of divergence is the infinity-locking domain. On the flip boundary at least one of the eigenvalues is equal to - 1. Each point on the flip boundary corresponds to a two-periodic cycle, and movement outside the domain of stability generates the Feigenbaum type period doubling sequence, leading to chaos (Feigenbaum [7]). On the flutter boundary 1,~~ 1= 1~121 = 1. It is easy to describe the type of bifurcations in all points on the flutter boundary. The condition Ij.+( = 1~121 = 1 means that p1 = &R, OGQGl ~(2= eei2Y and therefore TrJ* = ,u~+ j,~ = 2cos27rQ.
(14)
If 52 is a rational fraction: Q = p/q, then we have q-periodic (resonance) fixed points; between them there are fixed points of strong resonance with Q = l/3, 52= l/4. Other rational fractions 52= p/q represent points of weak resonance. The same periodic behavior is also observed in a small domain of s2 near p/q. This domain, the mode-locking domain, is the image of the Arnold tongue from the corresponding domain of change in eigenvalues in the complex plane (Arnold [S]). For strong resonance, the mode-locking domain starts within the domain of stability (Kogan [9]). Thus, the movement of the equilibria - travels of the equilibria - in the space of orbits along the segments of straight lines and the crossing the boundaries of the stability domain reveal a plethora of possible ways from stability, periodicity, Arnold horns and quasiperiodicity to chaos. A numerical procedure of the description of such phenomena includes the construction of the spatial bifurcation diagrams in which the bifurcation parameter is the equilibrium itself. Organizing the movement of equilibria in the space of orbits along segments of straight lines can be achieved in the following way: it is possible to parametrize each segment of the straight line between the equilibria (x1, yl) and (x2, y2), as x(j) = 1 - f X1 + $x2 ( I j = 0, 1, . * *, T,
r(i)=(1- fI YI+
(15)
f~2
where j is a bifurcation parameter and T is the number of bifurcation steps. The usual bifurcation diagram can be obtained from eqn (15) by fixing x(j) or y(j).
2218
M. SONlS
Next, the ebmeMs of analysis of bifurcations procedure will be used for the study of the bifurcations of H&on map (1) with the purpose to show that under different parameter specifications this map can produce any preset dynamic behavior including stability, periodic motion, quasi-periodicity and various forms of chaotic movement. Our findings are in good agreement with previous results of numerical calculations and theoretical studies (see Simo [lo], Huang [lilt Alligood and Sauer [12], Devaney [13]. Part 2, Peitgen et al. 1141Ch. 12).
2.1.
Domain 0.f stability of equilibrium
for the Heizon map
The fixed points (x, y) of the HCnon map xr+.i = 1 .- ax; + yt .V*-7 i
=
lx,.
1 =
0.
1.
2,
.,
a, b f 0
satisfy the following quadratic system: .s =- 1 .- ax2 + )’ I
J’ = Iw
(16)
Thus, there are two equilibria satisfying a\.: .+ (1 - b).u -- 1 LT(I. The conditions
(16) imply that x f 0 and -9,,/.rz_ Ll - 1I -. f -+ _,
b -= y !.r .
(IX)
Therefore, instead of change of the external bifurcation parameters a and b one can change the location of equilibrium (x, y) in the space of orbits: in such a way the coordinates x and v can play a role of internal bifurcation parameters. The Jacobi matrix J(t. i i 11 of the H&on map is
and its value J” at the fixed point X. v ic
such that TrJ*
= -2ax
= -2(1 - .r + y)/.x (21)
detJ* = -h = -,y/:w.
Therefore. (10) implies that the domain of stability of equilibria (x. y) is given by the inequalities: -1 2 2(1 - x -!- y)/.u c: -v/x
<‘: 1,
x + 0
(22)
or. in a simpler form: 213
(.. .Y - ?; 5::2.
x + .\: ‘> 0.
