Chaos synchronization in coupled systems is often characterized by a map φ between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set—by which we mean graph(φ)—can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. In brief, synchronization sets can in general become nondifferentiable, and in the more severe case of noninvertible dynamics, they might even be multivalued. We suggest two different ways to quantify these features, and we discuss possible failures in detecting chaos synchrony using standard continuity-based methods when these features are present.
Topics
Chaos synchronization
REFERENCES
1.
2.
V.
Afraimovich
, N. N.
Verichev
, and M. I.
Rabinovich
, Radiophys. Quantum Electron.
29
, 795
(1986
).3.
M.
Rosenblum
, A.
Pikovsky
, and J.
Kurths
, Phys. Rev. Lett.
76
, 1804
(1996
);4.
F.
Varela
, J. P.
Lachaux
, E.
Rodriguez
, and J.
Martinerie
, Nat. Rev. Neurosci. 2, 229 (2001);E.
Rodriguez
, N.
George
, J-P.
Lachaux
, J.
Martinerie
, B.
Renault
, and F. J.
Varela
, Nature (London)
397
, 430
(1999
);5.
H. D. I. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, New York, 1996).
6.
S. J.
Schiff
, P.
So
, T.
Chang
, R. E.
Burke
, and T.
Sauer
, Phys. Rev. E
54
, 6708
(1996
).7.
N. F.
Rulkov
, M. M.
Sushchik
, L. S.
Tsimring
, and H. D. I.
Abarbanel
, Phys. Rev. E
51
, 980
(1995
).8.
P.
So
, E.
Barreto
, K.
Josić
, E.
Sander
, and S. J.
Schiff
, Phys. Rev. E
65
, 046225
(2002
).9.
The choice of unidirectional coupling is for ease of analysis. With bidirectional coupling, crises may occur and additional dynamical changes might obscure the discussion. However, in certain cases, unidirectionally coupled systems are locally equivalent to bidirectionally coupled systems (Ref. 10). Most importantly, we expect the complications (such as wrinkling, cusps, and fractal structures) in the synchronization set discussed here to persist in the bidirectional case.
10.
11.
The definition of generalized synchrony given in Ref. 7 requires a continuous map between trajectories in the phase spaces of the two component systems. Most authors (e.g., Refs. 5, 16, 18) define generalized synchrony via a stronger but more practical condition requiring the existence of a continuous map between states of the component phase spaces.
12.
The synchronization set is a graph over but attractors for this coupled system may only be subsets of this graph. This is true for both invertible and noninvertible systems.
13.
L. M.
Pecora
, T. L.
Carroll
, G. A.
Johnson
, D. J.
Mar
, and J. F.
Heagy
, Chaos
7
, 520
(1997
);14.
H. D. I.
Abarbanel
, N. F.
Rulkov
, and M. M.
Sushchik
, Phys. Rev. E
53
, 4528
(1996
).15.
16.
17.
E.
Barreto
, P.
So
, B. J.
Gluckman
, and S. J.
Schiff
, Phys. Rev. Lett.
84
, 1689
(2000
).18.
19.
20.
For a typical point (x, y) in the synchronization set, consider an orbit segment consisting of the points for The time- Lyapunov exponents over the orbit segment emanating from can be calculated. If, for this spectrum of Lyapunov exponents remains the same for almost all (x, y) with respect to the natural measure, then we call these the past-history Lyapunov exponents of the system.
21.
The Hölder exponent of φ at is if this quantity is less than 1, and equal to 1 otherwise.
22.
23.
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1977), Vol. 583.
24.
See http://complex.gmu.edu/gallery5.html
25.
C. E.
Frouzakis
, L.
Gardini
, I. G.
Kevrekidis
, G.
Millerioux
, and C.
Mira
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
7
, 1167
(1997
).26.
27.
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993);
W. L.
Ditto
, S. N.
Rauseo
, and M. L.
Spano
, Phys. Rev. Lett.
65
, 3211
(1990
);G.
Matsumoto
, K.
Aihara
, M.
Ichikawa
, and A.
Tasaki
, J. Theor. Neurobiol.
3
, 1
(1984
).28.
H.
Steinlein
and H.-O.
Walther
, J. Dyn. Diff. Eqs. 2, 325 (1990).29.
This is similar to the classification of the invariant set within a dynamical horseshoe (see Ref. 30).
30.
K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York, 1997).
31.
J. C.
Chubb
, E.
Barreto
, P.
So
, and B. J.
Gluckman
, Int. J. Bifurcation Chaos Appl. Sci Eng.
11
, 2705
(2001
).32.
Reference 33 describes a 1:2 multivalued state that results from a bifurcation of the response system as the coupling is decreased. This is fundamentally different from the mechanism described in the present work, which is based on the intrinsic noninvertibility of the drive. One should note that the examples in Ref. 33 are not globally asymptotically stable. This means that for a given history of the drive, two different initial response states do not necessarily have the same asymptotic state on the branches in this case). In contrast, the situation considered here is more coherent in the sense that for a given history of the drive, there is only one unique asymptotic response state despite the Cantor-type structure of the synchronization set.
33.
34.
35.
A. Arneodo, E. Bacry, and J. F. Muzy, Wavelets in Physics (Cambridge University Press, Cambridge, 1999).
36.
S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, New York, 1999).
37.
38.
S.
Jaffard
, SIAM (Soc. Ind. Appl. Math.) J. Math. Anal.
28
, 944
(1997
);S.
Jaffard
, SIAM (Soc. Ind. Appl. Math.) J. Math. Anal.
28
, 971
(1997
).39.
See, for example, C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993).
40.
Since in this paper we are only interested in the Hölder exponent we only require Ψ to be orthogonal to first degree polynomials. If the synchronization manifold is differentiable, the regularity of its higher derivatives can be studied using wavelets orthogonal to higher order polynomials (Ref. 35).
41.
R. de la Llave and N. Petrov, Exp. Math. (in press).
This content is only available via PDF.
© 2003 American Institute of Physics.
2003
American Institute of Physics
You do not currently have access to this content.