Yuri Maistrenko

Chaotic synchronization : applications to living systems

Coupled Nonlinear Oscillators Transverse Stability of Coupled Maps Unfolding the Riddling Bifurcation Time-Continuous Systems Coupled Pancreatic Cells Chaotic Phase Synchronization Population Dynamic Systems Clustering of Globally Maps Interacting Nephrons Coherence Resonance Oscillators.

Loss of coherence in dynamical networks: spatial chaos and chimera states.

We discuss the breakdown of spatial coherence in networks of coupled oscillators with nonlocal interaction. By systematically analyzing the dependence of the spatiotemporal dynamics on the range and strength of coupling, we uncover a dynamical bifurcation scenario for the coherence-incoherence transition which starts with the appearance of narrow layers of incoherence occupying eventually the whole space. Our findings for coupled chaotic and periodic maps as well as for time-continuous Rössler systems reveal that intermediate, partially coherent states represent characteristic spatiotemporal patterns at the transition from coherence to incoherence.

Transition from spatial coherence to incoherence in coupled chaotic systems.

We investigate the spatio-temporal dynamics of coupled chaotic systems with nonlocal interactions, where each element is coupled to its nearest neighbors within a finite range. Depending upon the coupling strength and coupling radius, we find characteristic spatial patterns such as wavelike profiles and study the transition from coherence to incoherence leading to spatial chaos. We analyze the origin of this transition based on numerical simulations and support the results by theoretical derivations, identifying a critical coupling strength and a scaling relation of the coherent profiles. To demonstrate the universality of our findings, we consider time-discrete as well as time-continuous chaotic models realized as a logistic map and a Rössler or Lorenz system, respectively. Thereby, we establish the coherence-incoherence transition in networks of coupled identical oscillators.

Spectral properties of chimera states.

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

Imperfect chimera states for coupled pendula.

The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimeras cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newtons laws that are ubiquitous in many physical and engineering systems.

Laser chimeras as a paradigm for multistable patterns in complex systems

A chimera state is a rich and fascinating class of self-organized solutions developed in high-dimensional networks. Necessary features of the network for the emergence of such complex but structured motions are non-local and symmetry breaking coupling. An accurate understanding of chimera states is expected to bring important insights on deterministic mechanism occurring in many structurally similar high-dimensional dynamics such as living systems, brain operation principles and even turbulence in hydrodynamics. Here we report on a powerful and highly controllable experiment based on an optoelectronic delayed feedback applied to a wavelength tuneable semiconductor laser, with which a wide variety of chimera patterns can be accurately investigated and interpreted. We uncover a cascade of higher-order chimeras as a pattern transition from N to N+1 clusters of chaoticity. Finally, we follow visually, as the gain increases, how chimera state is gradually destroyed on the way to apparent turbulence-like system behaviour.

Bifurcation structure of a model of bursting pancreatic cells

One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic beta-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other. The transition from this structure to the so-called period-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-III intermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which (n+1)-spike bursting behavior is born, slightly overlapping with a subcritical period-doubling bifurcation in which n-spike bursting behavior loses its stability.

Partial synchronization and clustering in a system of diffusively coupled chaotic oscillators

We examine the problem of partial synchronization (or clustering) in diffusively coupled arrays of identical chaotic oscillators with periodic boundary conditions. The term partial synchronization denotes a dynamic state in which groups of oscillators synchronize with one another, but there is no synchronization among the groups. By combining numerical and analytical methods we prove the existence of partially synchronized states for systems of three and four oscillators. We determine the stable clustering structures and describe the dynamics within the clusters. Illustrative examples are presented for coupled Rossler systems. At the end of the paper, synchronization in larger arrays of chaotic oscillators is discussed.

Loss of synchronization in coupled Rössler systems

This paper considers the loss of synchronization for a system of two coupled Rossler oscillators. Bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the synchronized chaotic state are obtained, and we show that desynchronization for a pair of symmetrically coupled, identical Rossler systems is associated with different orbits undergoing transverse pitchfork or period-doubling bifurcations. The transverse destabilization of the period-1 orbit is examined in detail, and we follow the sequence of bifurcations that the asynchronous periodic cycles undergo. In the presence of an asymmetry in the coupling, the transverse period-doubling bifurcation remains essentially the same. The transverse pitchfork bifurcation, on the other hand, is transformed into a transcritical riddling bifurcation. If the interacting Rossler oscillators have different parameter values, the non-generic character of the pitchfork bifurcation leads it to be replaced by a saddle-node bifurcation occurring off the symmetric sub-space. Finally, we show how the transverse stability properties of the equilibrium point can be used to obtain approximative analytical results for the transverse stability of the coupled chaotic oscillators.

ON STRONG AND WEAK CHAOTIC PARTIAL SYNCHRONIZATION

We study coupled nonlinear dynamical systems with chaotic behavior in the case when two or more (but not all) state variables synchronize, i.e. converge to each other asymptotically in time. It is ...