Dynamic system driven by a retarded force acting as colored noise

Statistical properties of deterministic chaos are investigated in systems with a variational structure, driven by a delayed force. The delayed force is chosen to be periodic with respect to the amplitude of the motion. When the period is much smaller than the response time of the system, ${\mathit{T}}_1$, the chaotic solutions are shown to be noiselike, i.e., they have statistical properties very similar to those of random chaos resulting from nonlinear Langevin equations. As the period increases to ${\mathit{T}}_1$, the delay term is shown to act in a manner similar to a colored noise in all cases of large delay, but with different statistics. However, when the delay is about equal to ${\mathit{T}}_1$, very special effects like phase transitions appear, giving rise to new peaks in the probability distribution. The ‘‘phase transition’’ is shown to be well described by an approximate Fokker-Planck equation. An analytical expression is proposed for the drift term in this equation, which agrees with numerical computations.