Coupling induced topological transition in a ring of chaotic oscillators

A ring of diffusively coupled R?ssler oscillators, which can develop the conventional rotating wave from high-dimensional chaos by increasing the coupling ɛ continuously is studied. The chaotic generator for the rotating wave emerges around ɛ = ɛ, where the topological transition induced by the coupling not only changes the topological structure of all the oscillators, which share a common strange attractor, but also changes them into being different from each other. Starting from this transition, infinitely long range temporal correlation and spatial order in the style of antiphase state are established gradually, which gives rise to the chaotic generator of the rotating wave. Received 15 March 2001     

Globally coupled systems with prescribed synchronized dynamics

A system of globally coupled maps whose synchronized dynamics differs from the individual (chaotic) evolution is considered. For nonchaotic synchronized dynamics, the synchronized state becomes stable at a critical coupling intensity lower than that of the fully chaotic case. Below such critical point, synchronization is also stable in a set of finite intervals. Moreover, the system is shown to exhibit multistability, so that even when the synchronized state is stable not all the initial conditions lead to synchronization of the ensemble. Received 22 October 1999     

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes N are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions all attractors of the system become identical in the thermodynamical limit up to variations of order 1/, and thus replica symmetry is recovered for N→∞. In contrast to this, when fluctuating-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit. Received 9 July 2001     

The effect of the asymptotic response dynamics on the generalized synchronization

Generalized synchronization in a drive-response Chua circuits is investigated. A cascade of transitions to GS is observed with increasing the interaction strength. The mechanism on the transitions to GS is given based on the asymptotic behaviors of response dynamics.     

Adaptive synchronization of weighted complex dynamical networks through pinning

This paper considers the problem of controlling weighted complex dynamical networks by applying adaptive control to a fraction of network nodes. We investigate the local and global synchronization of the controlled dynamical network through the construction of a master stability function and a Lyapunov function. Analytical results show that a certain number of nodes can be controlled by using adaptive pinning to ensure the synchronization of the entire network. We present numerical simulations to verify the effectiveness of the proposed scheme. In comparison with feedback pinning, the proposed pinning control scheme is robust when tested by noise, different weighting and coupling structures, and time delays.     

Optimal synchronizability of networks

We numerically investigate how to enhance synchronizability of coupled identical oscillators in complex networks with research focus on the roles of the high level of clustering for a given heterogeneity in the degree distribution. By using the edge-exchange method with the fixed degree sequence, we first directly maximize synchronizability measured by the eigenratio of the coupling matrix, through the use of the so-called memory tabu search algorithm developed in applied mathematics. The resulting optimal network, which turns out to be weakly disassortative, is observed to exhibit a small modularity. More importantly, it is clearly revealed that the optimally synchronizable network for a given degree sequence shows a very low level of clustering, containing much fewer small-size loops than the original network. We then use the clustering coefficient as an object function to be reduced during the edge exchanges, and find it a very efficient way to enhance synchronizability. We thus conclude that under the condition of a given degree heterogeneity, the clustering plays a very important role in the network synchronization.     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

Multiphase patterns in self-phase-locked optical parametric oscillators

Multiphase patterns are found in a mean-field model of a singly-resonant optical parametric oscillator that converts a pump field at frequency 3ω into signal and idler fields at frequencies 2ω and ω. A complex Ginzburg-Landau equation without diffusion and with a quadratic phase-sensitive nonlinear term is derived under single-longitudinal and paraxial propagation approximations. Owing to the phase-matched multistep parametric process ω + ω = 2ω, phase locking of the resonated signal field is possible with three distinct phase states. Three-armed rotating spirals, target patterns and light filamentation are found by a numerical analysis of the mean-field equation. Received 19 April 2001 and Received in final form 21 June 2001     

Clusters in an assembly of globally coupled bistable oscillators

We study the dynamics of an assembly of globally coupled bistable elements. We show that bistability of elements results in some new features of clustering in the assembly when there is global coupling. We provide conditions for the existence of stable amplitude-phase clusters and splay-phase states. Received 12 June 1998 and Received in final form 30 November 1998     

Controlling chaos by a modified straight-line stabilization method

By adjusting external control signal, rather than some available parameters of the system, we modify the straight-line stabilization method for stabilizing an unstable periodic orbit in a neighborhood of an unstable fixed point formulated by Ling Yang et al., and derive a more simple analytical expression of the external control signal adjustment. Our technique solves the problem that the unstable fixed point is independent of the system parameters, for which the original straight-line stabilization method is not suitable. The method is valid for controlling dissipative chaos, Hamiltonian chaos and hyperchaos, and may be most useful for the systems in which it may be difficult to find an accessible system parameter in some cases. The method is robust under the presence of weak external noise. Received 10 January 2001     

Spike-burst and other oscillations in a system composed of two coupled, drastically different elements

