Tassos Bountis

The Painlevé property and singularity analysis of integrable and non-integrable systems

Abstract We present a review of results of the so-called Painleve singularity approach to the investigation of the integrability of dynamical systems with finite and infinite number of degrees of freedom. Rigorous results based on the theorems of Yoshida and Ziglin concerning proofs of non-integrability are also presented, as well as an application of the new “poly-Painleve” method due to Kruskal. Finally a section is devoted to the singularity analysis of the solutions of non-integrable dynamical systems.

Proton transfer in hydrogen-bonded systems

Proceedings of the May 1991 workshop, held in Crete, Greece, within the program of activities of the NATO Special Program on Chaos, Order, and Patterns. Participants were physicists, biologists, and chemists, theoreticians and experimentalists, exchanging perspectives and data on proton transfer, th

ACTIVE CONTROL AND GLOBAL SYNCHRONIZATION OF THE COMPLEX CHEN AND LÜ SYSTEMS

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 Lu. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and...

Remerging Feigenbaum trees in dynamical systems

Abstract Finite sequences of remerging period-doubling bifurcations have been recently observed in a variety of physically interesting dynamical systems. We show here that such remerging Feigenbaum trees are quite common in models with more than one parameter and discuss a number of criteria under which they are generally observed. These criteria are applied to simple mappings as well as the conservative Duffings equation where the formation of a primary “bubble” is seen to lead to higher-order bubbles and hence to remerging Feigenbaum sequences. In the case of Duffings equation, we follow the development of one such sequence, with the aid of the variation of the winding number along a symmetry axis of the problem.

Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

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 (GALIk) 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, GALIk tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALIk 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: ∝ t −2(k−N)+m if 0 ≤ m < k − N, and ∝ t −(k−N) if m ≥ k − N. The GALIk is a generalization of the Smaller Alignment Index (SALI) (GALI2 ∝ SALI). However, GALIk 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.

Detecting order and chaos in Hamiltonian systems by the SALI method

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 behaviour 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 behaviour 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 σ1 ,σ 2 i.e. SALI ∝ e −(σ

On the complete and partial integrability of non-Hamiltonian systems

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painleve property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painleve transcendents. Relaxing the Painleve property we find many partially integrable cases whose movable singularities are poles at leading order, with In(t-t0) terms entering at higher orders. In an Nth order, generalized Rossler model a precise relation is established between the partial fulfillment of the Painleve conditions and the existence of N - 2 integrals of the motion.

Chimera States in Networks of Nonlocally Coupled Hindmarsh–Rose Neuron Models

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.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

THE DYNAMICS OF SYSTEMS OF COMPLEX NONLINEAR OSCILLATORS: A REVIEW

Dynamical systems in the real domain are currently one of the most popular areas of scientific study. A wealth of new phenomena of bifurcations and chaos has been discovered concerning the dynamics of nonlinear systems in real phase space. There is, however, a wide variety of physical problems, which, from a mathematical point of view, can be more conveniently studied using complex variables. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. In this survey, we shall focus on such classes of autonomous, parametrically excited and modulated systems of complex nonlinear oscillators. We first describe appropriate perturbation approaches, which have been specially adapted to study periodic solutions, their stability and control. The stability analysis of these fundamental periodic solutions, though local by itself, can yield considerable information about more global properties of the dynamics, since it is in the vicinity of such solutions that the largest regions of regular or chaotic motion are observed, depending on whether the periodic solution is, respectively, stable or unstable. We then summarize some recent studies on fixed points, periodic solutions, strange attractors, chaotic behavior and the problem of chaos control in systems of complex oscillators. Some important applications in physics, mechanics and engineering are mentioned. The connection with a class of complex partial differential equations, which contains such famous examples, as the nonlinear Schrodinger and Ginzburg–Landau equations is also discussed. These complex equations play an important role in many branches of physics, e.g. fluids, superconductors, plasma physics, geophysical fluids, modulated optical waves and electromagnetic fields.