Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions

Pages: 941 - 952, Issue Special, September 2011

 Abstract        Full Text (358.0K)              

Monica Lazzo - Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari, Italy (email)
Paul G. Schmidt - Department of Mathematics and Statistics, Parker Hall, Auburn University, AL 36849-5310, United States (email)

Abstract: We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of $\mathbb{R}^n$, with $n\in\mathbb{N}$, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. In Part 1 of the paper we showed that, in every dimension $n\<=4$, all forward-bounded solutions converge to one of two equilibria (one stable, the other unstable). Here we establish the existence of nontrivial periodic orbits in every dimension $n\>=12$. For $12\<=n\<=16$, we prove that such orbits arise via Hopf bifurcation from the unstable equilibrium. Additional results suggest that at least one Hopf bifurcation occurs whenever $n\>=12$, followed by at least a second one if $n\>=62$, and that the number of successive bifurcations increases without bound as $n\to\infty$.

Keywords:  Positive-feedback systems, Poincar√©-Bendixson theory, Routh-Hurwitz algorithm, Hopf bifurcation, periodic solutions.
Mathematics Subject Classification:  Primary: 34C12; Secondary: 34C23; 34C25, 35B44

Received: August 2010;      Revised: April 2011;      Published: October 2011.