Examples of Lorenz-like Attractors in Hénon-like Maps

Гонченко, С. В.; Gonchenko, A. S.; Ovsyannikov, Ivan; Turaev, Dmitry

doi:10.1051/mmnp/20138504

Cited by 33 publications

(42 citation statements)

References 116 publications

(186 reference statements)

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“…We should notice that our paper continues the study of dynamics of three-dimensional Hénon-like maps started in a circle of works [23,27,31,28]. The possibility of birth of discrete Lorenz attractors at local bifurcations was established in [24]; the first example of such attractor was found in [23]; scenaria of emergence of the discrete Shilnikov and Lorenz attractors in multidimensional maps were described and discussed in [3,24,27,28]; conditions of existence of the discrete Lorenz attractors were derived in [24,23,31] as well as various their examples were found [31]. We would like to note papers [32,33,34], where the discrete Lorenz and figure-8 attractors were discovered in systems of rigid body dynamics.…”

Section: Discussionmentioning

confidence: 58%

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: Discussionmentioning

confidence: 58%

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

Gonchenko

Гонченко

2016

Physica D: Nonlinear Phenomena

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“…is fulfilled. Therefore, by the same arguments as for the map (8), Theorem 3 provides a formal justification for the claim of [24] (see Lemma 3.1 there) about the existence of pseudohyperbolic attractors for an open set of parameters ε 1,2,3 in maps (10) whose coefficients satisfy condition (10).…”

Section: Shilnikov Criterionmentioning

confidence: 80%

“…Since the map (8) is obtained from (7) just by a change of coordinates, we have the following Note that the same conclusion holds for larger classes of three-dimensional maps. It was shown in [24] that the normal form for the bifurcations of the zero fixed point of any map of the typē…”

Section: Shilnikov Criterionmentioning

confidence: 99%

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Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system

2018

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We prove that the Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

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“…Therefore, for such spiral attractors, we proposed in [42] the generalizing title: "Shilnikov attractor" for flows (these include, in particular, the mentioned above Roesler attractor and ACT-attractors); or "Shilnikov discrete attractor" for maps. Various examples of such discrete attractors were found in three-dimensional Hénon maps [41,42,43,44,45] for which very interesting results were obtained, and we plan to review them in the next part of our paper.…”

Section: Introductionmentioning

confidence: 99%

Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

Gonchenko

Гонченко

Kazakov

et al. 2018

Int. J. Bifurcation Chaos

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The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a cycle of three our papers), elements of the theory of pseudo-hyperbolic attractors of multidimensional maps are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We give a mathematical definition of a pseudo-hyperbolic attractor for the case of multidimensional maps, from which we derive the necessary conditions for its existence in the three-dimensional case, formulated using the Lyapunov exponents. We also describe some phenomenological scenarios for the appearance of pseudo-hyperbolic attractors of various types in one-parameter families of three-dimensional diffeomorphisms, we propose new methods for studying such attractors (in particular, a method of saddle charts and a modified method of Lyapunov diagrams). We consider also three-dimensional generalized Hénon maps as examples.In the second part, we plan to review the theory of spiral attractors, which compose an important class of attractors often meeting in applications. The third part will be devoted to mixed dynamics -a new type of chaos, which is characteristic, in particular for reversible systems, i.e. systems invariant with respect to the time reversal. It is well known that such systems are met in many problems of mechanics, electrodynamics and other fields of natural science.

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Examples of Lorenz-like Attractors in Hénon-like Maps

Cited by 33 publications

References 116 publications

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

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

Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system

Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

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