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Chaos / Volume 18 / Issue 1 / FOCUS ISSUE: MIXED MODE OSCILLATIONS: EXPERIMENT, COMPUTATION, AND ANALYSIS

Chaos 18, 015109 (2008); http://dx.doi.org/10.1063/1.2779847 (9 pages)

Nonlocal branches of cycles, bistability, and topologically persistent mixed mode oscillations

E. Bouse1, A. Krasnosel’skii2, A. Pokrovskii1, and D. Rachinskii1

1Department of Applied Mathematics, University College Cork, Cork, Ireland
2Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bol’shoi Karetny, Moscow, Russia

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(Received 26 January 2007; accepted 30 May 2007; published online 27 March 2008)

A possible mechanism for generating mixed mode oscillations is based on an appropriate S-shaped structure, which graphs the relation between the parameter and the collection of periodic oscillations existing for a particular parameter value in the product of parameter and phase spaces. This natural scenario should be supplemented by simple and constructive criteria of existence, and methods of localization, of such S-shaped structures. These criteria are the main focus of the paper.

© 2008 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. S-SHAPED BRANCHES OF CYCLES
  3. PROOFS
    1. Proof of Theorem 1
    2. Proof of Corollary 1
    3. Proof of Theorem 2
      1. Contracting property of Ar
      2. Invariance of the cylinder math
      3. Lipschitz continuity of the branch of cycles
    4. Proof of Theorem 3

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KEYWORDS and PACS

Keywords

bifurcation

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.2779847

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

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$abstract.society.value is a Member of CrossRef American Institute of Physics

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