Discrete chaos in Banach spaces  

作　　者：SHI Yuming CHEN Guanrong 

机构地区：[1]Department of Mathematics,Shandong University,Jina 250100,China Department of Electronic Engineering,City University of Hong Kong,China

出　　处：《Science China Mathematics》2005年第2期222-238,共17页中国科学：数学（英文版）

基　　金：This work was partially supported by the National Natural Science Foundation of China(Grant No.10071043);the Hong Kong Research Grants Council under the CERG grant CityU 1115/03E;the NSF Shandong Research Funds for Young Scientists(Grant No.03BS094);the Shandong University Scientific Research Funds for Young Staff.The authors are very happy to have this opportunity to thank the referees for helpful remarks.

摘　　要：This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t 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 differentiate 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.This paper is concerned with chaos in discrete dynamical systems governed by continuously Frechet 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.

关 键 词：chaos  DISCRETE DYNAMICAL system  BANACH space  Marotto's theorem. 

分 类 号：N[自然科学总论]