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2011, 2011(Special): 941-952. doi: 10.3934/proc.2011.2011.941

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

1. 

Dipartimento di Matematica, Università di Bari, via Orabona 4, 70125 Bari

2. 

Department of Mathematics and Statistics, Parker Hall, Auburn University, AL 36849-5310, United States

Received  August 2010 Revised  April 2011 Published  October 2011

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 $n\to\infty$.
Citation: Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 2. Periodic solutions. Conference Publications, 2011, 2011 (Special) : 941-952. doi: 10.3934/proc.2011.2011.941
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