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In this work, we consider systems that are subjected to intermittentinstabilities due to external stochastic excitation. These intermittentinstabilities, though rare, have a large impact on the probabilistic responseof the system and give rise to heavy-tailed probability distributions. Bymaking appropriate assumptions on the form of these instabilities, which arevalid for a broad range of systems, we formulate a method for the analyticalapproximation of the probability distribution function (pdf) of the systemresponse (both the main probability mass and the heavy-tail structure). Inparticular, this method relies on conditioning the probability density of theresponse on the occurrence of an instability and the separate analysis of thetwo states of the system, the unstable and stable state. In the stable regimewe employ steady state assumptions, which lead to the derivation of theconditional response pdf using standard methods for random dynamical systems.The unstable regime is inherently transient and in order to analyze this regimewe characterize the statistics under the assumption of an exponential growthphase and a subsequent decay phase until the system is brought back to thestable attractor. The method we present allows us to capture the statisticsassociated with the dynamics that give rise to heavy-tails in the systemresponse and the analytical approximations compare favorably with direct MonteCarlo simulations, which we illustrate for two prototype intermittent systems:an intermittently unstable mechanical oscillator excited by correlatedmultiplicative noise and a complex mode in a turbulent signal with fixedfrequency, where multiplicative stochastic damping and additive noise modelinteractions between various modes.

Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes