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Strange Attractors for Periodically Forced Parabolic Equations


About this Title

Kening Lu, School of Mathematics, Sichuan University, Chengdu, China — and — Department of Mathematics Brigham Young University, Provo, Utah 84602, Qiudong Wang, Department of Mathematics, University of Arizona, Tuscon, Arizona 85721 and Lai-Sang Young, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 224, Number 1054
ISBNs: 978-0-8218-8484-3 (print); 978-1-4704-1005-6 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00669-1
Published electronically: November 19, 2012
Keywords:Hopf bifurcations, periodic forcing, parabolic PDEs, strange attractors, SRB measures.
MSC: Primary 37L30; Secondary 37D45

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Basic definitions and facts
  • Chapter 3. Statement of theorems
  • Chapter 4. Invariant manifolds
  • Chapter 5. Canonical form of equations around the limit cycle
  • Chapter 6. Preliminary estimates on solutions of the unforced equation
  • Chapter 7. Time-$T$ map of forced equation and derived 2-D system
  • Chapter 8. Strange attractors with SRB measures
  • Chapter 9. Application: The Brusselator
  • Appendix A. Proofs of Propositions 3.1–3.3
  • Appendix B. Proof of Proposition 7.5
  • Appendix C. Proofs of Proposition 8.1 and Lemma 8.2

Abstract


We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

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