# American Institute of Mathematical Sciences

July  2006, 14(3): 549-578. doi: 10.3934/dcds.2006.14.549

## Global attractivity, I/O monotone small-gain theorems, and biological delay systems

 1 Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, United States, United States

Received  October 2004 Revised  March 2005 Published  December 2005

This paper further develops a method, originally introduced by Angeli and the second author, for proving global attractivity of steady states in certain classes of dynamical systems. In this approach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed.
Citation: G. A. Enciso, E. D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 549-578. doi: 10.3934/dcds.2006.14.549
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