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Renormalisation in implicit complex maps
Physica D 39 (1989) 149-162 North-Holland, Amsterdam
RENORMALISATION
IN IMPLICIT COMPLEX
MAPS
B,.D. MESTELa and A.H. OSBALDESTINb “Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK hDepartment of Mathematical Sciences, Loughborough University of Technology, Loughborough,
LEll
3TU, UK
Received 2 May 1989 Accepted 1 July 1989 Communicated by A.V. Holden
We study the dynamics of a family of implicit complex maps restricted to an invariant circle. The renormalisation analysis for golden mean rotation number is given. We find a one-parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renorrnalisation transformation. These period-2 points are pairs of fractional linear maps.
1. Introduction
The polynomial equation
g(z, w) = 0
(1-l)
in the two complex variables z and w defines a many-to-many map f : z * w if the pair z, w solves eq. (1.1). This map is called an implicit complex map. Such maps were introduced by Bullett et al. [l] as they may possess time-reversal symmetries. They may also be regarded as an extension of the theory of rational maps and Kleinian groups [2]. They may incorporate rational-map behaviour, dissipative and Hamiltonian behaviour and are therefore of considerable interest. The purpose of this paper is to reveal the scaling behaviour of certain quasiperiodic orbits in a particular family of implicit maps discovered by Bullett [2]. We begin by defining the bi-quadratic family in which we shall work and by recalling some basics of implicit maps [2]. In particular, there appears to be an invariant circle and it is
dynamics on this that we study in this paper. We note that the implicit map restricted to this circle induces an “almost-c’” circle map - a term which we shall define in section 3. Some numerical observations for orbits with golden mean rotation number are presented in section 4. In section 5 we review the renormalisation analysis of circle maps, paying special attention to maps with discontinuous derivatives. Next, we note the existence of an invariant “modulus” for almost-C’ maps under the renormalisation transformation and in section 7 deduce an explicit expression for it for the bi-quadratic maps. We show that in fact the fixed points of the square of the renormalisation operator govern our observed behaviour and, moreover, that the fixed points are fractional linear mappings*. The universality of the scaling that we find is investigated in section 9, where we consider a simple almost-C’ map for which we can directly prescribe the modulus. We end with some remarks and open questions.
0167-2789/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
*We thank Ya. G. Sinai for pointing
this out to us
150
B. D. Mestel and A. H. Oshaldestin / Renormalisation in implicit complex maps
2. The bi-quadratic family of implicit complex maps We say that most quadratic family
in each of z and w separately. implicit
of the invariant
bi-quadratic
g was considered will in general
h(w)
complex
the
The case of general in ref. [2]. For such
have two distinct
and two distinct preimages. Bullett [2] introduces a two-parameter implicit
A
maps was
circle that bounds
Siegel disc was investigated. g, points
complex
in ref. [l] and the golden mean scaling
structure
images family of
maps. It is given by
= h(z),
(2.1)
iz* + mz m3z+i miz + 1 -1(1+m2)(miz+l)’
=
Here
I and
z++w*
-z_,
As
a result
m are real parameters,
(2.2)
0 I I I 1 and
h(z) = h(5) . For I= 1 this is the family of maps studied in ref. [l] (with different parameters) while I = 0 gives
(2.5)
iz2 f mz = miz+l ’
of
the
symmetry
quadratic family induces maps f from the right These
maps
For z E H, f(z) continuous
(2.5)
(2.3)
are given
z~w*w*I.
(2.4)
This symmetry was introduced by Bullett et al. in ref. [l] in order to mimic Hamiltonian systems. Secondly, the bi-quadratic family has a timepreserving symmetry with respect to reflection in
bi-
is defined
by f = (h)-‘h.
by taking
the solution
that lies in H. The maps f are
everywhere
in H except
along
an arc
of a semi-circle that extends from the imaginary axis [2]. This arc is called the discontinuity line, D disconti-
An important feature of the dynamics of f is an invariant semi-circle, C, that extends outwards from the imaginary axis into H. If it exists this semi-circle is unique and has end points at z, and z- respectively and goes through D at the discontinuity point zd [2] *. zd maps to two points on the imaginary axis, z+ and zP (which are complex conjugates) and which themselves map to the same point in H. In terms of the parameters m and I, we have
zi=
_ti(l
- 1)l’*,
(24
zd >
=
[l
+
I +
m2 - m4(1 - 1) - m6(1 - 1)2]1’2 2
m2+1 We shall call the family given by (2.2) the biquadratic farnib. The bi-quadratic family has two symmetries. Firstly it has a time-reversal symmetry with respect to reflection in the real axis. This says that
the
a family of single-valued half-plane, H, to itself.
f: H-H
w of h(w) = h(z)
Re( h(z)
-w.
(see fig. 1). The map f has a simple jump nuity as D is crossed.
where
h(z)
axis. This says that
g is bi-quadratic if g(z, w) is at
of bi-quadratic
introduced
the imaginary
Im( zd) =
m[(l
- l)m2+ m2+l
l]
(2.7) ’
If we identify the points z i (writing z+ - z_) then this invariant semi-circle becomes an invariant circle and we may look at the map induced by f on this circle. This map is well defined and continuous because f( z +) = f (z _) since h (z +) = *This fact makes the numerical ant circle particularly easy.
determination
of the invari-
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
(
\:
.
2;. ::/-y-Y :s_,&i ,,..,,.. :.: ,y: :
‘.,
/k
.,,: -*..-... . .. ., ..,,,___, ,,, ,,
j,.
.2.
j
_&yy’
i
:_.,
‘..,,,.,
.i r--\ \ ..___..
._,..:
..
\.
.’
,‘-*,
i.
..
:
..
,i
__,____-----‘.
.....’ ..
:
.
.__
..’
a
b
Fig. 1. Phase portrait of the quadratic family (2.2) for three values of 1. The value of m 1s “tuned” so that the rotation number on the invariant semi-circle is 1 - 0, (I = $(1/5 - 1). The figures contain the invariant circle and the part of the discontinuity line that starts on the imaginary axis and ends on the invariant circle. (a) I = 0, m = 0.289737. (b) I = 1, m = 0.25375. (c) I = 0.5, m = 0.27144.
h (z _). We shall henceforth refer to the invariant semi-circle in H as the invariant circle, with it being tacitly assumed that the points t * are identified. We shall study this map in section 3. For 1= 0 the invariant circle is the unit circle C = {eieI - a/2 I 8 I ~/2} with the points
{ + i, -i} identified. The parameter m determines the rotation number of the induced circle map. Fig. la shows several orbits of the map f for I= 0 and m = 0.289737 when the rotation number on C is u’ = 1 - u, where u = t(fi - l), the golden mean.
152
B. D. Mesrel and A. H. Osbaldestin / Renormalisation
For I = 1 we get the map studied in ref. [l]. For m = 0.25375, there appears to be an invariant circle of rotation number u’. This circle meets the imaginary axis in a right-angled “beak” at the origin. Orbits of the map for I = 1 and m = 0.25375 are shown in fig. lb. The numerical studies in ref. [l] suggest that this is a circle that is C’ (except at the origin) but not C2. For 0 < I < 1, there appears (at least numerically) to be a curve m = m (1) in the (m, 1) parameter space on which the map has an invariant semi-circle in the right half plane with rotation number u’. Fig. lc shows the invariant circle for I = 0.5 and m = 0.27144 on which the rotation number is u’. We stress that it is not proved that such a circle exists for 0 < I _<1 but the numerical evidence is very suggestive. In fact, there appears to be an “Arnol’d tongue” structure in the (m, I) parameter space for 0 I I I 1 [3]. For fixed I there is phase locking on nontrivial intervals of the parameter m as is the case with dissipative maps. One important feature of the invariant semicircles is that they are perpendicular to the imaginary axis for 0 I 1~ 1. This is clearly true for I = 0 when C is the unit semi-circle and appears plausible for 0 < I < 1 from fig. lc. This property is a consequence of the symmetry (2.5) and the uniqueness of the invariant circle in H. The implicit function theorem applied to C guarantees its extension to the left half-plane. The reflection of this extension in the imaginary axis is also an invariant curve in H and therefore, since C is unique, the reflection of the extension of C is C itself. This implies that C is perpendicular to the imaginary axis. Note that when 1= 1 the end points z + of C meet at the origin and the circle crosses over to form a “figure eight”. In this singular case the circle is not perpendicular to the imaginary axis (as is clear from fig. lb). Note also that the symmetry (2.4) implies that C is symmetric with respect to reflection in the real axis. We parametrise C with polar coordinates about the origin, This is clearly convenient for I = 0 in
in implicit complex maps
which C is the unit semi-circle. Moreover, for 0 I I < 1 it means that the circle extends from -IT/~ to 71/2 in e-space: C= {~(B)rr(B)e’~I-a/21B1a/2}.
