On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems

2007/06/09 by Enciso, German A.
#34C12 #35K57 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.0706.1331

Abstract

A standard result by Smale states that n dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane. An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria.

Related

• Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems
• A cooperative system which does not satisfy the limit set dichotomy
• Longtime behaviors of a reducible cooperative system with nonlocal diffusions and free boundaries
• Instability of equilibrium and convergence to periodic orbits in strongly 2-cooperative systems
• Convergence to periodic orbits in 3-dimensional strongly 2-cooperative systems