Generation of chaos by dynamical systems defined in ordered banach spaces*

Abstract

We investigate the existence of solutions with “complicated” dynamics in continuous non-smooth many (finite or infinite) - dimensional models with non-negative variables. We assume that a discrete-time dynamical system is defined in a closed solid normal cone K of a Banach space. This cone induces an ordering on K. The dynamical system is defined by the superposition of continous functions F: k × K→k and in: x ε k→(x,x) as follows F(.)=F(in(.)). The mapping F(.,.) is a monotone increasing positively homogeneous function in the first variable and a monotone decreasing function in the second variable. We prove the existence of chaos under the following main assumptions: (i) the origin is an ejecting fixed point of the mapping F(.,0), (ii) there exist x εK-{0} such that F n x)=0 for some n, and (iii) the mapping F(.) is locally monotone at the origin. The effectiveness of the method is demonostrated on the model of population dynamics with age structure and non-overlapping generations and the model of an ecological system of many cooperative species.