No abstract

We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N -dimensional tori. More specifically we introduce the Generalized Alignment Index of order k (GALI k ) as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N degree of freedom Hamiltonian systems that, for chaotic orbits, GALI k tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALI k fluctuates around non-zero values for 2 ≤ k ≤ N and goes to zero for N < k ≤ 2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus:The GALI k is a generalization of the Smaller Alignment Index (SALI) (GALI 2 ∝ SALI). However, GALI k provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.

Chaos synchronization is a very important nonlinear phenomenon, which has been studied to date extensively on dynamical systems described by real variables. There also exist, however, interesting cases of dynamical systems, where the main variables participating in the dynamics are complex, for example, when amplitudes of electromagnetic fields are involved. Another example is when chaos synchronization is used for communications, where doubling the number of variables may be used to increase the content and security of the transmitted information. It is also well-known that similar generalization of the Lorenz system to one with complex ODEs has been introduced to describe and simulate the physics of a detuned laser and thermal convection of liquid flows. In this paper, we study chaos synchronization by applying active control and Lyapunov function analysis to two such systems introduced by Chen and Lü. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and exhibit strange attractors. We calculate numerically the values of the parameters at which these attractors exist. Active control and global synchronization techniques are then applied to study the phenomenon of chaos synchronization. Analytical criteria concerning the stability of these techniques are implemented and excellent agreement is found upon comparison with numerical experiments. In particular, studying the time evolution of "errors" (or differences between drive and control dynamics), we show that both techniques are very effective for controlling the behavior of these systems, even in regimes of very strong chaos.

We have identified the occurrence of chimera states for various coupling schemes in networks of two-dimensional and three-dimensional Hindmarsh-Rose oscillators, which represent realistic models of neuronal ensembles. This result, together with recent studies on multiple chimera states in nonlocally coupled FitzHugh-Nagumo oscillators, provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications. Moreover, our work verifies the existence of chimera states in coupled bistable elements, whereas to date chimeras were known to arise in models possessing a single stable limit cycle. Finally, we have identified an interesting class of mixed oscillatory states, in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion.

Abstract. We use the Smaller Alignment Index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behavior of the SALI for the two cases: the index fluctuates around non-zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behavior for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents σ 1 , σ 2 i.e. SALI ∝ e −(σ1−σ2)t . Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of 2 and 3 degrees of freedom.