The dynamics of a system composed of two nonlinearly coupled, drastically different nonlinear and eventually oscillatory elements is studied. The rich variety of attractors of the system is studied with the help of phase space analysis and Poincare maps. Received 19 March 1999 and Received in final form 1 November 1999     

Enhancing synchronizability by weight randomization on regular networks

In weighted networks, redistribution of link weights can effectively change the properties of networks, even though the corresponding binary topology remains unchanged. In this paper, the effects of weight randomization on synchronization of coupled chaotic maps is investigated on regular weighted networks. The results reveal that synchronizability is enhanced by redistributing of link weights, i.e. coupled maps reach complete synchronization with lower cost. Furthermore, we show numerically that the heterogeneity of link weights could improve the complete synchronization on regular weighted networks.     

Wave propagation along interacting fiber-like lattices

A fiber-like lattice with resistively coupled electronic elements mimicking a 1-D discrete reaction-diffusion system is considered. The chosen unit or element in the fiber is the paradigmatic Chua's circuit, capable of exhibiting bistable, excitable, oscillatory or chaotic behavior. Then the dynamics of a structure of two such interacting parallel active fibers is studied. Suitable conditions for the interaction to yield synchronization and other forms of collective behavior involving both fibers are obtained. They include wave front propagation, pulse reentry and pulse propagation failure, overcoming of propagation failure, and the appearance of a source of synchronized pulses. The possibility of designing controlled dynamic contacts by means of one or a few inter-fiber couplings is also discussed. Received 12 December 1998     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback

We consider the deterministic dynamics of a semiconductor laser with saturable absorber that is subject to delayed optical feedback. Alone, both the saturable absorber and delayed feedback cause the CW output to become unstable to periodic output via Hopf bifurcations. We examine the combined effects of these two destabilizing mechanisms to determine new conditions for the Hopf bifurcations. We also describe the transient as the unstable CW output evolves to the oscillatory state. A main result is that the presence of a saturable absorber can increase the sensitivity of the laser to delayed feedback. Received 1st August 2001 and Received in final form 28 November 2001     

Birth and long-time stabilization of out-of-equilibrium coherent structures

We study an analytically tractable model with long-range interactions for which an out-of-equilibrium very long-lived coherent structure spontaneously appears. The dynamics of this model is indeed very peculiar: a bicluster forms at low energy and is stable for very long time, contrary to statistical mechanics predictions. We first explain the onset of the structure, by approximating the short time dynamics with a forced Burgers equation. The emergence of the bicluster is the signature of the shock waves present in the associated hydrodynamical equations. The striking quantitative agreement with the dynamics of the particles fully confirms this procedure. We then show that a very fast timescale can be singled out from a slower motion. This enables us to use an adiabatic approximation to derive an effective Hamiltonian that describes very well the long time dynamics. We then get an explanation of the very long time stability of the bicluster: this out-of-equilibrium state corresponds to a statistical equilibrium of an effective mean-field dynamics. Received 28 February 2002 / Received in final form 24 July 2002 Published online 31 October 2002 RID="a" ID="a"e-mail: Thierry.Dauxois@ens-lyon.fr RID="b" ID="b"UMR-CNRS 5672 RID="c" ID="c"UMR 5582     

Time delay as a key to apoptosis induction in the p53 network

A feedback mechanism that involves the proteins p53 and mdm2, induces cell death as a controlled response to severe DNA damage. A minimal model for this mechanism demonstrates that the response may be dynamic and connected with the time needed to translate the mdm2 protein. The response takes place if the dissociation constant k between p53 and mdm2 varies from its normal value. Although it is widely believed that it is an increase in k that triggers the response, we show that the experimental behaviour is better described by a decrease in the dissociation constant. The response is quite robust upon changes in the parameters of the system, as required by any control mechanism, except for few weak points, which could be connected with the onset of cancer. Received 8 May 2002 / Received in final form 9 July 2002 Published online 17 September 2002     

Multi-affinity and multi-fractality in systems of chaotic elements with long-wave forcing

Multi-scaling properties in quasi-continuous arrays of chaotic maps driven by long-wave random force are studied. The spatial pattern of the amplitude X(x,t) is characterized by multi-affinity, while the field defined by its coarse-grained spatial derivative exhibits multi-fractality. The strong behavioral similarity of the X- and Y-fields respectively to the velocity and energy dissipation fields in fully-developed fluid turbulence is remarkable, still our system is unique in that the scaling exponents are parameter-dependent and exhibit nontrivial q-phase transitions. A theory based on a random multiplicative process is developed to explain the multi-affinity of the X-field, and some attempts are made towards the understanding of the multi-fractality of the Y-field. Received 16 November 1998     

Power injected in dissipative systems and the fluctuation theorem

We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the “fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of “temperature" for a dissipative system out of equilibrium. We consider how this “temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration. Received 26 June 2000 and Received in final form 24 October 2000     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.