(2.8)
The points 8 = k IT/~ correspond to the points z i on the imaginary axis. We identify the points zk so that the semi-circle becomes a circle. Note that, since C is perpendicular to the imaginary axis, r( 0) is stationary at 19= & 71/2 and so r’( &n/2) = 0, or, equivalently, z’( 6) is real. This fact will be important in section 7. For I= 1 we parametrise with an angle ranging between -IT/~ and +a/4.
3. The induced circle map The map f restricted to the invariant circle C induces a circle map in &space. We shall denote this map by +: [-IT/~, IT/~] -+ [-a/2, IT/~]. Thus
Recall that circle C goes through the end of the discontinuity line at the point zd = ~(8,) where 0, E [ -~/2, a/2]. Although the map f (and hence +) is continuous on C, the derivative of 9 is discontinuous at 6, and at the point -a/2 - n/2. Elsewhere + appears to be C’. We define an almost-c’ circle map to be a circle map which is continuously differentiable except possibly at two points, one being the image of the other, where the left and right derivatives may differ. In terms of the map f we have
e-+4
- -fw))
e + e, + =q(z(e))
+z+,
+
z_.
(3.2)
Fig. 2 shows the induced circle maps for the three
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
-1.5
-0.5
0.0
1.0
153
00
1.5
THETA
THETA I
I
,
,
I
I’ -1.5
I -I B
7
I’ -0.5
I’ 00
-1.5 I’ 05
I’ 10
I 1.5
THETA
Fig. 2. The maps induced on the invariant circles for the cases (a)-(c)
cases in fig. 1. Note that the curves are symmetric in the line (p = - 0 because of the time-reversal symmetry, (2.4) with respect to the real axis and that they appear to be C’ except at the points 19, and +(e,), where the left and right derivatives are different.
4. Golden mean scaling-numerical
observations
continued fraction expansion of u’ is [2,1,1,1,. . .] and its rational convergents are given by F,_,/F,,. For various values of the parameter I we found the value of m = m(l) for which the rotation number on the circle was u’ and looked at the scaling of the Fibonacci iterates of the discontinuity point rd. We observed that IP+‘(r,) IP”(%)
We shall now investigate the scaling structure around 0, for the case of rotation number u’. We denote by F, the sequence of Fibonacci numbers F,= 1, FI = 1, Fn+l = F, + F,_,. The
of fig. 1
- .Ql - KJ”, - %I - KY”
(4.la) (4.lb)
for constants K,, K, and y depending on 1. y appears to vary smoothly from 1 = 0 to 1= 1. At 1 = 1 we have y = u2, the value corresponding to
154
B. D. Mestel and A. H. Osbaldestin/
Renormalisation
in implicit complex maps
Table 1 /=o s = 0.303274
I=1 s = 1
I= 0.5 s = 0.459603
y implicit map y exact y test map (sect. 9)
0.37006895 0.37006894 0.37008
0.38196 0.38196601 0.38196601
0.3768519 0.37685189 0.37685
6 implicit map 6 exact 6 test map (sect. 9)
7.0017126 7.00171411 7.001714
6.854104 6.85410197 6.85410197
6.9162415 6.91624138 6.916241
the scaling behaviour of diffeomorphisms. This was observed in ref. [l]. Furthermore, for fixed I we calculated the parameter values m, for which the discontinuity point z,, is periodic with period F, with rotation number F,,_,/F,. We find
m2ni1
-mzn-1
m2n+2
-
mzn
-
-
Loa-“,
(4.2a)
Lea-“.
(4.2b)
The values of L,, L, and S also depend on the value of the parameter I. Table 1 lists the values of y and 6 for the three values of 1. At I= 1, 6 is the square of the eigenvalue - l/a2 of the circle-map renormalisation transformation at the simple fixed point [4]. Thus the scaling behaviour, also geometric, is not completely universal for these types of maps as the values of ?3 and y depend on 1. Nevertheless there is a single one-parameter family of universality
classes
for circle maps
of this
typeFig. 3 shows the maps I#+,, I#B~(obtained in the limit n -+ cc by scaling the F2,,+rth and F,,th iterates about 0, by y”) for the case 1= 0, m = 0.289737. Note that while $B& and & are discontinuous at 0,, the left and right limits of $, at 19,, respectively, equal the right and left limits of +L, respectively. The scaling behaviour (4.1) and (4.2) is unusual in that it involves only every alternate rational convergent to u’. This period-two scaling suggests that an explanation may be found in period-two
Fig. 3. The maps %,, $I~ (obtained in the limit n + co by scaling the F2,,+ 1th and F&h iterates of the induced circle map 9 by y” about Bd) for the case I = 0, m = 0.289737.
points of the circle map renormalisation transformation. We consider this in the next section. We note that Wilbrink [5] has observed similar period-2 behaviour in the golden mean circles of piecewise linear standard-like mapping. We believe our analysis here will be of some help in understanding this.
5. Renormalisation analysis for maps with discontinuities in the derivative The renormalisation analysis for circle maps given in ref. [4] involves the renormalisation trunsformation, T, acting on pairs (6,~) of functions that, on appropriate domains, “glue together” to form a map on the circle. This device is somewhat artificial in the case of smooth circle maps (and indeed may be dispensed with [6]) but in the case of our almost-C’ maps it is completely natural. Such a pair (<, 7) is illustrated in fig. 4. Given a pair (&, q), we define a circle map $(, II to be
+&4
= E(x)
if q(O)
= q(x)
ifO
(5.1)
B.D. Mestel and A. H. Osbaldestin/Renormalisation in implicit complex maps
155
critical hxed points. The simple fixed point determines the scaling behaviour of diffeomorphisms and is the rigid rotation through angle u. The critical fixed point determines the scaling behaviour of circle maps with a single cubic critical point at 0 and is observed in the transition from quasiperiodicity to weak turbulence in dissipative dynamical systems [4]. The fixed point equations for T are 4(x) = P-‘n(PxL
(5.5a)
n(x) = P%@@x)).
(5.5b)
The simple fixed point is the solution Fig. 4. Definition of the pair of maps (I, q). The modulus s is determined by the slopes of 6 and 1 at 0 and t(O) - q(O).
& II is a well-defined homeomorphism of the circle [q(O), t(O)] identifying the points t(O) and ~(0) provided <(n(O)) = 17(5(O)) and 0 < 5(O) < 1 [4]. We normalise the pair (5, n) by setting the length of the circle to 1, i.e.,
Any differentiable circle map $J may be written as a pair of differentiable maps (&, n+) in this way by defining
E,(x) = c4
@J(O) -1 IXlO,
q*(x)=f$(x)-1,
OSx0(0).
(5.3)
Moreover any circle map which has discontinuities in the derivative at 0 and +(O) and which is C’ elsewhere may also be written in this way. The golden-mean renormalisation transformation T is given by
&(x)=x+e,
(5.6a)
Vi(X) =x - a*
(5.6b)
of (5.5). The analysis of the spectrum of the derivative of T at (
(5 *4] Here j? is chosen to preserve the normalisation condition (5.2) so that j3 = n(0) - 17(5(O)). The renormalisation analysis [4, 6-81 is given in terms of two fixed points of T: the simple and
((17 - VU(O) = 0,
(5.7)
((9 - VNO)
(5.8)
= 0.
For almost-C’ maps (5.8) is not satisfied and the extra - 1 eigenvalue cannot be suppressed in this
B. D. Mestel and A.H. Osbaldestin/ Renormalisation in implicit complex maps
156
manner. However, (5.7) is satisfied by almost-C’ circle maps and we therefore restrict to the space of pairs that satisfy (5.7). This eliminates the eigenvalue - l/u. The remaining eigenvalue of modulus greater than 1 is the so-called “essential eigenvalue” 6 = l/a2. This is the eigenvalue that governs the scaling of the parameter values as in eq. (4.2). We denote by S the second iterate of the transformation T:
This modulus is the ratio of the slopes of [ and 71 at 0 and E(O) - ~(0) as illustrated in fig. 4. We claim that, provided* $(77(5(O))) f 0, and (5.7) holds,
(6.2) and thus, provided* holds,
(T&J)(v([(~)))
s( S(& 17)) = s&17). (5.9)
Z 0, and (5.7)
(6.3)
Hence, the modulus s is preserved by the renormoperator S. This is what makes the modulus s a natural parameter for the line of hxed points L. Eq. (6.2) follows from a simple calculation: aiisation
where y is given by y =
v(‘m) - m17tw
(5.10)
so that the normalisation (5.2) is enforced. (ti, vi) is, of course, a fixed point of S. dS([,, vi) has two eigenvalues of modulus greater than or equal to 1, viz., the essential eigenvalue l/u4 and the noncommuting eigenvalue + 1 corresponding to violation of (5.8). The eigenvalue + 1 indicates that there is a line, L, of fixed points of S through (Zi, qi). We shall see that L provides a one-parameter family of universality classes that corresponds to the scaling (4.1) and (4.2) observed above. Each of these fixed points has a scaling factor y and essential eigenvalue 6. Each fixed point is a period-two point for T and hence we see the period-two scaling behaviour in (4.1) and (4.2). In section 8 we shall give the exact form for this line of fixed points of s.
