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

Pages: 931 - 940, Issue Special, September 2011

 Abstract        Full Text (338.2K)              

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. 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:  Positive-feedback systems, Poincar√©-Bendixson theory, monotone flows, Ljapunov functions.
Mathematics Subject Classification:  Primary: 34C12; Secondary: 34C23; 34C25, 35B44.

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