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Strange Attractors for Periodically Forced Parabolic Equations
Kening Lu, Brigham Young University, Provo, UT, Qiudong Wang, University of Arizona, Tucson, AZ, and Lai-Sang Young, Courant Institute of Mathematical Sciences, New York University, NY
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Memoirs of the American Mathematical Society
2013; 85 pp; softcover
Volume: 224
ISBN-10: 0-8218-8484-0
ISBN-13: 978-0-8218-8484-3
List Price: US$69 Member Price: US$55.20
Order Code: MEMO/224/1054

The authors 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.

• Introduction
• Basic definitions and facts
• Statement of theorems
• Invariant manifolds
• Canonical form of equations around the limit cycle
• Preliminary estimates on solutions of the unforced equation
• Time-$$T$$ Map of forced equation and derived $$2$$-D system
• Strange attractors with SRB measures
• 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
• Bibliography