(23)
This system of inequalities geometrically represents the infinite half-strip (see Fie. 1)
2219
Analysis of bifurcations
Divergence boundary
Fig. 1. A half-strip of stability of equilibrium for the Htnon map.
such that if the fixed point (x, y) lies within the half-strip (23) then its become an attractor. The three sides of this half-strip represent the flip boundary with the equation x - y = 2/3; the divergence boundary with the equation x - y = 2; and the flutter boundary with the equation x + y = 0. If, starting within the half-strip of stability, the equilibrium (x, y) crosses the divergence line x - y = 2 then it is transformed from an attractor to a repeller and the orbits became divergent. The most interesting bifurcation behavior appears when equilibrium approach the flutter line x + y = 0. Only a segment of this straight line with the end points (l/3, -l/3) and (1, - 1) belongs to the boundary of the domain of stability of equilibrium. The points on this segment generate a multitude of different bifurcation phenomena. It is easy to calculate the coordinates of equilibria generating a q-periodic orbit. The conditions (14) and (21), together with condition x = - y imply that TrJ* = -2(1 - x + y)/x = 2cos- 23i-P 9
or
Table 1 represents the coordinates of equilibria generating a q-periodic orbits for q c 12; this table can be easily extended. The important feature of this periodic behavior is the existence on the flutter segment of mode-locking domains - the Arnold tongues - interwoven within each other (Arnold [8]). For strong resonance, the mode-locking domain starts within the domain of stability
(Wan PI).
Figure 2 represents the planar bifurcation diagram describing a travel of equilibria
M. SONIS Table 1. Coordinates of equilibria eerating Type of periodic&y 4 ..- -...--.--_.____
Arguments of eigenvahes Q=2np 4
periodicity of H&non map
Values of trace TrJ * = 2cos.9 4 -----.._-.
Coordinates of equilibria .t =: .- \ ___.. --.-__
2
i:
-2
0.333 33
3
2n 3 i; 1‘ 27i -.5 477 $
-1
0.4
4 5
0
0.5
0.61803
Cl.59137
1.61803
0.356
._--.
0.666 67 27 .7 4, hi;
f
::
10
4 2a 9 47 (4 x-I4 0 I:
4 .?o? .-.. 5
II
1.24698
0.72647
- 0.44504
0.44994
--1.x01 94
(1.34471
1.41421
0.773 46
I.53209
0.8104
0.347 30
0.547 54
--1.87939
1.h1803 -0.61803
0.341) 17 n.tw 64 0.43309 0.X6.7 0.63251 0.46678
towards the point of three periodic bifurcation (strong resonance p/q = l/3) along the segment with the end points: (0.46, -0.34) -+ (0.4, -0.4). This planar bifurcation diagram includes 500 orbits, each starting from the initial position: x(O) = y(0) = 1; each orbit includes 1000 points, 200 of them hidden. The three-periodic mode-locking domain is started within the domain of stability; moreover, its interlacement with the nine-periodic mode-locking domain is clearly visible. It is easy with the help of Table 1 to produce analogical planar bifurcation diagrams for other periodic bifurcation phenomena.
Analysis of bifurcations
:!221
1.5
x, = 0.46 y, = -0.34 x2 = 0.4
y2
= -0.4
x0 = I ( Yg= 1 t= 1000 q=200 Points = 500 I=
1.5
-1.
Fig. 2. Planar bifurcation diagram representing three-periodic flutter.
Further, for the amplitudes 52 of eigenvalues which are not rational one obtains the quasi-periodic orbits and different ways to chaos. The mode-locking domains of different quasi-periodicities, chaos and divergence also appeared. Figure 3 represent the bifurcations appearing when equilibrium is moving along the flutter boundary segment. It is interesting to note that on the flutter segment there are the mode-locking domains of divergence; after their appearance the periodicity, quasi-periodicity and chaos appeared once more. Figure 4 represents such an appearance of 5-period hyper-cycle of periodic and quasi-periodic curves. 2.2
Second iterate of the H&non map
If the equilibrium (x, y) crosses the flip line x - y = 2/3 then it enters the domain of stability of two-periodic cycle. The complete description of behavior of two-periodic bifurcations can be offered with the help of analysis of the second iterate of the H&on map. The second iterate of the Henon map is: x,+2
= 1 - ax:+1 + yt+l = 1 - a(1 - ax: + yt)2 + bx, (25)
Yt+2 = bxt+, Let
x, = u,,
x,+2 =
ut+1;
Yt
= w
-
a-d
+ Yth
then one obtains the following map:
= “t,
Yt+2 = Ut+1,
ut+1
= 1 - a(1 - auf + u,)~ + but
ut+l = b(1 - auf + u,).
(26)
2222
M. SONIS 3
I i
'r
/"
/
,//
x, = 0.4474
;,
i' J
i'
L
yt = -0.4474
*'
,, " _/'
./
/
,' ,,"
,I'
x2 = 0.55 y2 = -0.55 .x0 = I 0 Y* = 1 t = 1000 q = 200 Points = 20 r--3
-3 :J
-3
3
Fig. 3. Periodic and quasi-periodic flutter
1, = 0.562 y, = --0.562 x2 = 0.566 y2 = --0.566 x0=
?
p()= 1 I = 1000 g=200 Points = 10 r=3
-3
0
Fig. 4. Speriodic hyper-cycle.
