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October  2005, 12(5): 959-972. doi: 10.3934/dcds.2005.12.959

A generalization of Desch--Schappacher--Webb criteria for chaos

Jacek Banasiak 1, and  Marcin Moszyński 2,

1. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa
2. 

Wydział Matematyki Informatyki i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland

Received  May 2004 Revised  July 2004 Published  February 2005

In [8] the authors proved that a linear dynamical system $\mathcal T$ on a Banach space $X$ is topologically chaotic if there exists a selection of eigenvectors of the generator of $\mathcal T$, that is analytic in some open set of a complex plane that meets the imaginary axis, and such that a non-degeneracy condition holds. In this paper we show that if we drop the last assumption, then $\mathcal T$ is still chaotic albeit in a possibly smaller, but still infinite-dimensional, $\mathcal T$-invariant subspace of $X$. Such kind of chaotic behaviour we shall call subspace chaos. We also present criteria that allow to rule out subspace chaos in certain dynamical systems and discuss simple but instructive examples where these criteria are applied to the birth, as well as the death, type systems of population dynamics.
Keywords: Topological chaos, infinite-dimensional linear systems, hypercyclic operators, birth-and-death systems..
Mathematics Subject Classification: 34G10, 37B99, 47A16, 47D0.
Citation: Jacek Banasiak, Marcin Moszyński. A generalization of Desch--Schappacher--Webb criteria for chaos. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 959-972. doi: 10.3934/dcds.2005.12.959
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