Simple Scenarios of Onset of Chaos in Three-Dimensional Maps

Gonchenko, A. S.; Гонченко, С. В.; Kazakov, Alexey O.; Turaev, Dmitry

doi:10.1142/s0218127414400057

Cited by 61 publications

(78 citation statements)

References 47 publications

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“…Note that it was shown in [27], see also [28], that discrete Lorenz attractors can arise as a result of simple and universal bifurcation scenarios which naturally occur in one-parameter families of three-dimensional maps. A sketch of such scenario for oneparameter families T µ of three-dimensional diffeomorphisms is shown in Fig.…”

Section: Introductionmentioning

confidence: 98%

Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps

Gonchenko

Гонченко

2016

Physica D: Nonlinear Phenomena

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In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has a positive maximal Lyapunov exponents and this property is robust, i.e. it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized Hénon maps of form x = y,ȳ = z,z = Bx + Az + Cy + g(y, z), where A, B and C are parameters (B is the Jacobian) and g(0, 0) = g ′ (0, 0) = 0.

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Section: Introductionmentioning

confidence: 98%

Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps

Gonchenko

Гонченко

2016

Physica D: Nonlinear Phenomena

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“…Trajectories on a homoclinic attractor can pass arbitrarily close to the saddle orbit belonging to it. The dynamics near this saddle orbits and, as a result, on the whole homoclinic attractor, significantly depends on the multipliers of the corresponding saddle orbit [20]. In particular, in the small neighborhood of a homoclinic attractor containing a saddle-focus periodic orbit with two-dimensional unstable manifold, two-dimensional areas are expanded and Lyapunov exponents on the whole attractor "can feel" this expansion.…”

Section: Introductionmentioning

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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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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“…Especially that such attractors are not exotic, and they can appear in dynamical systems (e.g. in three-dimensional maps) as a result of simple and universal bifurcation scenarios [13,14].…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, in [13,14], there was proposed a simple and universal scenario of emergence of discrete Lorenz attractors in three-dimensional orientable maps. This scenario can be observed even in one parameter families of maps and starts with values of the parameter for which the map has a fixed (periodic) point; then, at varying parameter, this point loses the stability under the soft (supercritical) period doubling bifurcation.…”

Section: Introductionmentioning

confidence: 99%