Computer Physics Communications 121–122 (1999) 429–431www.elsevier.nl/locate/cpc

Bistable generalized synchronization of chaotic systems

J.M. González-Miranda1

Departamento de Física Fundamental, Universidad de Barcelona, Avenida Diagonal 647, 08028 Barcelona, Spain

Abstract

Chaotic driving of systems that exhibit phase-space symmetries may result in multistable generalized chaotic synchronization.This multistabily is studied numerically, and used in a computer simulation study of communications through chaos. 1999Elsevier Science B.V. All rights reserved.

The behavior under chaotic driving of dissipativechaotic systems that are symmetric in phase spaceis studied. The case when a chaotic system, calledthe drive, provides a signal which acts on anotherchaotic system, called the response, is considered. Thedriving scheme proposed by Pecora and Carroll [1],which allows two identical chaotic systems to besynchronized, is used. Here, synchronization meansthat the distance between the two systems in phasespace converges to zero when one of them acts as thedrive and the other as the response.

In the case of spatially symmetric chaotic sys-tems, multistable chaotic synchronization can be ob-served [2,3]. This means that, depending on the ini-tial conditions, the response can synchronize in a va-riety of forms to the drive, being each of these dif-ferent forms of synchronization a case of generalizedsynchronization [4], characterized by a functional thatdetermines the time evolution of the response from thetime evolution of the drive. The number of generalizedsynchronization states available is equal to the num-ber of different and independent symmetry transfor-mations that leave the system invariant [2].

1 E-mail: jgm@hermes.ffn.ub.es.

Computer simulations have been performed withseveral mathematical models, given by chaotic flowsin three dimensions, which show invariance underrotations ofπ/2 radians around thez-axis. Projectionsof the chaotic attractors onto thex–y plane for twoexample attractors [2,5] are displayed in Fig. 1. It wasobtained that they had two different synchronizationstates available: the regular one in which the responsefairly reproduced the drive, and a generalized one inwhich the response reproduces a copy of the driverotated byπ/2 radians. Calculations of the Lyapunovexponents of the response showed that it is possibleto find systems in which these synchronization statesare asymptotically stable. Systematic calculations ofthe basins of attraction to these synchronization states,examples of which are shown in Fig. 2, provedthat such basins change smoothly with the initialconditions, and are robust under small changes of theparameters of the system.

Computer simulations of the synchronization phe-nomena under external noise showed two relevant ef-fects:(i) weakening of the synchronization in which the re-

sponse fluctuates close to the synchronized state,and

0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S0010-4655(99)00375-6

430 J.M. González-Miranda / Computer Physics Communications 121–122 (1999) 429–431

Fig. 1. Examples of chaotic flows showing invariance under a rotation ofπ/2 radians. Systems given by (a) are from Ref. [2], and (b) Ref. [5].

Fig. 2. Examples of basins of attraction to the regular (black) and generalized (white) synchronization states for the systems in Fig. 1 (a) and (b),respectively.

(ii) jumps between the two different synchronizationstates.

This second effect was found to be strongly dependenton the particular system studied, being practicallyirrelevant for a topology of the attractor with the shapeof a ring, like the one in Fig. 1(a).

The potential applications of that kind of systemshave been proven by the design and analysis, by com-puter simulation, of a method for secure transmissionof information [6]. The basic ideas are to use:(i) the information contents of the basins of attraction

of the symmetric chaotic systems as a tool formessage encoding, and

(ii) the dependence on the initial conditions as a toolfor encryption.

Computer experiments made with the system inFig. 1(a) have proven the feasibility of these ideas. Inparticular, systematic calculations simulating imper-fect communicating devices (because of the presenceof noise or mismatch between drive and response,)have proven that the communication scheme is robustenough to be feasible in the laboratory.

Acknowledgements

This research was supported by DGICYT, throughproject PB96-0392.

J.M. González-Miranda / Computer Physics Communications 121–122 (1999) 429–431 431

References

[1] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821–824.[2] J.M. González-Miranda, Phys. Rev. E 53 (1996) 5656–5669.[3] J.M. González-Miranda, in: Proceedings of the VIII Spanish

Meeting on Statistical Physics, Anales de Física, MonografíasRSEF 4 (1998) pp. 125–128.

[4] N.F. Rulkov, M.M. Sushchick, L.Sh. Tsimring, H.D.I. Abar-banel, Phys. Rev. E 51 (1995) 980–994.

[5] B. Deng, Int. J. Bifurc. Chaos 4 (1994) 823–841.[6] J.M. González-Miranda, Phys. Lett. A 251 (1999) 115–120.