Discrete chaos in Banach spaces
Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 22_Publication in policy or professional journal
Related Research Unit(s)
|Journal / Publication||Science in China, Series A: Mathematics|
|Publication status||Published - Feb 2005|
|Link to Scopus||https://www.scopus.com/record/display.uri?eid=2-s2.0-15044354626&origin=recordpage|
This paper is concerned with chaos in discrete dynamical systems governed by continuously Frechét differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiable map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.
- Banach space, Chaos, Discrete dynamical system, Marotto's theorem