Oscillation modes of a linear oscillator, induced by frequency fluctuations in the form of non-Markovian dichotomous noise

Background and Objectives: A set of differential equations is derived for the probability density functions of the phase coordinates of dynamic systems featuring parametric fluctuations in the form of non-Markovian dichotomous noise having arbitrary distribution functions for life at the states $\pm 1$. As an example, the first moment of the phase coordinate of an oscillator was calculated, its perturbed motion being described by a stochastic analogue of the Mathieu – Hill equation. It is intended to show that linear dynamical systems subjected to parametric fluctuations are capable of producing states not appropriate to deterministic modes. Materials and Methods: The problem is solved using the method of supplementary variables which facilitates, through an expansion of the phase space, transformation of the non-Markovian dichotomous noise into a Markovian one. Results: It has been established that sustained beating oscillations of the amplitudes are observed provided the dichotomous noise structure contains the life time distribution function as a sum of two weighted exponents describing two states of the system, i.e. $\pm 1$. Conclusion: As a matter of fact, a Markovian simulation of the oscillator features only damped oscillations. Properties of the process in question being delta-correlated or Gaussian are not utilized. The calculations are made using ordinary differential equations with no integral operators being involved.