3
Analysis of bifurcations
2223
The fixed points (u, u) of the map (26) are two-period cycles of the H&on map. The coordinates u, u satisfy the equations: u = 1 - a(1 - au2 + u)* + bu (27)
u = b(1 - au* + u). It is possible to prove that u satisfies the following equation: a3u4 - 2a2u2 + (1 - b)3~ + [a - (1 - b)*] = 0.
(28)
It is easy to check that u3u4 - 2a2u2 + (1 - b)% + [a - (1 - b)2] = ( a*u* - a(1 - b)u + [(l - b)* - a])[uu* + (1 - b)u - 11. (29)
The equation au* + (1 - b)u - 1 = 0 gives the fixed points of the H&on map (see eqn (17)). Therefore the equations a214- a*(1 - b)u + [(l - b)* - a] = 0,
u = 6(1 - au* + u)
give the coordinates of the proper two-periodic cycle of the H&on conditions (30) imply that
(30)
map. Note that
u*uu = b[(l - !7)2 - a].
(31) The discriminant of the quadratic equation (30) gives the conditions of the existence of a proper two-periodic cycle of the Henon map: a*(1 - b)* - 4a2[(1 - 6)2 - a] 2 0, or 4a a 3(1 - by.
(32)
Using the conditions (18) one obtains 4(1 - x + y) 3 3(x - y)2
or -2 c x - y < 213
(33)
This means that if the equilibrium (x, y) crosses the flip line x - y = 2/3 then the two-periodic cycle (u, u) appears. Next we define the domain of stability of two-periodic cycle (u, u) which is a part of the strip (33). The Jacobi matrix JJ(t, t + 1) of the second iterate of Henon map is
JJ(t, t + 1) =
b + 4a2u,u,+l/b -2abu,
--2au,+db b
1
(34)
and its value JJ* at the two-periodic cycle (u, u) JJ* =
such that one obtains, using (31)
b + 4a*uu/b -2abu
-2au/b b
1
(35)
TrJJ* = 4a*uu/b + 2b = 4[(1 - b)* - a] + 26 detJJ* = b*.
(36)
M. SONS
2224
Therefore, (10) implies that the domain of stabihty of two-periodic cycle (u, u) is given by the inequalities: -1 + (4[(1 - b)’ - (~1+ 26) < b’ e: 1.
(37)
or. using (18), -2
?I [4(x - y)Z - 4(1 - x f y)] + 2xy < y2 < x2.
(38)
Further, one obtains the inequalities 3(x - y)” - 4(1 - x + y) < 0 4(x - y)” - 4(1 - x c y) + (x + yy > 0 x? - p > 0. This system of inequalities is equivalent to two systems, defining a right strip of stability I) <. x - “y < 2/x
s + y ‘*’ 0
(39)
without the part inside the ellipse 4(x - yj2 - 4(1 - x -t- y) + (X +
yj2 > 0
(4)
and the left strip of stability of two-periodic cycles -2 -c .Y - y i 0.
x + ,y < 0
(41)
without the part inside the same ellipse (see Fig. 5). The domain of stability of two-periodic cycles includes the divergence boundary x y = 2/3 belonging to the right half-strip of stability and the divergence boundary x - y = -2 belonging to the left half-strip of stability: the flutter boundaries are x = y, x + y = 0 and the flip boundary is 4(x -- .\‘)2 - 4(1 - .Y + y) -t (x + y)’ = 0
Flutter bowday
Fig. 5. Left and right half-strips of stability of two-periodic cycle for the H&on map.
