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Гонченко, С. В.; Meiss, James D.; Ovsyannikov, Ivan

doi:10.1070/rd2006v011n02abeh000345

Cited by 59 publications

(15 citation statements)

References 32 publications

(51 reference statements)

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“…The elegant theory of Smale horseshoe is often applied to the study of the Hénon or generalized Hénon maps [5]. There exist many results on the polynomial maps with horseshoes, for instance, the investigation of Devaney and Nitecki on the two-dimensional Hénon map [3]; Friedland and Milnor's study on the n-fold horseshoes of two-dimensional polynomial diffeomorphisms [5]; the work of Dullin and Meiss on two-dimensional cubic Hénon maps [4]; the investigations of Gonchenko et al on several types of three-dimensional polynomial maps [6,7,8,9]; some discussions on polynomial maps of any dimensions [25,26].…”

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confidence: 99%

Polynomial maps with hidden complex dynamics

Zhang

Chen²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Hénon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

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confidence: 99%

Polynomial maps with hidden complex dynamics

Zhang

Chen²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…for suitable values of the parameters ϕ and a. This map can be regarded as a 3D analogue of the Hénon map ( 5) since (a) it is a quadratic truncation of the unfolding of the normal form near a triple-one multiplier [20], (b) its inverse is also a quadratic volume-preserving map [35], and (c) it appears as a truncation of the return map near a homoclinic quadratic tangency [25]. The map ( 9) is a discretization of the well-known Michelson ODEs [43]…”

Section: The Hopf-one Bifurcation In Volume-preserving Mapsmentioning

confidence: 99%

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

et al. 2018

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Angle-action maps that are periodic in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a volume-preserving family of maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map.

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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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Cited by 59 publications

References 32 publications

Polynomial maps with hidden complex dynamics

Polynomial maps with hidden complex dynamics

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

Anti-Integrability for 3-Dimensional Quadratic Maps

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