Li-Yorke chaos in weak topology of the n-dimensional linear systems

Abstract

The paper deals with the complicated dynamics of the linear systems on the n-dimensional Euclidean space with the weak topology. A weak topology defined by a family of semi-norms is found to systematically investigate the chaotic dynamics in weak topology. It is rigorously proved that the one-dimensional linear systems are not Li-Yorke chaotic in weak topology and the n-dimensional linear systems with n>1 exhibit weak Li-Yorke chaos in weak topology. Moreover, some necessary and sufficient conditions for weak Li-Yorke chaos in weak topology of n-dimensional linear systems with n≥2 are established by proving the existence of a semi-irregular or an irregular vector in weak topology. As an application, the dynamics in weak topology for the three-dimensional linear systems with real distinct, real repeated and complex conjugate eigenvalues are classified.