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    Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium
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

Monica Lazzo 1, and  Paul G. Schmidt 2,

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 $n\to\infty$.
Keywords: Poincaré-Bendixson theory, periodic solutions., Positive-feedback systems, Routh-Hurwitz algorithm, Hopf bifurcation.
Mathematics Subject Classification: Primary: 34C12; Secondary: 34C23; 34C25, 35B44.
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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