Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Delshams, Amadeu; Гонченко, С. В.; Gonchenko, V. S.; Lázaro, J. Tomás; Sten'kin, O V

doi:10.1088/0951-7715/26/1/1

Cited by 46 publications

(78 citation statements)

References 33 publications

(88 reference statements)

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“…Chaos in dissipative systems is quite different and is associated with strange attractors. Our goal in this paper is to attract attention to one more type of chaos, the third one, which was called "mixed dynamics" in [1,2]. This type of behavior is characterized by inseparability of attractors, repellers, and conservative elements in the phase space [3].…”

Section: Introductionmentioning

confidence: 99%

On three types of dynamics and the notion of attractor

Гонченко

Turaev

2017

Proc. Steklov Inst. Math.

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We propose a theoretical framework for an explanation of the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes which are observed in dynamical systems with attractors as defined in a work by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior -conservative and dissipative, while the attractors of the third type, the reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal, i.e., it displays dynamics of maximum possible richness and complexity.

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Section: Introductionmentioning

confidence: 99%

On three types of dynamics and the notion of attractor

Гонченко

Turaev

2017

Proc. Steklov Inst. Math.

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“…In this section we describe two stochastic lattice models, the simplest tropical climate model developed in [15,30,33] and the stochastic skeleton model for the MJO developed and applied in [34,35,41,42]. They both consist of an ODE system U t , which describes the dry dynamics based on continuum thermal-dynamical PDEs, and a stochastic jump process Á t , which describes the intermittent tropical variability.…”

Section: Stochastic Lattice Models For the Tropicsmentioning

confidence: 99%

“…In [15,30,33], the simplest tropical climate model is derived to capture the impact of tropical moisture variability. Here we discuss a simplified setup of flows above the equator, which follows a PDE:…”

Section: The Simplest Tropical Climate Model Deterministic Modelmentioning

confidence: 99%

“…While it is possible for the same result to hold under other conditions, (2.5) has probably the simplest form. Moreover, condition (2.5) is rather mild, as it holds for most models in [15,29,30] and especially when d Â C d sh D d q , since 0 < x Q < 1 and˛ 0.…”

Section: Geometric Ergodicitymentioning

confidence: 99%

“…In Section 2 we formulate and present two stochastic lattice models for moist tropical convection in climate science as motivation for further developments in the paper. One model is the stochastic skeleton model for the MJO [41,42] with small nonzero damping; the second model is a stochastic lattice model designed to capture intermittent fluctuations [29,31,32] in the simplest tropical climate models [15,30,33]. With this background, the general theorem on contracting PDMPs is formulated and proved in Sections 3 and 4 using Lyapunov functions and perturbation analysis.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Ergodicity for Piecewise Contracting Processes with Applications for Tropical Stochastic Lattice Models

Majda

Tong

2015

Comm Pure Appl Math

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Stochastic lattice models are increasingly prominent as a way to capture highly intermittent unresolved features of moist tropical convection in climate science and as continuum mesoscopic models in material science. Stochastic lattice models consist of suitably discretized continuum partial differential equations interacting with Markov jump processes at each lattice site with transition rates depending on the local value of the continuum equation; they are a special case of piecewise deterministic Markov processes but often have an infinite state space and unbounded transition rates. Here a general theorem on geometric ergodicity for piecewise deterministic contracting processes is developed with full generality to apply to stochastic lattice models. A highly nontrivial application to the stochastic skeleton model for the Madden-Julian oscillation (Thual et al., 2013) is developed here where there is an infinite state space with unbounded and also degenerate transition rates. Geometric ergodicity for the stochastic skeleton model guarantees exponential convergence to a unique invariant measure that defines the statistical tropical climate of the model. Another application of the general framework is developed here for stochastic lattice models designed to capture intermittent fluctuation in the simplest tropical climate models. Other straightforward applications to models motivated by material science are mentioned briefly here.

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