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A generalization of Desch-Schappacher-Webb criteria for chaos. (English) Zbl 1084.47033

Contrary to popular perception, linear dynamical systems can be chaotic, but for this they must be infinite-dimensional. This area has recently received a lot of attention and was found to be closely related to the theory of hypercyclic operators. In fact, topologically chaotic dynamical systems (in the sense of Devaney) are exactly those having an orbit dense in the state space (and thus hypercyclic) and having a dense set of periodic points. In the seminal paper of W. Desch, W. Schappacher and G. F. Webb [Ergodic Theory Dyn. Syst. 17, No. 4, 793–819 (1997; Zbl 0910.47033)], the authors formulated useful sufficient conditions for chaoticity of a \(C_0\)-semigroup of linear operators in terms of the structure of the point spectrum and eigenvectors of its generator.
In the present paper, it is shown that by discarding one (and the most cumbersome to check) condition from the Desch-Schappacher-Webb criterion, one still obtains a chaotic semigroup but with chaotic behaviour occurring only in a certain infinite-dimensional subspace of the original state space. Such semigroups are termed sub-chaotic. Since any ball centred at the origin intersects the subspace in which the semigroup is chaotic, sub-chaotic semigroups display sensitive dependence on initial conditions which is a trademark of chaos.
The authors proceed by giving conditions which prevent a semigroup from being chaotic in any subspace and conclude the paper providing a comprehensive analysis of sub-chaoticity of the birth as well as the death systems of population dynamics, which are related to forward and backward shift operators.

MSC:

47D06 One-parameter semigroups and linear evolution equations
92D25 Population dynamics (general)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
47N60 Applications of operator theory in chemistry and life sciences
47A16 Cyclic vectors, hypercyclic and chaotic operators
37N25 Dynamical systems in biology

Citations:

Zbl 0910.47033
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