Volodymyr L. Maistrenko

CYCLES OF CHAOTIC INTERVALS IN A TIME-DELAYED CHUA'S CIRCUIT

We study the bifurcations of attractors of a one-dimensional 2-segment piecewise-linear map. We prove that the parameter regions of existence of stable point cycles γ are separated by regions of existence of stable interval cycles Γ containing chaotic everywhere dense trajectories. Moreover, we show that the period-doubling phenomenon for cycles of chaotic intervals is characterized by two universal constants δ and α, whose values are calculated from explicit formulas.

BIFURCATIONS OF ATTRACTING CYCLES FROM TIME-DELAYED CHUA’S CIRCUIT

We study the bifurcations of attracting cycles for a three-segment (bimodal) piecewise-linear continuous one-dimensional map. Exact formulas for the regions of periodicity of any rational rotation number (Arnold’s tongues) are obtained in the associated three-dimensional parameter space. It is shown that the destruction of any Arnold’s tongue is a result of a border-collision bifurcation, and is followed by the appearance of a cycle of intervals with the same rotation number, whose dynamics is determined by a skew tent map. Finally, for the interval cycle the merging bifurcation corresponds to a homoclinic bifurcation of some point cycle.

On period-adding sequences of attracting cycles in piecewise linear maps

Abstract We study numerically bifurcations in a family of bimodal three-piecewise linear continuous one-dimensional maps. Attention is paid to the attracting cycles arising after the bifurcation ‘from unimodal map to bimodal map’. It is found that this type of bifurcation is accompanied by the appearance of period-adding cascades of attracting cycles γ ( a 11 + a 12 k )/( a 21 + a 22 k ) which are characterized by ρ k = ( a 11 + a 12 k )/( a 21 + a 22 k ), k = 0, 1, …

Chimera states in three dimensions

The chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. In this paper, the first evidence of three-dimensional chimera states is reported for the Kuramoto model of phase oscillators in 3D grid topology with periodic boundary conditions. Systematic analysis of the dependence of the spatiotemporal dynamics on the range and strength of coupling shows that there are two principal classes of the chimera patterns which exist in large domains of the parameter space: (I) oscillating and (II) spirally rotating. Characteristic examples from the first class include coherent as well as incoherent balls, tubes, crosses, and layers in incoherent or coherent surrounding; the second class includes scroll waves with incoherent, randomized rolls of different modality and dynamics. Numerical simulations started from various initial conditions indicate that the states are stable over the integration time. Videos of the dynamics of the chimera states are presented in supplementary material. It is concluded that three-dimensional chimera states, which are novel spatiotemporal patterns involving the coexistence of coherent and incoherent domains, can represent one of the inherent features of nature.

PHASE CHAOS IN THE DISCRETE KURAMOTO MODEL

The paper describes the appearance of a novel, high-dimensional chaotic regime, called phase chaos, in a time-discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It arises from the nonlinear interaction among the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional time-discrete Kuramoto model, we outline the region of phase chaos in the parameter plane and determine the regions where phase chaos coexists with different periodic attractors. We also study the subcritical frequency-splitting bifurcation at the onset of desynchronization and demonstrate that the transition to phase chaos takes place via a torus destruction process.

Multiple scroll wave chimera states

Abstract We report the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that the number of incoherent chimera domains can grow additively under appropriate variations of the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for the Kuramoto model of N3 identical phase oscillators placed in the unit 3D cube with periodic boundary conditions, parameters being the coupling radius r and phase lag α. To obtain the multiheaded chimeras, we perform the so-called ‘cloning procedure’ as follows: choose a sample single-headed 3D chimera state, make appropriate scale transformation, and put some number of copies of them into the unit cube. After that, start numerical simulations with slightly perturbed initial conditions and continue them for a sufficiently long time to confirm or reject the state existence and stability. In this way it is found, that multiple scroll wave chimeras including those with incoherent rolls, Hopf links and trefoil knots admit this sort of multiheaded regeneration. On the other hand, multiple 3D chimeras without spiral rotations, like coherent and incoherent balls, tubes, crosses, and layers appear to be unstable and are destroyed rather fast even for arbitrarily small initial perturbations.

CHAOTIC SYNCHRONIZATION AND ANTISYNCHRONIZATION IN COUPLED SINE MAPS

This paper investigates different types of chaotic synchronization in a system of two coupled sine maps. Due to the bimodal nature of the individual map, there is a range of parameters in which two synchronized chaotic states coexist along the main diagonal. In certain parameter regions, various (regular or chaotic) asynchronous states coexist with the synchronized chaotic states, and the basins of attraction become quite complicated. We determine the regions of stability for the so-called principal cycles that arise through transverse period-doubling bifurcations of synchronized saddle cycles. Particular emphasis is paid to the occurrence of chaotic antisynchronization, the coexistence of antisynchronous chaotic states, and the presence of narrow regions of parameter space in which states of chaotic synchronization and antisynchronization exist simultaneously. For each of these cases we provide detailed pictures of the associated basin structures.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.