(421
2225
Analysis of bifurcations
which is an ellipse with center (-l/4, l/4) and main diameters x + y = 0 and x - y = -l/2. Let us start from the description of bifurcation phenomena appearing in the right half-strip. Figure 6 represents the movement of equilibria along the segment (5, 4.2) -+ (5, 5). This movement starts within the domain of stability of equilibrium in which the orbits attracted to the equilibrium. When the equilibrium crosses the flip boundary, the stable two-periodic cycle appeared; when equilibrium touches the flutter boundary for two-periodic cycles, the two-periodic hyper-cycle of quasi-periodic orbits appears. The analysis of the bifurcations in the form of the two-periodic hyper-cycle of quasi-periodic orbits can be done with the help of movement of equilibria along the flutter line x = y (see Fig. 7). As in the case of fixed point it is possible to calculate the coordinates of all bifurcation phenomena. It is interesting to note that when one moves the equilibria left along the flutter line x = y the two-periodic hyper-cycles of chaotic attractors appear (see Fig. 8). The movement to the right along the flutter line x = y smoothes out invariant curves incorporated in 2-hyper-cycles (see Fig. 9). When the equilibrium crosses the elliptic flip boundary of the left half-strip of stability of two-periodic cycles it enters the domain of stability of four-periodic cycles. This domain is also bounded by flip, flutter and divergence curves. This opens the periodic doubling way to chaos. Figure 10 represents the planar bifurcation diagram of the Feigenbaum way to chaotic Henon attractor by the movement of equilibria through the domains of stability of different double-periodic cycles along the segment (1, 0.3) + (0.62, 0.186). This pheno-
x, = 5
y, = 4.2 x* = 5 Y2= 5 x0= 1 a Yo= 1 t= 1000 q = 200 Points = 150 r= 10
-10
0
10
Fig. 6. Movement of equilibrium, starting within its domain of stability, crossing the domain of stability of two-periodic cycles and touching the flutter boundary of this domain.
M. SONIS
2226
X)
3
=
Y, =3 x2 = s Y, = 5 x0 = 1 Y”=
1
1= 1000 q=2w
Points = 20 r= to
IQ
0
Fig. 7. Movement of equilibrium along the flutter curve of two-periodic cycle.
x1 =
2.44
y, =
2.44
x7 =
2.44
yz =
2.44
x0=
I
y*=
I
f := 1000
q=o Points = 1
-6
Q
Fig. 8. Two-periodic hyper.cycle of chaotic flutter phenomena.
6
Analysis of bifurcations
2227
220
x, = 1
Y, = 1
x2 = 100
y* = 100
x0 = 1
0 Y()=
1
t=20oa q = 200 Points = 100 r=220
-22C Fig. 9. Two-hyper-cycles on the flutter boundary belonging to the right half-strip of the domain of stability of two-periodic cycles.
menon is much more visible if one considers the usual bifurcation diagram (see Fig. 11) representing the well-known Feigenbaum period-doubling cascade. We will start the description on bifurcations generated in the right half-strip by the example of movement (on a big distance from the point (-2.5, -2.5) to the point (-1000, -1000)) of equilibrium (x, y) along the flutter boundary x = y (see Fig. 12). The same phenomena (as on Fig. 9) of the 2-hyper-cycle of periodicity, quasi-periodicity and chaos appeared. The crossing of the left half-strip into direction to the flip boundary ellipse generates consequently two-periodic flutter, stable two-periodic cycles and two-hyper cycle of the period-doubling way to chaos. Figure 13 represents such a set of bifurcations starting from two-hyper-cycle of 6-periodic resonances, followed by two-hyper-cycle of periodic-doubling and finishing by the moving two-hyper-cycle of the H&on strange attractors. 3. BIFURCATIONS GENERATED BY THE MOVEMENT OF SECOND EQUILIBRIUM
As has been mentioned above, the H&on map has two equilibria: a primary equilibrium (x, y) and the second - dual - equilibrium (x’, y ‘) satisfying the quadratic system: x=1--&+y,
y = bx.
The Vietta conditions immediately imply that x + x’ = (b - 1)/u, XX’ = -l/a means that x’ = -l/ax
= -x/(1
-x + y),
y’ = -y/(1
- X + y).
which (43)
M. SONIS
2228 1.8
/ Right half-strip of &main of stability of two-periodic cycle
x, = I y, = 0.3
yz = 0.186
x0 = 1
Yo = I
1=1000
q=mo Points = 50 r=
1.8
Fig. 10. Planar bifurcation diagram of the period-doubling way to the chaotic H&non attractor. 1.8
y, = 0.3
y7 = 0.186
x0 = I (I, yo=
I
!=
1000
0.3)
(0.62. 0.186)
y = 200
Points
= 50
rz1.8
Fig. Il. Bifurcation diagram FOIthe Feigenbaum way to chaos when equilibrium crosses boundaries of domains of stability of various period-doubling cycles.
