The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

Granados, Albert; Hogan, S. J.; Seara, Tere M.

doi:10.1016/j.physd.2013.11.008

Cited by 13 publications

(8 citation statements)

References 41 publications

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“…We hope to come back to this problem. Another problem where one has scattering maps between two different normally hyperbolic invariant manifolds is the problem of two rocking blocks under periodic forcing [GHS14].…”

Section: Resultsmentioning

confidence: 99%

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.

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Section: Resultsmentioning

confidence: 99%

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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“…The study of HIS combines the features of Hamiltonian dynamics and those of piecewise smooth dynamical systems [7,19,22], which are specific examples of hybrid systems (e.g. [16,14]). Utilizing the Hamiltonian structure, one hopes to gain information on global scales.…”

Section: Introductionmentioning

confidence: 99%

On Near Integrability of Some Impact Systems

Pnueli¹,

Rom‐Kedar²

2018

SIAM J. Appl. Dyn. Syst.

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“…Piecewise-smooth (piecewise-defined or non-smooth) systems are non-regular or discontinuous systems induced by dynamics associated with sharp changes in position, velocity, or other magnitudes undergoing a jump in their value. This type of systems provide more natural and simpler models in many applications, such as switching systems in power electronics [31,133,53,16], sliding-mode techniques in control theory [130,43,52,47], hybrid systems with resets in neuroscience [35,83,103,74] or impact systems in mechanics [78,71,72]. Using non-smooth modeling can reduce the dimension of the system, but may result in more complicated dynamics.…”

Section: Introductionmentioning

confidence: 99%

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

Granados¹,

Alsedà²,

Krupa³

2017

SIAM Rev.

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This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasicontractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and "rotation" numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.

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The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

Cited by 13 publications

References 41 publications

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

On Near Integrability of Some Impact Systems

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

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