On discrete Lorenz-like attractors

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

doi:10.1063/5.0037621

Cited by 24 publications

(9 citation statements)

References 47 publications

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“…Recall that, by definition (see, e.g., [19]), discrete Lorenz-like attractors contain a saddle fixed (periodic) point whose saddle value is greater than one. Along the pathway AB considered above, the resulting attractor contains a fixed point whose saddle value is less than one.…”

Section: Lorenz-shape Attractorsmentioning

confidence: 99%

Lorenz- and Shilnikov-Shape Attractors in the Model of Two Coupled Parabola Maps

Kuryzhov¹,

Efrosiniia

Mints

2021

Nelin. Dinam.

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We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.

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Section: Lorenz-shape Attractorsmentioning

confidence: 99%

Lorenz- and Shilnikov-Shape Attractors in the Model of Two Coupled Parabola Maps

Kuryzhov¹,

Efrosiniia

Mints

2021

Nelin. Dinam.

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“…Exponent k is i or j depending on which saddle point, O 1 or O 2 , satisfies the resonance condition. The angle variable ϕ is given by formula (21). Varying a small µ 2 near one of the zeros of the trigonometric function, and at the same time, keeping it away from zero, one get parameter M 2 taking arbitrary finite values.…”

Section: B Global Mapsmentioning

confidence: 99%

“…The properties of these attractors were studied in Ref. 21. Note that the second iterate of the map in this case has a fixed point with multipliers (−1, −1, +1), but, unlike the orientable case, they form three Jordan blocks instead of two, so the bifurcations that happen for B > 0 and B < 0 are principally different.…”

Section: Introductionmentioning

confidence: 99%

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Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors

Ovsyannikov

2021

Preprint

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Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. However, to date only partial cases were investigated. In this paper bifurcations of homoclinic and heteroclinic cycles with quadratic tangencies of invariant manifolds are studied. A full list of such bifurcations, leading to the appearance of discrete Lorenz attractors is provided. In addition, with help of numerical techniques, it was proved that if one reverses time in the diffeomorphisms described above, the resulting systems also have such attractors. This result is an important step in the systematic studies of chaos and hyperchaos. a

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“…Studying the dynamics of such maps is important in understanding Lagrangian mixing problems, with many applications [Hal15]. The volume-contracting case, |δ| < 1, arises as a normal form near homoclinic bifurcations of 3D maps [GMO06] and can give rise to discrete Lorenz-like attractors [GGKS21].…”

Section: Introductionmentioning

confidence: 99%

Anti-Integrability for 3-Dimensional Quadratic Maps

Hampton,

Meiss

2021

Preprint

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We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced thirty years ago for the Frenkel-Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit to become an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations and obtaining orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node or period doubling bifurcations have symbol sequences that differ in exactly one position.

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On discrete Lorenz-like attractors

Cited by 24 publications

References 47 publications

Lorenz- and Shilnikov-Shape Attractors in the Model of Two Coupled Parabola Maps

Lorenz- and Shilnikov-Shape Attractors in the Model of Two Coupled Parabola Maps

Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors

Anti-Integrability for 3-Dimensional Quadratic Maps

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