Active synchronization in nonhyperbolic hyperchaotic systemsElbert E. N. Macau,1Celso Grebogi,2and Ying-Cheng Lai31Brazilian National Institute for Space Research (INPE), Sa˜o Jose´dos Campos, Sa˜o Paulo, 12227-010, Brazil2Instituto de Fı´sica, Universidade de Sa˜o Paulo, Caixa Postal 66318, 05315-970 Sa˜o Paulo, Sa˜o Paulo, Brazil3Departments of Mathematics, Electrical Engineering, and Physics, Systems Science and Engineering Research Center,Arizona State University, Tempe, Arizona 85287-1804共Received 17 April 2001; published 22 January 2002兲We propose a methodology to address the outstanding problem of synchronization in nonhyperbolic hyper-chaotic physical systems. Our approach makes use of a controlling-chaos strategy that accomplishes the task bytransmitting only one scalar signal even in the presence of noise.DOI: 10.1103/PhysRevE.65.027202 PACS number共s兲: 05.45.Xt, 05.45.Gg, 05.45.VxThe inherent sensitive dependence on initial conditionsimplies that two trajectories starting from slightly differentinitial conditions diverge exponentially in time on the aver-age. Despite that, it has been known that two chaotic systemscan be synchronized 关1兴. Later on, Pecora and Carrol 关2兴gave a condition for the synchronization of two nearly iden-tical chaotic systems: Using appropriately chosen state vari-ables of a chaotic system 共the driver兲 as input to a replica ofthe original system, the replica subsystem 共the slave兲 mightsynchronize with the original system if its Lyapunov expo-nents are all negative. Since that work, synchronization inchaotic systems has become an area of intense activity 关4,3兴.In this report, we address the important problem of syn-chronization of nonhyperbolic hyperchaotic systems 共sys-tems with more than one positive Lyapunov exponent兲. De-spite the success of a previous method 关5兴 for synchronizingcertain hyperchaotic systems, so far, to our knowledge, thephysically relevant issue of nonhyperbolicity, which is typi-cally represented by unstable dimension variability and canbe extremely severe from the standpoint of shadowing 关6,7兴,has not been addressed. We present a general approach,based on the idea of controlling chaos 关8兴, to synchronizenonhyperbolic chaotic systems in high dimensions by utiliz-ing only one scalar transmitted signal. In particular, we ap-ply small perturbations to some parameter of the slave sys-tem to synchronize it with the driver system. In the slavesystem, the current state of the driver system is obtainedfrom the scalar transmitted signal by using the extended Kal-man filter 关9兴. We call this approach the active synchroniza-tion to emphasize its main difference in relation to the moretraditional passive synchronization process in which syn-chronization happens as a consequence of a proper couplingscheme 关1–5兴. To show that our ideas make sense, we usethe pole-placement method 关10兴 conveniently adapted toachieve robust active synchronization for the situations ofstrongly nonhyperbolic chaotic systems with more than onepositive Lyapunov exponents and in the presence of noise.We mention that the concept of utilizing the principle ofcontrolling chaos to achieve synchronization is, in fact, notnew. It was suggested in Ref. 关11兴. However, the strategyproposed there is applicable to low-dimensional systemsonly and requires the transmission of all the state variables ofthe system from the driver to the slave. In contrast, the strat-egy proposed in this report allows for the synchronization ofhigh-dimensional, nonhyperbolic chaotic systems with thetransmission of a single scalar signal.Intuitively, to achieve synchronization on hyperchaoticsystems, the number of variables to be transmitted should beequal to that of positive Lyapunov exponents in order toaccount for the same number of unstable directions along thechaotic trajectory 关12兴. It was shown in Ref. 关5兴 and in sub-sequent works introducing various improvements 关3,13,14兴that, this belief is incorrect. In general, all those proposedapproaches use feedback strategies whose parameters arefixed and are calculated using empirical strategies or optimi-zation algorithms. As a consequence, none of those strategiescan be considered to work for sure with any hyperchaoticsystem.In all the ideas previously discussed, the requirement of ahyperbolic structure for the systems to be synchronized isimplicit 关15兴. By its turn, nonhyperbolic systems can be clas-sified into two types. For the first type, the splitting of thephase space into expanding and contracting subspaces is in-variant along a trajectory except at the tangencies of thestable and unstable manifolds, where the angles betweensubspaces are zero 关16兴. The second type of nonhyperbolicityin hyperchaotic systems is due to unstable dimension vari-ability 关7兴. It is related to the presence of unstable periodicorbits with different numbers of unstable directions embed-ded within the chaotic attractor. As a consequence, a typicaltrajectory experiences different numbers of unstable andstable directions as it evolves. Thus, the continuous splittingof the phase space into expanding and contracting subspacesis no longer valid.Because of the global sensitivity, synchronization of cha-otic systems having unstable dimension variability in thepresence of noise or even small parameter mismatches isextremely difficult, if not impossible, to achieve. As synchro-nized trajectories move from one neighborhood to anotherhaving unstable periodic orbits with different number of ex-panding directions, they tend to separate exponentially fromeach other. What makes the situation hard for this type ofnonhyperbolicity is the fact that the sets of periodic orbitswith a different number of expanding directions are denselymixed 关7兴. Thus, regions where synchronization is highlysusceptible to being destroyed due to the presence of noiseextend over most of the attractor. As a result, previous meth-ods 关5,14兴 cannot be expected to work for hyperchaotic sys-PHYSICAL REVIEW E, VOLUME 65, 0272021063-651X/2002/65共2兲/027202共4兲/$20.00 ©2002 The American Physical Society65 027202-1tems with unstable dimension variability, especially for ex-perimental implementation where noise is always present. Ingeneral, the densely mixed sets of periodic orbits with dif-ferent numbers of expanding directions prevent the successof any passive feedback strategies of synchronization