Yu. Maistrenko

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, …

Cluster-splitting bifurcation in a system of coupled maps

Abstract We consider cluster-splitting bifurcations in a system of globally coupled maps as coupling parameter decreases. At these transitions the number of clusters, i.e., groups of elements with identical dynamics, increases. We demonstrate that different cascades of cluster-splitting can occur, depending on statistics of redistribution of the oscillators between new-born clusters.

Transcritical loss of synchronization in coupled chaotic systems

Abstract The synchronization transition is described for a system of two asymmetrically coupled chaotic oscillators. Such a system can represent the two-cluster state in a large ensemble of globally coupled oscillators. It is shown that the transition can be typically mediated by a transcritical transversal bifurcation. The latter has a hard brunch that dominates the global dynamics, so that the synchronization transition is normally hard. For a particular example of coupled logistic maps a diversity of transition scenaria includes both local and global riddling. In the case of small non-identity of the interacting systems the riddling is shown to turn into an exterior or interior crisis.

Role of asymmetric clusters in desynchronization of coherent motion

The transition from full synchronization (coherent motion) to two-cluster dynamics is studied for a system of N globally coupled logistic maps. When increasing the nonlinearity parameter of the individual map, new periodic and strongly asymmetric two-cluster states are found to emerge in the same order as the periodic windows arise in the logistic map. These strongly asymmetric two-cluster states are generally first to stabilize when reducing the coupling strength. Similar phenomena are also observed for a system of globally coupled Henon maps.

Chaos synchronization and riddled basins in two coupled one-dimensional maps

Abstract The connection between chaos synchronization and phenomena of riddled basins is discussed for a family of two-dimensional piecewise linear endomorphisms which consist of two linearly coupled one-dimensional maps. Under analytically given conditions, chaotic behavior in both maps can synchronize but synchronized state is characterized by different types of stability. The mechanism of occurrence of riddled basins is described in detail.

Riddling bifurcations in coupled piecewise linear maps

Abstract A mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described. We give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold. It is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation.

Clustering zones in the turbulent phase of a system of globally coupled chaotic maps.

The paper develops an approach to investigate the clustering phenomenon in the system of globally coupled chaotic maps first introduced by Kaneko in 1989. We obtain a relation between the transverse and longitudinal multipliers of the periodic clusters and prove the stability of these clusters for the case of symmetric, equally populated distributions between subclusters. Stable clusters emanate from the periodic windows of the logistic map and extend far into the turbulent phase. By numerical simulations we estimate a total basin volume of low-periodic clusters issued from the period-3 window and analyze the basin structure. The complement to the basin volume is ascribed to chaotic, very asymmetric high-dimensional clusters that are characterized by the presence of one or more leading clusters, accumulating about half of the oscillators while all the remaining oscillators do not cluster at all.

Synchronization and Clustering in Ensembles of Coupled Chaotic Oscillators

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained, providing a special kind of dynamical patterns called clusters. The simplest, coherent clusters arise when all oscillators display the same temporal behavior. Others, more complicated clusters are developed when population of the oscillators splits into subgroups such that all oscillators within a given group move in synchrony. Considering a system of mean-field coupled logistic maps, we study in details the transition from coherence to clustering and demonstrate that there are four different mechanisms of the desynchronization: riddling and blowout bifurcations, appearance of symmetric and asymmetric clusters. We also investigate the cluster-splitting bifurcation when the underlying dynamics is periodic. For the system of three and four coupled Rossler oscillators, we prove the existence of clusters and describe related bifurcations and in-cluster dynamics.