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Synchronization in Coupled Chaotic Systems [PPT]

 Because of its potential practical applications such as secure communication, the phenomenon of synchronization in coupled chaotic systems has become a field of intensive research. Our main concerns on the chaos synchronization are as follows.

(1) Bifurcation Mechanism for the Loss of Chaos Synchronization [PPT]
  When identical chaotic systems synchronize, a synchronous chaotic attractor (SCA) exists on an invariant subspace. If the SCA is stable against a perturbation transverse to the invariant subspace, it may become an attractor in the whole phase space. Such transverse stability of the SCA is intimately associated with transverse bifurcations of periodic saddles embedded in the SCA. If all periodic saddles are transversely stable, the SCA is strongly stable. However, as the coupling parameter passes a threshold value, a periodic saddle first becomes transversely unstable through a local bifurcation. Then, the SCA becomes weakly stable. We investigate the bifurcation mechanism for the transition from strong to weak synchronization in unidirectionally coupled systems. It is thus found that a direct transition to global riddling occurs via a transcritical contact bifurcation between a periodic saddle embedded in the SCA and a repeller on the basin boundary.  After such a riddling transition, the basin becomes globally riddled with a dense set of ``holes,'' leading to divergent trajectories. Note that this bifurcation mechanism is different from that in symmetrically coupled systems. The effect of asymmetry on the bifurcation mechanism for the loss of chaos synchronization is also studied in a system of asymmetrically coupled 1D maps that contains a parameter $\alpha$ tuning the ``degree'' of the asymmetry from the symmetrically-coupled case ($\alpha=0$) to the unidirectionally-coupled case ($\alpha=1$). Furthermore, mechanisms for the hard bubbling transition, giving rise to abrupt appearance of  intermittent bursts with large amplitude, are elucidated in symmetrically coupled systems. For more details, see the following publications:

[1] S.-Y. Kim and W. Lim, "Mechanism for the riddling transition in coupled chaotic systems," Phys. Rev. E 63, 026217 (2001).
[2] S.-Y. Kim, W. Lim, and Y. Kim, "New riddling bifurcation in asymmetric dynamical systems," Prog. Theor. Phys. 105, 187-196 (2001).
[3] W. Lim, S.-Y. Kim, and Y. Kim, "Riddling transition in asymmetric dynamical systems," J. Korean Phys. Soc. 38, 532-535 (2001).
[4] S.-Y. Kim and W. Lim, "Effect of asymmetry on the loss of chaos synchronization," Phys. Rev. E 64, 016211 (2001).
[5] W.-K. Lee, Y. Kim, and S. -Y. Kim, "Loss of periodic synchronization in unidirectionally coupled nonlinear systems," J. Korean Phys. Soc. 40, 788-792 (2002).
[6] W. Lim and S.-Y. Kim, "Global effect of transverse bifurcations in coupled chaotic systems," J. Korean Phys. Soc. 43, 193-201 (2003).
[7] S.-Y. Kim and W. Lim, "Mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems," J. Phys. A 36, 6951-6961 (2003).

(2) Effect of the parameter mismatch and noise on weak synchronization [PPT]
 
