The notion of distributional chaos has been rencently added to the study of the linear dynamics of operators and $C_0$-semigroups of operators.
A criterion for distributional chaos and the existence of a dense distributionally irregular manifold for a $C_0$-semigroup has been recently obtained in [1].
We apply it to several examples of $C_0$-semigroups that were already known to be chaotic in the sense of Devaney. In particular we will study distributional
chaos for birth-and-death processes with proliferations.
Joint work with J. Alberto Conejero.
[1] Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, and Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators .
Commun. Pure Appl. Anal., 12(5):2069--2082, 2013.
[2] Xavier Barrachina and José A. Conejero. Devaney chaos and distributional chaos in the solution of certain partial differential equations.
Abstr. Appl. Anal., 2012:Art. ID 457019, 11, 2012.