Determination of Lyapunov exponents by weak solutions of Fokker-Planck equations

Abstract

To determine the asymptotic stability of the zero position of a linear dynamic system one has to calculate the top Lyapunov exponent of the system. For this purpose, the state vector of the system is projected onto the unit hypersphere of Hasminskji's angle coordinates, which allows separation of the non-stationary solution part of the amplitude and representation of the top Lyapunov exponent in terms of the expectation values calculable from the density of the stationary angle processes. It turns out that these expected values are coefficients of orthogonal expansion series applied to the associated Fokker-Planck equation. Utilizing symbolic computation, the projection and the expansion can be performed automatically. The remaining numerical evaluation is restricted to the solution of weakly coupled linear equations.