6. An invariant modulus We now introduce a modulus, s, by which we parametrise this line of fixed points. Let (E, VI)be a pair of maps. We define s~
s(E,
II)
~
tvi?to> = 17’wN wa tw’to)
elto)) 77’(O) .
(64
(P-‘aw”rlP)‘(o) stT(~~d)
=
(p-l~pp-lq&q'(o)
hMW)(511)'(0) =
v'(MW(175)'(0)
1
=p
s(E, 9) ’
(6.4)
Note that s(E1, ql) = 1 (since there are no discontinuities in the derivatives for the simple fixed point) and that when s # 1 condition (5.8) is violated. Hence the line of fixed points may be parametrised by s and each fixed point of S on the line corresponds to precisely one value of S. The value of s determines the universality class of an almost-C’ map, i.e., the manifolds of constant s and rotation number u in the space of pairs are the stable manifolds of the fixed points on the line L. One interesting point is that s depends on both the left and right hand slopes at 0 and those at t(O) - q(O). It is therefore possible for the ratio (6.1) to equal 1, i.e., s([, q) = 1, and the map $(,? not to be C’. Therefore the universality class ‘This (61, %I.
will certainly
be true close to the simple fixed point
B. D. Mestel and A. H. Osbaldestin/ Renormalisation in implicit complex maps
151
Thus for a fixed value of the modulus s the scaling parameters y and 6 are universal. The ratio K,/K, in eq. (4.1) is also universal. If we consider every convergent instead of alternate convergents we find that
If%)
- ql- &,lfF”-‘(z,,)- zdl n odd, (6.5a)
Ifc(zd) - +I - PllfF”-~(~d)- zdl n even, (6.5b) Stable manifolds
Fig. 5. The geometry about the fixed point line L in the function space of almost-d maps with golden mean rotation number. The simple fixed point (6, nt) determines the scaling behaviour of diffeomorphisms. The fixed point (l,, a) determines the scaling behaviour of the almost-c’ maps with modulus equal to s. Each fixed point on L has a codimension-one stable manifold. These stable manifolds (which are the manifolds of constant s) foliate the function space in a neighbourhood of L. A generic one-parameter family +, of almost-c’ maps with golden mean rotation number cuts the stable manifold of (.$,, n,) in a unique point which will have the scaling parameters corresponding to (&, q,).
corresponding to the circle ditkomorphisms contains almost-C1 maps that are not C’. In section 8 we give explicit formulae for the fixed points of S in terms of the modulus. The dynamics of the renormalisation transformation S in the neighbourhood of the line of fixed points is illustrated in fig. 5. The line L of fixed points of S passes through the simple fixed point (.$i, ni). We denote by (t,, 11,) the tixed point with modulus equal to s. Each ([,, 17,) has a scaling parameter y = y(s) and a stable manifold that consists locally of all those almost-C’ maps of golden mean rotation number with the same modulus. Through each point there is a single unstable direction which has eigenvalue 6 = 6(s). These unstable directions are not shown in fig. 5. A typical one-parameter family 9, crosses each of the stable manifolds of the fixed points on L. As S is iterated the family approaches the line L giving the scaling behaviour (4.1) and (4.2) for the corresponding value of y = y(s) and 6 = S(s).
where f has rotation number c’ and &, j$ are the universal scaling numbers given in section 8 and which depend only on the value of the modulus. We have K/K,
= PO> PO& = Y.
(6.6)
7. Calculation of the modulus for the bi-quadratic family
We now calculate the modulus s for the biquadratic family given by eqs. (2.1) and (2.2). We consider the first case 0 I 1~ 1. The calculation is complicated by the fact that we have no analytic formula for the invariant circle C and are therefore unable to calculate the induced map $J directly. However we may use the fact that the invariant circle is perpendicular to the imaginary axis to calculate the modulus. We assume that the invariant circle C is parametrised by 8 and given by z = z(0) as in eq. (3.1). The renormalisation analysis is for circles of length 1 with a discontinuity in the slopes at 0 and +(O), and so we may make a linear change of coordinates x = (8 - Q/T. We denote by J/ the map in x-space. Then we may write JI as a pair of maps (6,~) as in eq. (5.3) with $J replaced by #. The modulus s is given now by eq. (6.1) and in terms of the map r#~by the ratio
+3’(71/2-wu4-) s = $I’(-a/2
+)$I’@, +) ’
(7.1)
158
B. D. Mestel and A. H. Osbaldestin / Renormalisation
where +‘(0 + ) denote the right and left limits respectively of # at 8. Now from the time-reversal symmetry (2.4) in the real axis we have #(IT/~ - ) = l/#( 19,f ) and +‘( - a/2 + ) = l/+‘( 8, - ) and so we obtain
#(ed-)
(
s= +‘(e,+)
2
i
(7.4
.
Now B + +(0) under the map +, and therefore (7.3)
G)(z(+(e)))
= h(z(e)).
Differentiating we get
this expression with respect to 8
(Q’(z(+(e)))
z++(e)>
+‘(e) = hYz(fl))
z’(e). (7.4)
Now consider the right-hand limit at 8,. As B + 0, + we have +( 0) + - ~/2 and hence
fw~d)) Z’Vd)
(7.5)
+'(B"+)=~)'(z(-n/2))z'(-n/2).
Similarly, considering the left-hand limit at /3,, we have +(e) + +n/2 as 8 + 0, - and so h’(z(ed))
G’(ed-)
= (h)‘(z(n/2))
z’(ed>
z’(a/2)
.
(7.6)
We combine (7.5) and (7.6) to obtain
in implicit complex maps
Thus
(h)‘(z_)
s=i (h)‘(z+)
2
1
(7.8)
Finally, an elementary calculation using (2.2) and (2.6) gives
.Y=
l
m(l
-ql’*-
1 2 (7.9)
m(1 - 1)l’* + 1 1 ’
which is an explicit expression for the modulus of the map + in terms of the parameters I and m. The above argument fails when I = 1 since the invariant circle C is no longer perpendicular to the imaginary axis. Parametrising C as z = z(0) with -a/4 I B I 1r/4, the modulus s is given by eq. (7.2)._Note that for 1 = 1 z( IT/~) = z( - a/4) = 0 and (h)‘(O) = 0. However, the parametrisation z(e) is singular at 8 = +~/4 so that ]z’(e)] -, cc as 8+ f~/4 and (h)‘(z(B))z’(@) remains finite as 8 + fir/4. This is seen in fig. 2b, where the induced circle map 4 appears smooth at e,. A direct calculation shows that for small E (i)(Ee’in’4)
= f2Ee+i”‘4+O(~2).
Now from the symmetry (2.5) we know that z’( - 71/4 + E) = - z’( IT/~ - E) and since the angle of the “beak” is 7r/2 [l], we have z’( -1r/4
+ E) = -]~‘(a/4
- E)(ei”/4,
z’( v/4 - E) = ]I’( a/4 - e) ]e-iq’4. s=
G)‘M-4)) z’(-n/2) * i m’(4~/w z’bv2) i .
(7.10)
(7.11a) (7.11b)
(7.7)
Now from the reflection symmetry in the real axis, (2.4), z(-e) =z(e) ) and SO z'(-B)= -m. In particular, z’( -~/2) = - z’( n/2) . However, as we observed in section 2, z’( *n/2) are real so that z’( - 71/2)/z’( ~/2) = - 1.
Since z’ is singular, ]z’(n/4 - E)] + M as E -+ 0. Combining eqs. (7.2), (7.10) and (7.11) and letting 0 go to f ~/4 in eq. (7.4), we see that for I = 1 we have s = 1. Note that the formula (7.9) holds in the case I= 1 although this was not obvious a priori. Therefore we expect to see simple scaling for the case I = 1. This was observed in ref. [l].
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
8.398
’
’
’
’
j
’
’
a
points is straightforward but tedious and we used the Reduce computer algebra program to perform the calculations. Recall that the renormalisation transformation T is given by 7% 11)= (8-‘nP,
8.5
1.0
1.5
s 7.16
’
’
’ b.
7 05 -
159
P-%8).