Analysis of bifurcations
2229
x, = -1000 y, = -1000 x2 = -2.5 y2 = -2.5 x0=-I y, = 0.5 I = 20.000 q=
1000
Points = 50 r = 2200
-22ot 10
2200
0
Fig. 12. Two-hyper-cycles generated by the travel of equilibria along the flutter boundary of left half-strip of stability.
x, =-2 y, = -2 x* = -1.1 Y, = 0
t = 1000 Movement
q = 500
of
Points = 100 r=3
-3 -3
0
Fig. 13. Two-hyper-cycle way to the moving two-hyper cycle of the H&on stranee attractors.
3
2230
M. SONlS
The movement of the equilibrium (x, y j implies the movement of the dual equilibrium (x’ . y ‘). This dual equilibrium has its own dual domain of stability defined by its own dual divergence, flip and flutter bountis. In the vicinity of these curves the different additional bifurcation phenomena oczm. The forn&as (43) &low us to represent the dual stability domain and dual bifurcations in the terms of coordinates of the primary equilibrium (X , y ) The Jacobi matrix J( t, r + 1) of the J%!Snonmap in the point of dual equilibrium (x’ . ,v‘) obtains a value
such that
TrJ ‘* = -2ffx’ = -2/x
(45)
det J’* = -b = -y/x.
Therefore, (10) implies that Ehe dom&n of. statiiity of dual equilibrium (x’, y ‘) is given in terms of the coordinates of the primary equifibrium (x. y) by the inequalities: -1 2 2/x < -y/s
cd 1.
f f 0
(46)
which are equivalent to the two systems of inequalities: y:>(), .-\‘-.- I“;2. I i- \’ -> 0: .K I-- y *- -2”
y *_ I)
(47)
.K + 1’ < 0.
(48)
These systems of inequalities geometrically represent two infinite straight angles (see Fig. 14) bounded by dual divergence straight line x’ - ; = 2, dual flip line x - ,y = -2 and dual flutter line x’ -+ y = 0. If, starting within the dual stability domain, the primary equilibrium (x. y) crosses the divergence line x - v = 2 then the corresponding dual equilibrium (x’. .v’> is transformed from an attractor to a repeller and orbits became divergent; if (x, y) crosses the flip boundary Y - y = -2 then the dual equilibrium is transformed into two-periodic cycle. When (x, y) approaches the flutter line .Y -t. y - 0 then the dual equilibrium generates a multitude of different bifurcation phenomena. It is easy to calculate the coordinates of equilibria generating a dual q-periodic orbit. The conditions (14) and (45), together with condition .Y ‘- -. \‘, imply that ‘Tr J”s
-‘T - 2 is
=
27ip
2 00s _- ._
or
,y II
.- 1’ -:- - 1 :cos
??!f.
LT -l/iTr
J’“,
(49)
ii
cl
Table 2 represents the coordmates of primary and dual equihbria generating a q-periodic orbits for q G 12; this table can be easily extended. Figures 15 and 16 give the examples of generating S-periodic and h-periodic phenomena while crossing the dual flutter boundary. The important feature of this periodic behavior is the existence on the flutter segment of mode-locking domains - the Arnold tongues. Next. the dual stability domain for 2-periodic cycle can be considered. As ascertained in Section 2.2 the domain of stability of two-periodic cycle (u. I>) is given by the inequalities: :, A (G((I -- !>\I -- [l] + 7))) -. /-I:
/
(SO)
Using the conditions (IX) and (43) one can see that the dual stability domain for 2-periodic cycle coincides with the stability domain for L&periodic cycle given by the inequalities (39)-(41). Therefore, it is possible to find the Feigenbaum period-doubling way
2231
Analysis of bifurcations
Dual stability
Fig. 14. Dual stability domain.
x, = 3.2341 y, = -3 x2 = 3.2341 y2 = -3.3 x0 = 0.5 y. = -0.5
t=
500
q = 200
Points = 100 r=2
--L
”
Fig. 15. Mode-locking of dual Speriodic cycles.
2232
M. SONIS
x, = 2 y, = -1.1 x* = 2 -y? = -2.01 X” = 0.5 0 1’” = -4.5 t = 500 q = 200 Points - 100
Fig. 16. Mode-locking of dual 6-periodic cycles.
