Title

Convergent Analytic Solutions For Homoclinic Orbits In Reversible And Non-Reversible Systems

Keywords

Homoclinic orbits; Reversible and non-reversible systems; Shil'nikov analysis; Traveling wave solution

Abstract

In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important non-linear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic settings. We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE. © 2013 Springer Science+Business Media Dordrecht.

Publication Date

8-1-2013

Publication Title

Nonlinear Dynamics

Volume

73

Issue

3

Number of Pages

1769-1782

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s11071-013-0902-z

Socpus ID

84880923617 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84880923617

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