On relationships between the Lyapunov spectrum and the Morse spectrum

Abstract

We consider dynamical systems on manifolds and explore the relationship between the Lyapunov and the Morse spectra. The concept of Morse spectrum is based on a study of the exponential growth rates associated with the t — T chains in the chain recurrent components of the flow on the projective bundle. It is known that the Morse spectrum contains the Lyapunov spectrum and the Morse spectrum is a union of closed intervals whose boundary points are Lyapunov exponents. Here we present a treatment for the case of a flow on a two dimensional compact manifold under the assumption that the chain recurrent components coincide with the limit sets. For this case we investigate the equality of the two spectra and provide methods to calculate them in practice. For planar flows there are only three possible types of nonwandering sets: fixed points, periodic orbits and cycles. We consider the case where these are isolated chain recurrent components. For these three types of sets we perform a case by case study by first linearizing the system over the solutions and then by computing the Lyapunov and the Morse spectra. The Lyapunov spectrum over a fixed point consists of the real parts of the eigenvalues of the Jacobian matrix of partial derivatives. For periodic orbit, they are the characteristic exponents. For a cycle we prove that the Lyapunov spectrum consists of the Lyapunov exponents for the fixed points in the cycle. Belgrade's theorem gives a relationship between the chain recurrent components in the base and the chain recurrent components for the flow in the projective bundle. We use this theorem to find Morse spectrum. We prove that for an isolated fixed point, a connected set of fixed points and a periodic orbit the Lyapunov and the Morse spectra coincide. For a