(8.1)
We choose to use the normalisation t(O) = 1 rather than (5.2) as this makes the algebra simpler. This means that j3 = n(O). A fixed point of S is a period-two orbit of T. It turns out to be easier algebraically to calculate both the period-two points rather than look for a hxed point of S. This is because then the equations are symmetric. We denote the two points of this period-two orbit by (&, no) and (
(51,%)= (Pi’llOPO~ twl0~0P0)~ Gt,?770) = (P3llPl?8hlMl).
(8.3)
We find that it is simpler to work with the functions &,, [i only. Eliminating no, q, in (8.3) we get s Fig. 6. The values of the scaling parameters y and 8 as a function of the modulus s. (a) shows the graph of y and (b) that of 6.
Table 1 lists the computed values of y and 6 for the maps considered earlier based on eq. (7.9) and fig. 6 shows how y and 6 vary as functions of the modulus.
8. Fractional linear fixed points
In this section we give expressions for the fixed points of S in terms of the modulus S. These fixed points are fractional linear transformations and the coefficients are algebraic functions of the modulus. The algebra required to calculate the fixed
with the normalisations &JO) = 1, [r(O) = 1. The functions TJ,,,n1 are given in terms of 5,,, 5i by no = PoS1~l?,
171=
PlEOK’.
(8.5)
We look for solutions of (8.4) which are fractional linear transformations. We therefore write the functions co, ,$i as fractional linear transformations with the normalisation to(O) = 6i(O) = 1.
50(x)=
El(X) = x c;x++b;o, qx + b, .
(8.6)
Here the coefficients b,, co, b,, ci are to be determined.
160
B. D. Mestel and A. H. Osbaldestin / Renormalisation
From (8.5), vo, nl are given by
x + PO&
The essential eigenvalue 6 of section 5 may also be calculated in terms of y. 6 is given by the root of the quadratic equation
x + P,bo
90(x)= cJ?;‘x + b, ’ 171(x) = cop;‘x + b.
in implicit complex maps
.
(8.7) Substituting for to and ,$i from (8.6) into (8.4) and equating coefficients in x we obtain equations which can be solved for the coefficients b,, co, b,, ci in terms of the two scaling parameters PO and pi. The scaling parameter y from (5.9) is given by y = fio&. Writing _Z= Do + pi, the coefficients are given by
ys* - (2y2 + y + 2)6 + y = 0,
(8.12)
which corresponds to 6 = l/a4 when s = 1, i.e., when y = a*. We now give y in terms of the modulus s. Writing
u=&+J-
(8.13)
6’
y satisfies the quartic equation b
=
0
co =
b, =
PO-Y+1 y-&-l
y4-y3-(u+2)y2-y+l=O.
(8.14)
y2+(1-X)y-1 MY-Pi-l)
’
pi-v+1
(8.8)
Y-PO-l'
y = a (4u + 17)l’* (
y2+(1-X)y-1 c1=
This may be solved (using the fact that y = l/e* when s = 1) to give
MY-PO-1)
’
- ( 2 I( 424+ 17)“2+
2u+ 1])1’2+
(8.15)
where y and E satisfy the relation y3-zy*+2y-l=O,
(8.9)
so that E’= y3+2Y-l Y2
(8.10) ’
Since PO& = y and PO+ Pi = 2, we see that PO,PI satisfy the quadratic equation /?’ - Z/3 + y = 0. This gives p = z+ 0
(82-4y)1’2 2
)
p = _z- (8*-4y)1’2 1 2
.
1).
(8.11a) (8.11b)
Using eqs. (8.8), (8.10), (8.11) we may express b,, co, b 1’ Cl, /3, and /3i in terms of the single parameter y.
From (8.13) and (8.15) to, Ei, no, nl, PO, Pi, Y and 6 may all be expressed in terms of the modulus s. Note that u is invariant under the transformation s + l/s. The choice of roots in eq. (8.11) is determined from the conditions s = s(Eo, no). 6 may also be calculated directly from s using the quartic equation 64-4463-(11+4u)6*-46+1=0
(8.16)
and the fact that 6 = l/a4 when s = 1. In table 1 we list the exact values of y and 6 corresponding to values of s in the implicit maps considered earlier. It should be emphasised here that the fixed points found here are “simple” fixed points rather than “critical” ones in the sense of section 5. We have evidence that there are analogous critical fixed points too.
B. D. Mestel and A. H. Osbaldestin/ Renormalisation in implicit complex maps
9. Another family of almost-C’
maps
As a test of the universality of scaling for almost-Cl maps we calculated the scaling of a simple example of an almost-C1 map. The formula (7.9) has the disadvantage that it is dependent on the “tuning” parameter m which must be calculated for a given value of 1. We therefore consider the following map: Let q(x) = m + Ix + (1 - Z)xp
(9.1)
and define t(x)=q(x+l)-1,
m-l
(9.2)
n(x)=&)-1,
O-cx-cm.
(9.3)
(Here p is a positive integer, which may be varied to obtain a quick check on universality.) Now &v(O)) = 5(m - 1) = n(m) = 11(5(O)) and so the pair ($‘, 7) defines a map $, 1)on the circle. It is easy to check that this circle map is almost-C’ and that the modulus s is given by
s=s(Z)=
p(l-I)+1 I
.
(9.4)
Table 1 compares the numerical values of y and 6 for this map (with p = 2) with those calculated earlier. Fig. 6 is indistinguishable from graphs we can draw from the results of this section. The good agreement suggests that the family of maps (9.1)-(9.3) are indeed transverse to the stable manifolds of the fixed points on L.
10. Conclusion
The study of implicit complex maps is at a very early stage of development but it has already generated a growing amount of interest. The complexity of their dynamical behaviour (as shown by the beautiful pictures contained in refs. [l, 21) will make them an important and interesting area of research. Their combination of complex analytic, dissipative and Hamiltonian dynamics may make
161
them an important tool for developing greater understanding of these well-established fields. It seems that one of the most significant aspects of their dynamical behaviour is the discontinuity line and the intricate structures that it generates. This is true of the scaling behaviour where the discontinuity line and the associated discontinuities in the derivative (at the discontinuity point and its images) are the important features. It is somewhat surprising that the complex analytic structure is not relevant to the study of scaling on the invariant circle. It is likely that the theory of renormalisation strange sets [6] for circle maps can be extended to the almost-C’ case and that there exists a oneparameter family of strange invariant sets in the space of pairs (E, 1)) that determines the scaling behaviour for arbitrary rotation numbers. There has recently been a great deal of interest in the range of scaling exponents as measured by the f(a) spectrum [6, 91 and it should be possible to obtain expressions for this spectrum here. The question of proving the existence of the invariant circle C may well be subtle. It appears that the Amol’d tongue structure is somewhat more complicated than that for circle diffeomorphisms [3]. However, it may well be possible to develop a Newton method small divisor proof for these maps. Furthermore, the stable manifolds of the fixed points on L appear to be co-extensive. This indicates that it might be possible to find a global theorem similar to the Herman-Yoccoz theorem for smooth circle maps. Recently, Stark [lo] has obtained a proof of the Herman-Yoccoz theorem using renormalization techniques and it may well be possible to adapt this method to the almost-C’ case. This may well prove to be an opening for the general study of piecewise-C’ circle maps.
Acknowledgements
B.D.M. would like to acknowledge the support of the UK SERC. The computer algebra was
162
B. D. Mestel and A. H. Osbaldestin / Renormalisation
implemented using the Reduce 3.2 program on the Queen Mary College Mathematics computers. We would like to thank Shaun Bullett and Jaroslav Stark for extremely helpful discussions.
References [l] S. Bullett, A.H. Osbaldestin and I.C. Percival, An iterated implicit complex map, Physica D 19 (1986) 290-300. (21 S. Bullett, Dynamics of quadratic correspondences, Nonlinearity 1 (1988) 27-50. [3] S. Bullet, private communication. [4] S. Ostlund, D.A. Rand, J. Sethna and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica D 8 (1983) 303-342.
in implicit complex maps
[51 J. Wilbrink,
Erratic behavior of invariant circles in standard-like mappings, Physica D 26 (1987) 358-368. [61D.A. Rand, Universality and renormalisation in dynamical systems, in: New Directions in Dynamical Systems, T. Bedford and J. Swift, eds., London Math. Sot. Lecture Notes 127 (C.U.P., Cambridge, 1988). L.P. Kadanoff and S.J. Shenker, [71 M.J. Feigenbaum, Quasi-periodicity in dissipative systems: a renormalisation group analysis, Physica D 8 (1982) 370-386. PI L. Jonker and D.A. Rand, Universal properties of maps of the circle with e-singularities, Comm. Math. Phys. 90 (1983) 273-292. I. Procaccia [91 T.C. Halsey, M.H. Jensen, L.P. KadanolI, and B.I. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A 33 (1986) 1141-1151. and renormalisation for WI J. Stark, Smooth conjugation diffeomorphisms of the circle, Nonlinearity 1 (1988) 541-575.
