Numerical Experiments on Trajectory Instabilities in Dissipative Systems

It is confirmed by computer experiments that a probabilistic approach is necessary for description of the dynamics of dissipative systems with unstable trajectories, although given equations of motion are deterministic. How initially close points in phase space do spread with time is examined. From this it is shown that the conditional probabilities can be defined naturally by introducing the concept of coarse graining in phase space and time, and moreover they satisfy a definition of a Markov process. The effect of external random forces is investigated by adding the Langevin forces to the equations of motion. It is suggested that the statistical properties of unstable dissipative systems are determined mainly by the inherent stochasticity due to the nonlinearity of the systems, unlike equilibrium systems whose statistical properties are governed by external random forces coming from contact with a heat reservoir.