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 behaviour. 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.
Kening Lu, Brigham Young University, Provo, UT, USA Qiudong Wang, University of Arizona, Tucson, AZ, USA Lai-Sang Young, Courant Institute of Mathematical Sciences, New York University, NY, USA
IntroductionBasic Definitions and Facts Statement of TheoremsInvariant ManifoldsCanonical Form of Equations Around the Limit CyclePreliminary Estimates on Solutions of the Unforced EquationTime-$T$ Map of Forced Equation and Derived $2$-D SystemStrange Attractors with SRB Measures Application: The Brusselator Appendix A. Proofs of Propositions 3.1-3.3Appendix B. Proof of Proposition 7.5Appendix C. Proofs of Proposition 8.1 and Lemma 8.2Bibliography