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

Abstract

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 ± 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. ±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.

Keywords

• non-markovian processes
• supplementary variables
• linear oscillator
• induced oscillations
• beating oscillations