A numerical method for calculating Lyapunov exponents of discontinuous dynamical systems

Discontinuous dynamical systems can be divided into three types: systems with a discontinuous Jacobi matrix, with a discontinuous vector field or with states discontinuities (jumps). The first two kinds are also called as Filippov systems. For such systems, the transitional matrix is deduced from the cocycle at the discontinuous instants, and then a numerical method for calculating Lyapunov exponents is introduced. The numerical method is based on the conventional algorithm for calculating Lyapunov exponents of smooth systems; therefore it is adequate numerical calculation. As an example, the dynamical behavior of an oscillator with dry friction is analyzed by calculating the Lyapunov exponents. Since the oscillator system has discontinuity both in the Jacobi matrix and in the vector field, it is a complex discontinuous system.