RENORMALISATION
IN IMPLICIT COMPLEX
MAPS
B,.D. MESTELa and A.H. OSBALDESTINb “Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK hDepartment of Mathematical Sciences, Loughborough University of Technology, Loughborough,
LEll
3TU, UK
Received 2 May 1989 Accepted 1 July 1989 Communicated by A.V. Holden
We study the dynamics of a family of implicit complex maps restricted to an invariant circle. The renormalisation analysis for golden mean rotation number is given. We find a one-parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renorrnalisation transformation. These period-2 points are pairs of fractional linear maps.
1. Introduction
The polynomial equation
g(z, w) = 0
(1-l)
in the two complex variables z and w defines a many-to-many map f : z * w if the pair z, w solves eq. (1.1). This map is called an implicit complex map. Such maps were introduced by Bullett et al. [l] as they may possess time-reversal symmetries. They may also be regarded as an extension of the theory of rational maps and Kleinian groups [2]. They may incorporate rational-map behaviour, dissipative and Hamiltonian behaviour and are therefore of considerable interest. The purpose of this paper is to reveal the scaling behaviour of certain quasiperiodic orbits in a particular family of implicit maps discovered by Bullett [2]. We begin by defining the bi-quadratic family in which we shall work and by recalling some basics of implicit maps [2]. In particular, there appears to be an invariant circle and it is
dynamics on this that we study in this paper. We note that the implicit map restricted to this circle induces an “almost-c’” circle map - a term which we shall define in section 3. Some numerical observations for orbits with golden mean rotation number are presented in section 4. In section 5 we review the renormalisation analysis of circle maps, paying special attention to maps with discontinuous derivatives. Next, we note the existence of an invariant “modulus” for almost-C’ maps under the renormalisation transformation and in section 7 deduce an explicit expression for it for the bi-quadratic maps. We show that in fact the fixed points of the square of the renormalisation operator govern our observed behaviour and, moreover, that the fixed points are fractional linear mappings*. The universality of the scaling that we find is investigated in section 9, where we consider a simple almost-C’ map for which we can directly prescribe the modulus. We end with some remarks and open questions.
0167-2789/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
*We thank Ya. G. Sinai for pointing
this out to us
150
B. D. Mestel and A. H. Oshaldestin / Renormalisation in implicit complex maps
2. The bi-quadratic family of implicit complex maps We say that most quadratic family
in each of z and w separately. implicit
of the invariant
bi-quadratic
g was considered will in general
h(w)
complex
the
The case of general in ref. [2]. For such
have two distinct
and two distinct preimages. Bullett [2] introduces a two-parameter implicit
A
maps was
circle that bounds
Siegel disc was investigated. g, points
complex
in ref. [l] and the golden mean scaling
structure
images family of
maps. It is given by
= h(z),
(2.1)
iz* + mz m3z+i miz + 1 -1(1+m2)(miz+l)’
=
Here
I and
z++w*
-z_,
As
a result
m are real parameters,
(2.2)
0 I I I 1 and
h(z) = h(5) . For I= 1 this is the family of maps studied in ref. [l] (with different parameters) while I = 0 gives
(2.5)
iz2 f mz = miz+l ’
of
the
symmetry
quadratic family induces maps f from the right These
maps
For z E H, f(z) continuous
(2.5)
(2.3)
are given
z~w*w*I.
(2.4)
This symmetry was introduced by Bullett et al. in ref. [l] in order to mimic Hamiltonian systems. Secondly, the bi-quadratic family has a timepreserving symmetry with respect to reflection in
bi-
is defined
by f = (h)-‘h.
by taking
the solution
that lies in H. The maps f are
everywhere
in H except
along
an arc
of a semi-circle that extends from the imaginary axis [2]. This arc is called the discontinuity line, D disconti-
An important feature of the dynamics of f is an invariant semi-circle, C, that extends outwards from the imaginary axis into H. If it exists this semi-circle is unique and has end points at z, and z- respectively and goes through D at the discontinuity point zd [2] *. zd maps to two points on the imaginary axis, z+ and zP (which are complex conjugates) and which themselves map to the same point in H. In terms of the parameters m and I, we have
zi=
_ti(l
- 1)l’*,
(24
zd >
=
[l
+
I +
m2 - m4(1 - 1) - m6(1 - 1)2]1’2 2
m2+1 We shall call the family given by (2.2) the biquadratic farnib. The bi-quadratic family has two symmetries. Firstly it has a time-reversal symmetry with respect to reflection in the real axis. This says that
the
a family of single-valued half-plane, H, to itself.
f: H-H
w of h(w) = h(z)
Re( h(z)
-w.
(see fig. 1). The map f has a simple jump nuity as D is crossed.
where
h(z)
axis. This says that
g is bi-quadratic if g(z, w) is at
of bi-quadratic
introduced
the imaginary
Im( zd) =
m[(l
- l)m2+ m2+l
l]
(2.7) ’
If we identify the points z i (writing z+ - z_) then this invariant semi-circle becomes an invariant circle and we may look at the map induced by f on this circle. This map is well defined and continuous because f( z +) = f (z _) since h (z +) = *This fact makes the numerical ant circle particularly easy.
determination
of the invari-
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
(
\:
.
2;. ::/-y-Y :s_,&i ,,..,,.. :.: ,y: :
‘.,
/k
.,,: -*..-... . .. ., ..,,,___, ,,, ,,
j,.
.2.
j
_&yy’
i
:_.,
‘..,,,.,
.i r--\ \ ..___..
._,..:
..
\.
.’
,‘-*,
i.
..
:
..
,i
__,____-----‘.
.....’ ..
:
.
.__
..’
a
b
Fig. 1. Phase portrait of the quadratic family (2.2) for three values of 1. The value of m 1s “tuned” so that the rotation number on the invariant semi-circle is 1 - 0, (I = $(1/5 - 1). The figures contain the invariant circle and the part of the discontinuity line that starts on the imaginary axis and ends on the invariant circle. (a) I = 0, m = 0.289737. (b) I = 1, m = 0.25375. (c) I = 0.5, m = 0.27144.
h (z _). We shall henceforth refer to the invariant semi-circle in H as the invariant circle, with it being tacitly assumed that the points t * are identified. We shall study this map in section 3. For 1= 0 the invariant circle is the unit circle C = {eieI - a/2 I 8 I ~/2} with the points
{ + i, -i} identified. The parameter m determines the rotation number of the induced circle map. Fig. la shows several orbits of the map f for I= 0 and m = 0.289737 when the rotation number on C is u’ = 1 - u, where u = t(fi - l), the golden mean.
152
B. D. Mesrel and A. H. Osbaldestin / Renormalisation
For I = 1 we get the map studied in ref. [l]. For m = 0.25375, there appears to be an invariant circle of rotation number u’. This circle meets the imaginary axis in a right-angled “beak” at the origin. Orbits of the map for I = 1 and m = 0.25375 are shown in fig. lb. The numerical studies in ref. [l] suggest that this is a circle that is C’ (except at the origin) but not C2. For 0 < I < 1, there appears (at least numerically) to be a curve m = m (1) in the (m, 1) parameter space on which the map has an invariant semi-circle in the right half plane with rotation number u’. Fig. lc shows the invariant circle for I = 0.5 and m = 0.27144 on which the rotation number is u’. We stress that it is not proved that such a circle exists for 0 < I _<1 but the numerical evidence is very suggestive. In fact, there appears to be an “Arnol’d tongue” structure in the (m, I) parameter space for 0 I I I 1 [3]. For fixed I there is phase locking on nontrivial intervals of the parameter m as is the case with dissipative maps. One important feature of the invariant semicircles is that they are perpendicular to the imaginary axis for 0 I 1~ 1. This is clearly true for I = 0 when C is the unit semi-circle and appears plausible for 0 < I < 1 from fig. lc. This property is a consequence of the symmetry (2.5) and the uniqueness of the invariant circle in H. The implicit function theorem applied to C guarantees its extension to the left half-plane. The reflection of this extension in the imaginary axis is also an invariant curve in H and therefore, since C is unique, the reflection of the extension of C is C itself. This implies that C is perpendicular to the imaginary axis. Note that when 1= 1 the end points z + of C meet at the origin and the circle crosses over to form a “figure eight”. In this singular case the circle is not perpendicular to the imaginary axis (as is clear from fig. lb). Note also that the symmetry (2.4) implies that C is symmetric with respect to reflection in the real axis. We parametrise C with polar coordinates about the origin, This is clearly convenient for I = 0 in
in implicit complex maps
which C is the unit semi-circle. Moreover, for 0 I I < 1 it means that the circle extends from -IT/~ to 71/2 in e-space: C= {~(B)rr(B)e’~I-a/21B1a/2}.
