American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 931-940. doi: 10.3934/proc.2011.2011.931

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

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. We describe the global dynamics of the system for arbitrary $n$ and prove that, in every dimension $n\<=4$, all forward-bounded solutions converge to one of two equilibria (one stable, the other unstable). In Part 2 of the paper, we will establish the existence of nontrivial periodic orbits in every dimension $n \>= 12$.
Keywords: Poincaré-Bendixson theory, monotone flows, Positive-feedback systems, Ljapunov functions..
Mathematics Subject Classification: Primary: 34C12; Secondary: 34C23; 34C25, 35B4.
Citation: Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931
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