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

Research output: Contribution to journalJournal article – Annual report year: 2014Researchpeer-review

Without internal affiliation

  • Author: Granados, Albert

    Inria Paris-Rocquencourt Research Centre

  • Author: Hogan, S. J.

    University of Bristol, United Kingdom

  • Author: Seara, T. M.

    Polytechnic University of Catalonia, Spain

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We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in R2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two C0 normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between C0 tori located at the same energy levels. By means of the impact map we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the scattering map, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnold diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring.
Original languageEnglish
JournalPhysica D: Nonlinear Phenomena
Volume269
Pages (from-to)1-20
ISSN0167-2789
DOIs
Publication statusPublished - 2014
Externally publishedYes
CitationsWeb of Science® Times Cited: No match on DOI

    Research areas

  • Arnold diffusion, Piecewise smooth systems, Rocking block

ID: 119551539