The stochastic Mathieu's equation

Abstract

In this manuscript, we consider generalizations of the classical Mathieu's equation to stochastic systems. Unlike previous works, we focus on internal frequencies that vary continuously between periodic and stochastic variables. By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide range of parameters to quantify how the growth rates vary in parameter space. In the nearly periodic limit, we recover the same growth rates as the classical Mathieu's equation. As the stochasticity increases, the maximum growth decreases and the unstable region broadens, which signifies that the stochasticity can both stabilize and destabilize a system in relation to the deterministic limit. In addition to the classical parametric modes, there is also an unstable stochastic mode that arises only for moderate stochasticity. The power spectrum of the solutions shows that an increase in stochasticity tends to narrow the width of the subharmonic peak and increase the decay away from this peak.