Description of Chaotic Motion by an Invariant Distribution at the Example of the Driven Duffing Oszillator

Summary

For chaotic motion the calculation of single trajectories under slightly different conditions results in completely different time series although the motion is bounded on the same strange attractor. Due to the exponential divergence of initially neighboring trajectories the state of the system becomes unpredictable in the long time behavior inspite of deterministic equations. According to this inherent property of chaotic systems the calculation of the system’s state for a certain time point contains little information about the variety of possible states. A more adequate description is given by a probability distribution for the state space variables within the area of attraction because it represents all possible trajectories. For autonomous chaotic systems this distribution is invariant. To approximate these probability distributions an additional stochastic excitation of small intensity is taken into account. This not only reduces problems concerning numerical convergence of the solution but is also more realistic because a finite level of noise is present everywhere in reality. Due to the additional noise excitation the calculation of the probability density can be achieved directly by the solution of a Fokker-Planck-Equation, if the excitation is given by white noise. For decreasing noise intensities the result tends to the distribution for the stochastically unperturbed system.