Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone

Gonchenko, A. S.; Гонченко, С. В.; Kazakov, Alexey O.; Samylina, E. A.

doi:10.1007/s11141-019-09935-4

Cited by 13 publications

(5 citation statements)

References 19 publications

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“…Discrete Shilinikov attractors were found in different dynamical systems such as the generalized three-dimensional Hénon maps, nonholonomic models of Chaplygin top [38] and Celtic stone [45], the model of coupled identical oscillators [39].However, not in all cases they were identified as hyperchaotic attractors. Apparently, in some cases the expansion of two-dimensional areas near a saddle-focus orbit with a two-dimensional unstable manifold is compensated by the volume contraction near some other saddle periodic orbits that also belong to the attractor, but which have only a one-dimensional unstable manifold.…”

Section: Transition To Chaos and Hyperchaosmentioning

confidence: 99%

“…It depends on the transition from the stable multi-round torus to chaotic regime. In all known models demonstrating the onset of the discrete Shilnikov attractor (in three-dimensional Hénon maps [19], nonholonomic models of Chaplygin top [38] and Celtic stone [45]) this inclusion happens in a soft manner by a smooth transformation of a torus-chaos attractors. However, we also suppose that a saddle-focus orbit can be included to the chaotic attractor sharply due to the crisis of multi-round torus-chaos attractor.…”

Section: Transition To Hyperchaos In System (1)mentioning

confidence: 99%

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Hyperchaos and multistability in the model of two interacting microbubble contrast agents

Garashchuk

Sinelshchikov

Kazakov

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study nonlinear dynamics of two coupled contrast agents that are micro-meter size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system various types of complex dynamics can occur, namely, we observe periodic, quasi-periodic, chaotic and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either Feigenbaum's cascade of period doubling bifurcations or Afraimovich-Shilnikov scenario of torus destruction. For the onset of hyperchaotic attractor we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the bubbles' dynamics can be multistable, i.e. various combinations of co-existence of the above mentioned attractors are possible. These cases include co-existence of hyperchaotic regime with any of the other remaining types of dynamics for different parameter values. Thus, the model of two coupled gas bubbles provide a new examples of physically relevant system with multistable hyperchaos.

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Section: Transition To Chaos and Hyperchaosmentioning

confidence: 99%

Section: Transition To Hyperchaos In System (1)mentioning

confidence: 99%

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

Garashchuk

Sinelshchikov

Kazakov

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…This in itself is not a new phenomenon. Flow-like chaotic attractors were previously observed in the three-dimensional Hénon map [36], in the Lorenz-84 model [15], in some class of 3D diffeomorphisms of the torus [14], in nonholonomic models of Celtic stone [28] and Chaplygin top [12], in models of identical globally coupled oscillators [42] and in other systems. In this section we give an explanation for this phenomenon suitable for the map (6) as well as for many other cases.…”

Section: Chaotic Attractors With Zero Second Lementioning

confidence: 76%

Scenarios for the creation of hyperchaotic attractors in 3D maps

et al. 2023

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We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

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“…It can lead to chaos with only one positive Lyapunov exponent. Such phenomenon was observed, for example, in nonholonomic models of Celtic stone [44] and Chaplygin top [43], in models of identical globally coupled oscillators [45] and in other systems. In this section we show that the same phenomenon can be observed in the map under consideration, i.e., depending on values of the Jacobian, discrete Shilnikov attractors in the three-dimensional Mirá map can have either two or only one positive Lyapunov exponent.…”

Section: Various Types Of Discrete Shilnikov Attractorsmentioning

confidence: 87%

“…Such transition to chaotic attractors (including Shilnikov ones), is observed quite often (see e.g. [38,39,40,41,42,43,44,45,46,47]) and can lead to the birth of "flow-like" chaotic attractors possessing one positive and one very close to zero Lyapunov exponent (for flow systems chaos with additional close to zero Lyapunov exponent in the spectrum is observed in this case). In Sec.…”

Section: (A)mentioning

confidence: 93%

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

Shykhmamedov¹,

Karatetskaia²,

Казаков³

et al. 2020

Preprint

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We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mirá map.

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Chaotic Dynamics and Multistability in the Nonholonomic Model of a Celtic Stone

Cited by 13 publications

References 19 publications

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

Scenarios for the creation of hyperchaotic attractors in 3D maps

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

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