(2.8)
The points 8 = k IT/~ correspond to the points z i on the imaginary axis. We identify the points zk so that the semi-circle becomes a circle. Note that, since C is perpendicular to the imaginary axis, r( 0) is stationary at 19= & 71/2 and so r’( &n/2) = 0, or, equivalently, z’( 6) is real. This fact will be important in section 7. For I= 1 we parametrise with an angle ranging between -IT/~ and +a/4.
3. The induced circle map The map f restricted to the invariant circle C induces a circle map in &space. We shall denote this map by +: [-IT/~, IT/~] -+ [-a/2, IT/~]. Thus
Recall that circle C goes through the end of the discontinuity line at the point zd = ~(8,) where 0, E [ -~/2, a/2]. Although the map f (and hence +) is continuous on C, the derivative of 9 is discontinuous at 6, and at the point -a/2 - n/2. Elsewhere + appears to be C’. We define an almost-c’ circle map to be a circle map which is continuously differentiable except possibly at two points, one being the image of the other, where the left and right derivatives may differ. In terms of the map f we have
e-+4
- -fw))
e + e, + =q(z(e))
+z+,
+
z_.
(3.2)
Fig. 2 shows the induced circle maps for the three
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
-1.5
-0.5
0.0
1.0
153
00
1.5
THETA
THETA I
I
,
,
I
I’ -1.5
I -I B
7
I’ -0.5
I’ 00
-1.5 I’ 05
I’ 10
I 1.5
THETA
Fig. 2. The maps induced on the invariant circles for the cases (a)-(c)
cases in fig. 1. Note that the curves are symmetric in the line (p = - 0 because of the time-reversal symmetry, (2.4) with respect to the real axis and that they appear to be C’ except at the points 19, and +(e,), where the left and right derivatives are different.
4. Golden mean scaling-numerical
observations
continued fraction expansion of u’ is [2,1,1,1,. . .] and its rational convergents are given by F,_,/F,,. For various values of the parameter I we found the value of m = m(l) for which the rotation number on the circle was u’ and looked at the scaling of the Fibonacci iterates of the discontinuity point rd. We observed that IP+‘(r,) IP”(%)
We shall now investigate the scaling structure around 0, for the case of rotation number u’. We denote by F, the sequence of Fibonacci numbers F,= 1, FI = 1, Fn+l = F, + F,_,. The
of fig. 1
- .Ql - KJ”, - %I - KY”
(4.la) (4.lb)
for constants K,, K, and y depending on 1. y appears to vary smoothly from 1 = 0 to 1= 1. At 1 = 1 we have y = u2, the value corresponding to
154
B. D. Mestel and A. H. Osbaldestin/
Renormalisation
in implicit complex maps
Table 1 /=o s = 0.303274
I=1 s = 1
I= 0.5 s = 0.459603
y implicit map y exact y test map (sect. 9)
0.37006895 0.37006894 0.37008
0.38196 0.38196601 0.38196601
0.3768519 0.37685189 0.37685
6 implicit map 6 exact 6 test map (sect. 9)
7.0017126 7.00171411 7.001714
6.854104 6.85410197 6.85410197
6.9162415 6.91624138 6.916241
the scaling behaviour of diffeomorphisms. This was observed in ref. [l]. Furthermore, for fixed I we calculated the parameter values m, for which the discontinuity point z,, is periodic with period F, with rotation number F,,_,/F,. We find
m2ni1
-mzn-1
m2n+2
-
mzn
-
-
Loa-“,
(4.2a)
Lea-“.
(4.2b)
The values of L,, L, and S also depend on the value of the parameter I. Table 1 lists the values of y and 6 for the three values of 1. At I= 1, 6 is the square of the eigenvalue - l/a2 of the circle-map renormalisation transformation at the simple fixed point [4]. Thus the scaling behaviour, also geometric, is not completely universal for these types of maps as the values of ?3 and y depend on 1. Nevertheless there is a single one-parameter family of universality
classes
for circle maps
of this
typeFig. 3 shows the maps I#+,, I#B~(obtained in the limit n -+ cc by scaling the F2,,+rth and F,,th iterates about 0, by y”) for the case 1= 0, m = 0.289737. Note that while $B& and & are discontinuous at 0,, the left and right limits of $, at 19,, respectively, equal the right and left limits of +L, respectively. The scaling behaviour (4.1) and (4.2) is unusual in that it involves only every alternate rational convergent to u’. This period-two scaling suggests that an explanation may be found in period-two
Fig. 3. The maps %,, $I~ (obtained in the limit n + co by scaling the F2,,+ 1th and F&h iterates of the induced circle map 9 by y” about Bd) for the case I = 0, m = 0.289737.
points of the circle map renormalisation transformation. We consider this in the next section. We note that Wilbrink [5] has observed similar period-2 behaviour in the golden mean circles of piecewise linear standard-like mapping. We believe our analysis here will be of some help in understanding this.
5. Renormalisation analysis for maps with discontinuities in the derivative The renormalisation analysis for circle maps given in ref. [4] involves the renormalisation trunsformation, T, acting on pairs (6,~) of functions that, on appropriate domains, “glue together” to form a map on the circle. This device is somewhat artificial in the case of smooth circle maps (and indeed may be dispensed with [6]) but in the case of our almost-C’ maps it is completely natural. Such a pair (<, 7) is illustrated in fig. 4. Given a pair (&, q), we define a circle map $(, II to be
+&4
= E(x)
if q(O)
= q(x)
ifO
(5.1)
B.D. Mestel and A. H. Osbaldestin/Renormalisation in implicit complex maps
155
critical hxed points. The simple fixed point determines the scaling behaviour of diffeomorphisms and is the rigid rotation through angle u. The critical fixed point determines the scaling behaviour of circle maps with a single cubic critical point at 0 and is observed in the transition from quasiperiodicity to weak turbulence in dissipative dynamical systems [4]. The fixed point equations for T are 4(x) = P-‘n(PxL
(5.5a)
n(x) = P%@@x)).
(5.5b)
The simple fixed point is the solution Fig. 4. Definition of the pair of maps (I, q). The modulus s is determined by the slopes of 6 and 1 at 0 and t(O) - q(O).
& II is a well-defined homeomorphism of the circle [q(O), t(O)] identifying the points t(O) and ~(0) provided <(n(O)) = 17(5(O)) and 0 < 5(O) < 1 [4]. We normalise the pair (5, n) by setting the length of the circle to 1, i.e.,
Any differentiable circle map $J may be written as a pair of differentiable maps (&, n+) in this way by defining
E,(x) = c4
@J(O) -1 IXlO,
q*(x)=f$(x)-1,
OSx0(0).
(5.3)
Moreover any circle map which has discontinuities in the derivative at 0 and +(O) and which is C’ elsewhere may also be written in this way. The golden-mean renormalisation transformation T is given by
&(x)=x+e,
(5.6a)
Vi(X) =x - a*
(5.6b)
of (5.5). The analysis of the spectrum of the derivative of T at (
(5 *4] Here j? is chosen to preserve the normalisation condition (5.2) so that j3 = n(0) - 17(5(O)). The renormalisation analysis [4, 6-81 is given in terms of two fixed points of T: the simple and
((17 - VU(O) = 0,
(5.7)
((9 - VNO)
(5.8)
= 0.
For almost-C’ maps (5.8) is not satisfied and the extra - 1 eigenvalue cannot be suppressed in this
B. D. Mestel and A.H. Osbaldestin/ Renormalisation in implicit complex maps
156
manner. However, (5.7) is satisfied by almost-C’ circle maps and we therefore restrict to the space of pairs that satisfy (5.7). This eliminates the eigenvalue - l/u. The remaining eigenvalue of modulus greater than 1 is the so-called “essential eigenvalue” 6 = l/a2. This is the eigenvalue that governs the scaling of the parameter values as in eq. (4.2). We denote by S the second iterate of the transformation T:
This modulus is the ratio of the slopes of [ and 71 at 0 and E(O) - ~(0) as illustrated in fig. 4. We claim that, provided* $(77(5(O))) f 0, and (5.7) holds,
(6.2) and thus, provided* holds,
(T&J)(v([(~)))
s( S(& 17)) = s&17). (5.9)
Z 0, and (5.7)
(6.3)
Hence, the modulus s is preserved by the renormoperator S. This is what makes the modulus s a natural parameter for the line of hxed points L. Eq. (6.2) follows from a simple calculation: aiisation
where y is given by y =
v(‘m) - m17tw
(5.10)
so that the normalisation (5.2) is enforced. (ti, vi) is, of course, a fixed point of S. dS([,, vi) has two eigenvalues of modulus greater than or equal to 1, viz., the essential eigenvalue l/u4 and the noncommuting eigenvalue + 1 corresponding to violation of (5.8). The eigenvalue + 1 indicates that there is a line, L, of fixed points of S through (Zi, qi). We shall see that L provides a one-parameter family of universality classes that corresponds to the scaling (4.1) and (4.2) observed above. Each of these fixed points has a scaling factor y and essential eigenvalue 6. Each fixed point is a period-two point for T and hence we see the period-two scaling behaviour in (4.1) and (4.2). In section 8 we shall give the exact form for this line of fixed points of s.