A transition from strong to weak synchronization occurs through a first transverse bifurcation of a periodic saddle embedded in the SCA. For this case, intermittent bursting or basin riddling takes place, depending on the global dynamics. If the global dynamics of the system is bounded and there are no attractors off the invariant subspace, locally repelled trajectories exhibit transient intermittent burstings. On the other hand, if the global dynamics is unbounded or there exists an attractor off the invariant subspace, repelled trajectories may go to another attractor (or infinity), and hence the basin becomes riddled with a dense set of ``holes,'' belonging to the basin of another attractor (or infinity). In a real situation, a small mismatch between the subsystems and a small noise are unavoidable. Hence, the effect of the parameter mismatch and noise must be taken into consideration for the study on the loss of chaos synchronization. In the regime of weak synchronization, a typical trajectory may have segments exhibiting positive local transverse Lyapunov exponents because of local repulsion of periodic repellers embedded in the SCA. For this case, any small mismatch or noise results in a permanent intermittent bursting and a chaotic transient with a finite lifetime for the bursting and riddling cases, respectively. These attractor bubbling and chaotic transient demonstrates the sensitivity of the weakly stable SCA with respect to the variation of the mismatching parameter and the noise intensity. To measure the ``degree'' of the parameter (noise) sensitivity, we introduce a quantifier, called the parameter (noise) sensitivity exponent [PSE (NSE)] in two coupled 1D maps. As the coupling parameter is changed away from the first transverse bifurcation point, successive transverse bifurcations of periodic saddles occur. Hence, the value of PSE (NSE) increases, because local transverse repulsion of periodic repellers becomes more and more strong. Thus, the parameter (noise) sensitivity of the weakly stable SCA is quantitatively characterized in terms of the PSEs (NSEs). Furthermore, the values of the PSE and NSE become the same, independently of the detailed properties of the paramter mismatch and noise, because the degree of the sensitivity is determined only by the same local transverse Lyapunov exponents of the ``unperturbed'' SCA in the absence of the parameter mismatch and noise. In terms of such PSE (NSE), we characterize the parameter-mismatching and noise effect on the bubbling and riddling. Thus, it is found that the scaling exponent $\mu$ for the average characteristic time (i.e., the average interburst interval and the average chaotic transient lifetime) for the bubbling and riddling cases is given by the reciprocal of the PSE (NSE). We also extend the method of quantitatively characterizing the parameter (noise) sensitivity in terms of the PSE (NSE) to the high-dimensional coupled invertible systems such as the coupled Henon maps and coupled pendula. It is thus found that the reciprocal relation between the scaling exponent $\mu$ and the PSE (NSE) seems to be ``universal,'' because it holds in typical coupled chaotic systems. For more details, see the following publications:

[1] S.-Y. Kim, W.Lim, and Y.Kim, "Effect of parameter mismatch and noise on weak synchronization," Prog. Theor. Phys. 107, 239-252 (2002).
[2] A. Jalnine and S.-Y. Kim, "Characterization of the parameter-mismatching effect on the loss of chaos synchronization," Phys. Rev. E 65, 026210 (2002).
[3] W. Lim and S.-Y. Kim, "Effect of parameter mismatch and noise on the loss of chaos synchronization," Sae Mulli (New Physics) 45, 1-6 (2002) (in Korean).
[4] S.-Y. Kim, W. Lim, A. Jalnine, and S.P. Kuznetsov, "Characterization of the noise effect on weak synchronization," Phys. Rev. E 67, 016217 (2003).
[5] S.-Y. Kim, A. Jalnine, W. Lim, and S. P. Kuznetsov, "Characterization of the parameter-mismatching and noise effect on weak synchronization," Applied Nonlinear Dynamics. 11, 81-86 (2003).
[6] W. Lim and S.-Y. Kim, "Parameter-mismatching effect on the attractor bubbling in coupled chaotic systems," J. Korean Phys. Soc. 44, 510-513 (2004).
[7] W. Lim and S.-Y. Kim, "Universality for the parameter-mismatching effect on weak synchronization in coupled chaotic systems," J. Phys. A 37, 8233-8244 (2004).
[8] W. Lim and S.-Y. Kim, "Noise effect on weak chaotic synchronization in coupled invertible systems," J. Korean Phys. Soc. 45, 287-294 (2004).

