# American Institute of Mathematical Sciences

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

 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 $n \>= 12$.
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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