Transverse instability and riddled basins in a system of two coupled logistic maps

Maistrenko, Yuri; Maistrenko, Volodymyr; Popovich, A.S.; Mosekilde, Erik

doi:10.1103/physreve.57.2713

Cited by 118 publications

(60 citation statements)

References 24 publications

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“…We remark that`generally' this occurs for a cycle of low period. This fact has been noticed in [13], qualitative reasons to explain this are given in [15], and it seems to be confirmed in our example.…”

Section: Definitionsupporting

confidence: 74%

“…We also remark that Case II has not been observed by the numerical explorations performed with our map (17), but we cannot exclude it, and examples of its occurrence are given in the literature ( [10,15]). …”

Section: Intermittency and Attracting Areamentioning

confidence: 99%

“…so that the restriction Fj Á f a is independent of the coupling parameter 4 (see [12,9,22,14,15]). In this case, the main bifurcations of the invariant sets of the two-dimensional map belonging to the subspace where synchronized dynamics occur, such as the bifurcation from topological attractor to a non topological one, can be studied as a function of the coupling parameter 4 without altering the dynamical properties of the attractors of the restriction, depending only on the parameter a.…”

Section: Definitionmentioning

confidence: 99%

“…The basic question examined in the recent literature is whether an attractor A of the one-dimensional map f is also an attractor located on the invariant submanifold Á, of the two-dimensional map T. In this context, new and interesting mathematical problems have been evidenced since measure theoretic, but not topological, attractors appear quite naturally in this context together with new striking properties of the basins (see [2,3,20,4,15]). …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Synchronization, intermittency and critical curves in a duopoly game

Bischi¹,

Stefanini²,

Gardini³

1998

Mathematics and Computers in Simulation

109

113

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The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x 1 x 2 , so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries. # 1998 IMACS/Elsevier Science B.V.

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Section: Definitionsupporting

confidence: 74%

Section: Intermittency and Attracting Areamentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Synchronization, intermittency and critical curves in a duopoly game

Bischi¹,

Stefanini²,

Gardini³

1998

Mathematics and Computers in Simulation

109

113

View full text Add to dashboard Cite

show abstract

“…The first situation generally arises when the riddling bifurcation is supercritical. 42 Therefore, riddling with infinity is expected for the parameters 0.736ϽdϽ0.8. Further evidence for this will be given in Sec.…”

Section: Influence Of Riddling Bifurcations On Riddled Basin Strucmentioning

confidence: 99%

Synchronization of time-continuous chaotic oscillators

Yanchuk

Maistrenko

Mosekilde

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed, respectively, in terms of the subcritical, supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled Rössler oscillators. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1496536͔The collective behavior of systems of coupled nonlinear oscillators is of significant interest in many areas of science and technology. It is likely, for instance, that transitions between different states of synchronization among the cells or functional units can play a major role in the regulation of many physiological systems. In the field of macroeconomics, synchronization phenomena may explain how the oscillatory dynamics of individual sectors or industries, rather than being averaged out in the aggregate, can lock together and contribute to the formation of the characteristic business and investment cycles. Particularly interesting is the case where the individual oscillators behave chaotically. In the present paper, we use a system of coupled Rössler oscillators to review some of the basic aspects of the theory of full synchronization for time-continuous chaotic oscillators. At the end of the paper we discuss the so-called cluster formation process by which the fully synchronized chaotic state loses its stability and breaks up into groups of oscillators with different dynamics, but such that the oscillators within a given group maintain synchrony.

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Nonlinear phase desynchronization in human electroencephalographic data

Breakspear¹

2002

Human Brain Mapping

109

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Ensembles of coupled nonlinear systems represent natural candidates for the modeling of brain dynamics. The objective of this study is to examine the complex signal produced by coupled chaotic attractors, to discuss their potential relevance to distributed processes in the brain, and to illustrate a method of detecting their contribution to human EEG morphology. Two measures of quantifying the behavior of coupled nonlinear systems are presented: a measure of phase synchrony and a novel measure of intermittent phase desynchronization. These are used to quantify the behavior of numerical examples of coupled chaotic attractors. Experimental evidence of their contribution to the morphology of the human alpha rhythm is then illustrated in a study of EEG recordings from 40 healthy human subjects. Amplitude-adjusted phase-randomized surrogate data is used to test the null hypothesis that the observed patterns of phase coherence can be described by purely linear methods. Statistical analysis reveals that this null hypothesis can be robustly rejected in a small number (approximately 4%) of EEG epochs. These findings are discussed with reference to the adaptive function and complex dynamics of the brain.

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Transverse instability and riddled basins in a system of two coupled logistic maps

Cited by 118 publications

References 24 publications

Synchronization, intermittency and critical curves in a duopoly game

Synchronization, intermittency and critical curves in a duopoly game

Synchronization of time-continuous chaotic oscillators

Nonlinear phase desynchronization in human electroencephalographic data

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