(3) Dynamical Consequence of Blowout Bifurcations [PPT]
 After the first transverse bifurcation of an unstable periodic orbit (UPO) embedded in the SCA, the UPOs are decomposed into two groups. The first group consists of transversely stable periodic saddles, while the second group is composed of transversely unstable periodic repellers. As the coupling parameter is changed from the first transverse bifurcation point, the ``weight'' of periodic repellers increases, because periodic saddles are transformed into periodic repellers via successive transverse bifurcations. Thus, bubbling and riddling effects becomes intensified. Eventually their weights of periodic saddles and repellers become balanced at a Blowout Bifurcation point, at which the transverse Lyapunov exponent of the SCA becomes zero. Consequently, the SCA becomes transversely unstable, and a complete desynchronization occurs. The global effect of the blowout bifurcation depends on the global dynamics. If the global dynamics is bounded and there is no attractor off the invariant synchronization subspace, then a new asynchronous attractor  appears via a supercritical blow-out bifurcation, and then it exhibits an intermittent bursting, called the on-off Intermittency. However, if the global dynamics is unbounded or there exists another attractor off the invariant synchronization subspace, then an abrupt collapse of the synchronized chaotic state occurs through a subcritical blow-out bifurcation, and then typical trajectories near the synchronization subspace are attracted to another attractor (or infinity). The dynamical consequence of the supercritical blowout bifurcation is particularly studied. First, we investigate the mechanism for the symmetry-preserving and -breaking blowout bifurcations, giving rise to the birth of symmetric and asymmetric asynchronous chaotic attractors, in symmetrically coupled 1D maps. Note that asynchronous attractors, born via blow-out bifurcations, may become chaotic or hyperchaotic. Hence, the dynamical origin for the occurrence of asynchronous hyperchaos and chaos is also investigated through competition between the laminar and bursting components of the newly-born asynchronous attractor exhibiting the on-off intermittency. When the ``strength'' of the bursting component is larger (smaller) than that of the laminar component, an asynchronous hyperchaotic (chaotic) attractor appears. For more details, see the following publications:

[1] W. Lim and S.-Y. Kim, "Symmetry-conserving and -breaking blow-out bifurcations in coupled chaotic systems," J. Korean Phys. Soc. 38, 528-531 (2001).
[2] S.-Y. Kim, W. Lim, E. Ott, and B. Hunt, "Dynamical origin for the occurrence of asynchronous hyperchaos and chaos via blowout bifurcations," Phys. Rev. E 68, 066203 (2003).
[3] W. Lim and S.-Y. Kim, "On the consequence of blow-out bifurcations," J. Korean Phys. Soc. 43, 202-206 (2003).

(4) Partial Synchronization in Coupled Chaotic Systems [PPT]
 
We investigate the dynamical mechanism for the partial synchronization in three coupled 1D maps. A completely synchronized attractor on the main diagonal becomes transversely unstable via a blowout bifurcation, and then a two-cluster state appears on an invariant subspace. If the two-cluster state becomes transversely stable, then a partial synchronization may occur on the invariant subspace. Otherwise, total desynchronization takes place. It is found that the transverse stability of the two-cluster state may be determined through the competition between its laminar and bursting components. When the ``strength'' of the laminar component is larger (smaller) than that of the bursting component, a partially synchronized (totally desynchronized) attractor appears through the blowout bifurcation. Particularly, the effect of the asymmetry, range, and time-delay in the coupling on the occurrence of the partial synchronization is investigated. In real situation, a small mismatch between the subsystems and a small noise are unavoidable. As in the case of the fully synchronized chaotic attractor, we introduce the parameter and noise sensitivity exponents to quantitatively measure the degree of the sensitivity of the partially synchronized chaotic attractor. In terms of such sensitivity exponents, we also characterize the parameter-mismatching and noise effect on the scaling behaviors associated with  the attractor bubbling. To examine the universality for the results obtained in the coupled 1D maps, we also investigate the coupled Henon maps and coupled pendula. For more details, see the following publication:

[1] W. Lim and S.-Y. Kim, "Mechanism for the partial synchronization in three coupled chaotic systems," Phys. Rev. E 71, 035221 (2005).
[2] W. Lim and S.-Y. Kim, "On the occurrence of partial synchronization in unidirectionally coupled maps" J. Korean Phys. Soc. 46, 638-641 (2005).
[3] W. Lim and S.-Y. Kim, "Coupling effect on the occurrence of partial synchronigation in four coupled chaotic systems," Phys. Lett. A 353, 398 (2006).

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