Table 2. Coordinates of primary and dual equilibria generating dual periodicity of Hhon map Type of periodicity 4
2
Arguments of eigenvalues Q = ??. Y
Values of trace TrJ ‘* = 2cos2nL (7
Coordinates of primary equilibria x”- v
Coordinates of dual equilibria x ’ = - ,2’
1
P
0.66667
0.61803 -1.61803
x
not exists
not exists
-3.23609
0.43309
1.23607
0.83964
7
0.4
I .24698
-- 1.60388
0.381 17
-0.445 04
4.49398
0.562 59
-- i .80194
1.10991
0.909 90
1.41421
--1.41421
0.36940
2233
Analysis of bifurcations Table 2. (continued) Type of periodicity
Arguments of eigenvalues 2s 9 4a 9
9
8a 9 IT 3
Coordinates of dual equilibria
1.53209
-1.30541
0.36153
0.347 30
-5.758 71
0.46006
1.064 18
0.943 12
-1.87939 1.61803
3a 5
-1.23607 3.23609
-0.61803
2a 11 4a ii 6n 11 ba 11 10T 11 12
Coordinates of primary equilibria
Values of trace
-1.18870
0.35196
0.830 83
-2.385 78
0.413 37
-0.284 63
7.026 67
0.53830
-1.30972
1.52704
0.74342
-1.91899
1.04221
0.96108
-1.15470
x, = -1.2357 y, = -1 x2 = -1.2357 Yz= 1 x0 = 0.5 y. = -0.5 r=500 p = 200 Points = 100 r=2
-2
0.59137
1.68251
1.73205
s 7
0.35600
”
Fig. 17. Feigenbaum’s way to chaos in the vicinity of dual equilibrium.
0.34892
2234
M. SONIS
to chaos in the vicinity of dual equilibrium via the movement of the primary equilibria. Figure 17 presents an example of such bifurcation events when the primary equilibrium crosses the dual flip boundary for two-periodic cycle. 4. CONCLUSION
In spite of the fact that we restricted ourselves by considering only the first and second iterate of the H&ton map, this gave a systematic and much more complete way for the analysis and control of bifurcations in Henon map. The study can be extended further by the consideration of other iterates and by more detailed travels of equilibria in the space of orbits. Three immediate conclusions can be outlined: first, the H&on map contains all possible bifurcation phenomena known for two-dimensional discrete maps; second, the existence of two equilibria generates the dual cascade of the bifurcation events in the vicinity of a second dual-equilibrium; and third, the analysis presented can serve as a prototype of the bifurcation analysis for finite-dimensional iterative processes with multiple equilibria. /\rknclM,IPdR~ments-The author is much indebted to Mr Eran Barzilai for his help with computer simulations
REFERENCES 1. M. H&ton, A two-dimensional mapping with a strange attractor. Commun Muth Whys 50, 69-77 (1078). I. Hao Bai-Lin, Chaos, Vol. II. World Scientific Publishing Co (1990). 3. M. Sonis, Behavior of iteration processes near the boundary of stability domain, with applications to the socio-spatial relative dynamics. in Functional-Differentia[ Equations. eds M. Drachlin and E. Litsyn. 198-227 (1993). 4. M. Sonis, Calculus of iterations and labor-capital core-periphery relative redistribution dynamics. Chaos, Solitons and Fractals U(4), 535-552 (1994). 5. P. A. Samuelson, Foundations of Economic Analysi.s. enlarged edition, Harvard Economic Studies. Vol. X0. Harvard University Press, Cambridge, MA (1983). h. D. S. Dendrinos and M. Sonis, Chaos and Sorio-Spntial Dpnamics. Applied Mathematical Sciences, Vol. 86. Springer, Berlin (1990). 7. M. Feigenbaum. Qualitative universality for a class of nonlinear transformations. J. Sfutivticul Whys. 19, 25-52 (1978). X. V. I. Arnold, Loss of stability of self-oscillations close to resonance and versa1 deformations of equivalent vector fields. Functional Anal. Appl. 11, X5-92 (1977). 9. V. Kogan, Loss of stability of auto-oscillations in the vicinity of resonances of order 3 and 4, in .Systems Dynamics. Gorky University Press, Gorky, Russia (1991). 10. C. Simo, On the H&cm-Pomeau attractor. .7. Stuti~l P&s. 21(4), 465-494 (1079). Il. Y-N. Huang, An raie ar@ytieal method for exploring periodic orbits of the Henon map. Scientiu Sinicu (series A) Z!#(lz), 1317(E&q. 12. K. T. ALligogd ad ‘P. Sauer, Rotation numbers of periodic orbits in the Henon’s map. C’ommun. Math Phys. 129, 105 (1988). 13. R. L. Devaney, An Introduction to Chaotic Dynamic Systems, 2nd edn, Part 2. Addison-Wesley, Reading, MA (1989). 14. H-O. Peitgen, H. Jurgens and D. Saupe. Chaos and Fractuls. Ch. 12. Springer, Berlin (1992).