6. An invariant modulus We now introduce a modulus, s, by which we parametrise this line of fixed points. Let (E, VI)be a pair of maps. We define s~
s(E,
II)
~
tvi?to> = 17’wN wa tw’to)
elto)) 77’(O) .
(64
(P-‘aw”rlP)‘(o) stT(~~d)
=
(p-l~pp-lq&q'(o)
hMW)(511)'(0) =
v'(MW(175)'(0)
1
=p
s(E, 9) ’
(6.4)
Note that s(E1, ql) = 1 (since there are no discontinuities in the derivatives for the simple fixed point) and that when s # 1 condition (5.8) is violated. Hence the line of fixed points may be parametrised by s and each fixed point of S on the line corresponds to precisely one value of S. The value of s determines the universality class of an almost-C’ map, i.e., the manifolds of constant s and rotation number u in the space of pairs are the stable manifolds of the fixed points on the line L. One interesting point is that s depends on both the left and right hand slopes at 0 and those at t(O) - q(O). It is therefore possible for the ratio (6.1) to equal 1, i.e., s([, q) = 1, and the map $(,? not to be C’. Therefore the universality class ‘This (61, %I.
will certainly
be true close to the simple fixed point
B. D. Mestel and A. H. Osbaldestin/ Renormalisation in implicit complex maps
151
Thus for a fixed value of the modulus s the scaling parameters y and 6 are universal. The ratio K,/K, in eq. (4.1) is also universal. If we consider every convergent instead of alternate convergents we find that
If%)
- ql- &,lfF”-‘(z,,)- zdl n odd, (6.5a)
Ifc(zd) - +I - PllfF”-~(~d)- zdl n even, (6.5b) Stable manifolds
Fig. 5. The geometry about the fixed point line L in the function space of almost-d maps with golden mean rotation number. The simple fixed point (6, nt) determines the scaling behaviour of diffeomorphisms. The fixed point (l,, a) determines the scaling behaviour of the almost-c’ maps with modulus equal to s. Each fixed point on L has a codimension-one stable manifold. These stable manifolds (which are the manifolds of constant s) foliate the function space in a neighbourhood of L. A generic one-parameter family +, of almost-c’ maps with golden mean rotation number cuts the stable manifold of (.$,, n,) in a unique point which will have the scaling parameters corresponding to (&, q,).
corresponding to the circle ditkomorphisms contains almost-C1 maps that are not C’. In section 8 we give explicit formulae for the fixed points of S in terms of the modulus. The dynamics of the renormalisation transformation S in the neighbourhood of the line of fixed points is illustrated in fig. 5. The line L of fixed points of S passes through the simple fixed point (.$i, ni). We denote by (t,, 11,) the tixed point with modulus equal to s. Each ([,, 17,) has a scaling parameter y = y(s) and a stable manifold that consists locally of all those almost-C’ maps of golden mean rotation number with the same modulus. Through each point there is a single unstable direction which has eigenvalue 6 = 6(s). These unstable directions are not shown in fig. 5. A typical one-parameter family 9, crosses each of the stable manifolds of the fixed points on L. As S is iterated the family approaches the line L giving the scaling behaviour (4.1) and (4.2) for the corresponding value of y = y(s) and 6 = S(s).
where f has rotation number c’ and &, j$ are the universal scaling numbers given in section 8 and which depend only on the value of the modulus. We have K/K,
= PO> PO& = Y.
(6.6)
7. Calculation of the modulus for the bi-quadratic family
We now calculate the modulus s for the biquadratic family given by eqs. (2.1) and (2.2). We consider the first case 0 I 1~ 1. The calculation is complicated by the fact that we have no analytic formula for the invariant circle C and are therefore unable to calculate the induced map $J directly. However we may use the fact that the invariant circle is perpendicular to the imaginary axis to calculate the modulus. We assume that the invariant circle C is parametrised by 8 and given by z = z(0) as in eq. (3.1). The renormalisation analysis is for circles of length 1 with a discontinuity in the slopes at 0 and +(O), and so we may make a linear change of coordinates x = (8 - Q/T. We denote by J/ the map in x-space. Then we may write JI as a pair of maps (6,~) as in eq. (5.3) with $J replaced by #. The modulus s is given now by eq. (6.1) and in terms of the map r#~by the ratio
+3’(71/2-wu4-) s = $I’(-a/2
+)$I’@, +) ’
(7.1)
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B. D. Mestel and A. H. Osbaldestin / Renormalisation
where +‘(0 + ) denote the right and left limits respectively of # at 8. Now from the time-reversal symmetry (2.4) in the real axis we have #(IT/~ - ) = l/#( 19,f ) and +‘( - a/2 + ) = l/+‘( 8, - ) and so we obtain
#(ed-)
(
s= +‘(e,+)
2
i
(7.4
.
Now B + +(0) under the map +, and therefore (7.3)
G)(z(+(e)))
= h(z(e)).
Differentiating we get
this expression with respect to 8
(Q’(z(+(e)))
z++(e)>
+‘(e) = hYz(fl))
z’(e). (7.4)
Now consider the right-hand limit at 8,. As B + 0, + we have +( 0) + - ~/2 and hence
fw~d)) Z’Vd)
(7.5)
+'(B"+)=~)'(z(-n/2))z'(-n/2).
Similarly, considering the left-hand limit at /3,, we have +(e) + +n/2 as 8 + 0, - and so h’(z(ed))
G’(ed-)
= (h)‘(z(n/2))
z’(ed>
z’(a/2)
.
(7.6)
We combine (7.5) and (7.6) to obtain
in implicit complex maps
Thus
(h)‘(z_)
s=i (h)‘(z+)
2
1
(7.8)
Finally, an elementary calculation using (2.2) and (2.6) gives
.Y=
l
m(l
-ql’*-
1 2 (7.9)
m(1 - 1)l’* + 1 1 ’
which is an explicit expression for the modulus of the map + in terms of the parameters I and m. The above argument fails when I = 1 since the invariant circle C is no longer perpendicular to the imaginary axis. Parametrising C as z = z(0) with -a/4 I B I 1r/4, the modulus s is given by eq. (7.2)._Note that for 1 = 1 z( IT/~) = z( - a/4) = 0 and (h)‘(O) = 0. However, the parametrisation z(e) is singular at 8 = +~/4 so that ]z’(e)] -, cc as 8+ f~/4 and (h)‘(z(B))z’(@) remains finite as 8 + fir/4. This is seen in fig. 2b, where the induced circle map 4 appears smooth at e,. A direct calculation shows that for small E (i)(Ee’in’4)
= f2Ee+i”‘4+O(~2).
Now from the symmetry (2.5) we know that z’( - 71/4 + E) = - z’( IT/~ - E) and since the angle of the “beak” is 7r/2 [l], we have z’( -1r/4
+ E) = -]~‘(a/4
- E)(ei”/4,
z’( v/4 - E) = ]I’( a/4 - e) ]e-iq’4. s=
G)‘M-4)) z’(-n/2) * i m’(4~/w z’bv2) i .
(7.10)
(7.11a) (7.11b)
(7.7)
Now from the reflection symmetry in the real axis, (2.4), z(-e) =z(e) ) and SO z'(-B)= -m. In particular, z’( -~/2) = - z’( n/2) . However, as we observed in section 2, z’( *n/2) are real so that z’( - 71/2)/z’( ~/2) = - 1.
Since z’ is singular, ]z’(n/4 - E)] + M as E -+ 0. Combining eqs. (7.2), (7.10) and (7.11) and letting 0 go to f ~/4 in eq. (7.4), we see that for I = 1 we have s = 1. Note that the formula (7.9) holds in the case I= 1 although this was not obvious a priori. Therefore we expect to see simple scaling for the case I = 1. This was observed in ref. [l].
B. D. Mestel and A. H. Osbaldestin / Renormalisation in implicit complex maps
8.398
’
’
’
’
j
’
’
a
points is straightforward but tedious and we used the Reduce computer algebra program to perform the calculations. Recall that the renormalisation transformation T is given by 7% 11)= (8-‘nP,
8.5
1.0
1.5
s 7.16
’
’
’ b.
7 05 -
159
P-%8).
