#### For citation:

Sirotkin O. L. Oscillation modes of a linear oscillator, induced by frequency fluctuations in the form of non-Markovian dichotomous noise. *Izvestiya of Saratov University. Physics *, 2021, vol. 21, iss. 4, pp. 343-354. DOI: 10.18500/1817-3020-2021-21-4-343-354, EDN: HCQQHD

# 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 ± 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.

**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.**

*Materials and Methods:**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.*

**Results:***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.*

**Conclusion:**- Van Kampen I. G.
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