(8.1)
We choose to use the normalisation t(O) = 1 rather than (5.2) as this makes the algebra simpler. This means that j3 = n(O). A fixed point of S is a period-two orbit of T. It turns out to be easier algebraically to calculate both the period-two points rather than look for a hxed point of S. This is because then the equations are symmetric. We denote the two points of this period-two orbit by (&, no) and (
(51,%)= (Pi’llOPO~ twl0~0P0)~ Gt,?770) = (P3llPl?8hlMl).
(8.3)
We find that it is simpler to work with the functions &,, [i only. Eliminating no, q, in (8.3) we get s Fig. 6. The values of the scaling parameters y and 8 as a function of the modulus s. (a) shows the graph of y and (b) that of 6.
Table 1 lists the computed values of y and 6 for the maps considered earlier based on eq. (7.9) and fig. 6 shows how y and 6 vary as functions of the modulus.
8. Fractional linear fixed points
In this section we give expressions for the fixed points of S in terms of the modulus S. These fixed points are fractional linear transformations and the coefficients are algebraic functions of the modulus. The algebra required to calculate the fixed
with the normalisations &JO) = 1, [r(O) = 1. The functions TJ,,,n1 are given in terms of 5,,, 5i by no = PoS1~l?,
171=
PlEOK’.
(8.5)
We look for solutions of (8.4) which are fractional linear transformations. We therefore write the functions co, ,$i as fractional linear transformations with the normalisation to(O) = 6i(O) = 1.
50(x)=
El(X) = x c;x++b;o, qx + b, .
(8.6)
Here the coefficients b,, co, b,, ci are to be determined.
160
B. D. Mestel and A. H. Osbaldestin / Renormalisation
From (8.5), vo, nl are given by
x + PO&
The essential eigenvalue 6 of section 5 may also be calculated in terms of y. 6 is given by the root of the quadratic equation
x + P,bo
90(x)= cJ?;‘x + b, ’ 171(x) = cop;‘x + b.
in implicit complex maps
.
(8.7) Substituting for to and ,$i from (8.6) into (8.4) and equating coefficients in x we obtain equations which can be solved for the coefficients b,, co, b,, ci in terms of the two scaling parameters PO and pi. The scaling parameter y from (5.9) is given by y = fio&. Writing _Z= Do + pi, the coefficients are given by
ys* - (2y2 + y + 2)6 + y = 0,
(8.12)
which corresponds to 6 = l/a4 when s = 1, i.e., when y = a*. We now give y in terms of the modulus s. Writing
u=&+J-
(8.13)
6’
y satisfies the quartic equation b
=
0
co =
b, =
PO-Y+1 y-&-l
y4-y3-(u+2)y2-y+l=O.
(8.14)
y2+(1-X)y-1 MY-Pi-l)
’
pi-v+1
(8.8)
Y-PO-l'
y = a (4u + 17)l’* (
y2+(1-X)y-1 c1=
This may be solved (using the fact that y = l/e* when s = 1) to give
MY-PO-1)
’
- ( 2 I( 424+ 17)“2+
2u+ 1])1’2+
(8.15)
where y and E satisfy the relation y3-zy*+2y-l=O,
(8.9)
so that E’= y3+2Y-l Y2
(8.10) ’
Since PO& = y and PO+ Pi = 2, we see that PO,PI satisfy the quadratic equation /?’ - Z/3 + y = 0. This gives p = z+ 0
(82-4y)1’2 2
)
p = _z- (8*-4y)1’2 1 2
.
1).
(8.11a) (8.11b)
Using eqs. (8.8), (8.10), (8.11) we may express b,, co, b 1’ Cl, /3, and /3i in terms of the single parameter y.
From (8.13) and (8.15) to, Ei, no, nl, PO, Pi, Y and 6 may all be expressed in terms of the modulus s. Note that u is invariant under the transformation s + l/s. The choice of roots in eq. (8.11) is determined from the conditions s = s(Eo, no). 6 may also be calculated directly from s using the quartic equation 64-4463-(11+4u)6*-46+1=0
(8.16)
and the fact that 6 = l/a4 when s = 1. In table 1 we list the exact values of y and 6 corresponding to values of s in the implicit maps considered earlier. It should be emphasised here that the fixed points found here are “simple” fixed points rather than “critical” ones in the sense of section 5. We have evidence that there are analogous critical fixed points too.
B. D. Mestel and A. H. Osbaldestin/ Renormalisation in implicit complex maps
9. Another family of almost-C’
maps
As a test of the universality of scaling for almost-Cl maps we calculated the scaling of a simple example of an almost-C1 map. The formula (7.9) has the disadvantage that it is dependent on the “tuning” parameter m which must be calculated for a given value of 1. We therefore consider the following map: Let q(x) = m + Ix + (1 - Z)xp
(9.1)
and define t(x)=q(x+l)-1,
m-l
(9.2)
n(x)=&)-1,
O-cx-cm.
(9.3)
(Here p is a positive integer, which may be varied to obtain a quick check on universality.) Now &v(O)) = 5(m - 1) = n(m) = 11(5(O)) and so the pair ($‘, 7) defines a map $, 1)on the circle. It is easy to check that this circle map is almost-C’ and that the modulus s is given by
s=s(Z)=
p(l-I)+1 I
.
(9.4)
Table 1 compares the numerical values of y and 6 for this map (with p = 2) with those calculated earlier. Fig. 6 is indistinguishable from graphs we can draw from the results of this section. The good agreement suggests that the family of maps (9.1)-(9.3) are indeed transverse to the stable manifolds of the fixed points on L.
10. Conclusion
The study of implicit complex maps is at a very early stage of development but it has already generated a growing amount of interest. The complexity of their dynamical behaviour (as shown by the beautiful pictures contained in refs. [l, 21) will make them an important and interesting area of research. Their combination of complex analytic, dissipative and Hamiltonian dynamics may make
161
them an important tool for developing greater understanding of these well-established fields. It seems that one of the most significant aspects of their dynamical behaviour is the discontinuity line and the intricate structures that it generates. This is true of the scaling behaviour where the discontinuity line and the associated discontinuities in the derivative (at the discontinuity point and its images) are the important features. It is somewhat surprising that the complex analytic structure is not relevant to the study of scaling on the invariant circle. It is likely that the theory of renormalisation strange sets [6] for circle maps can be extended to the almost-C’ case and that there exists a oneparameter family of strange invariant sets in the space of pairs (E, 1)) that determines the scaling behaviour for arbitrary rotation numbers. There has recently been a great deal of interest in the range of scaling exponents as measured by the f(a) spectrum [6, 91 and it should be possible to obtain expressions for this spectrum here. The question of proving the existence of the invariant circle C may well be subtle. It appears that the Amol’d tongue structure is somewhat more complicated than that for circle diffeomorphisms [3]. However, it may well be possible to develop a Newton method small divisor proof for these maps. Furthermore, the stable manifolds of the fixed points on L appear to be co-extensive. This indicates that it might be possible to find a global theorem similar to the Herman-Yoccoz theorem for smooth circle maps. Recently, Stark [lo] has obtained a proof of the Herman-Yoccoz theorem using renormalization techniques and it may well be possible to adapt this method to the almost-C’ case. This may well prove to be an opening for the general study of piecewise-C’ circle maps.
Acknowledgements
B.D.M. would like to acknowledge the support of the UK SERC. The computer algebra was
162
B. D. Mestel and A. H. Osbaldestin / Renormalisation
implemented using the Reduce 3.2 program on the Queen Mary College Mathematics computers. We would like to thank Shaun Bullett and Jaroslav Stark for extremely helpful discussions.
References [l] S. Bullett, A.H. Osbaldestin and I.C. Percival, An iterated implicit complex map, Physica D 19 (1986) 290-300. (21 S. Bullett, Dynamics of quadratic correspondences, Nonlinearity 1 (1988) 27-50. [3] S. Bullet, private communication. [4] S. Ostlund, D.A. Rand, J. Sethna and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica D 8 (1983) 303-342.
in implicit complex maps
[51 J. Wilbrink,
Erratic behavior of invariant circles in standard-like mappings, Physica D 26 (1987) 358-368. [61D.A. Rand, Universality and renormalisation in dynamical systems, in: New Directions in Dynamical Systems, T. Bedford and J. Swift, eds., London Math. Sot. Lecture Notes 127 (C.U.P., Cambridge, 1988). L.P. Kadanoff and S.J. Shenker, [71 M.J. Feigenbaum, Quasi-periodicity in dissipative systems: a renormalisation group analysis, Physica D 8 (1982) 370-386. PI L. Jonker and D.A. Rand, Universal properties of maps of the circle with e-singularities, Comm. Math. Phys. 90 (1983) 273-292. I. Procaccia [91 T.C. Halsey, M.H. Jensen, L.P. KadanolI, and B.I. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A 33 (1986) 1141-1151. and renormalisation for WI J. Stark, Smooth conjugation diffeomorphisms of the circle, Nonlinearity 1 (1988) 541-575.