A PDE approach for the invariant measure of stochastic oscillators with hysteresis

TL;DR

This paper introduces a PDE-based method to compute the invariant measure of stochastic oscillators with hysteresis, offering an efficient alternative to Monte Carlo simulations for complex nonlinear dynamics.

Contribution

It extends the PDE approach to three-dimensional bilinear elasto-plastic oscillators, enabling modeling of richer hysteretic behaviors and broadening computational tools.

Findings

Efficient computation of threshold crossing frequency.

Alternative to Rice's formula for probability estimation.

Establishment of invariant measure existence via Lyapunov functions.

Abstract

This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic elasto-plastic framework of Bensoussan et al. [SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396] from the two-dimensional elasto-perfectly-plastic oscillator to the three-dimensional bilinear elasto-plastic oscillator. By constructing an appropriate Lyapunov function, the existence of an invariant measure is established. This extension thus enables the modelling of richer hysteretic behavior and broadens the scope of PDE alternatives to Monte Carlo methods. Two applications demonstrate the method's efficiency: calculating the oscillator's threshold crossing frequency (providing an alternative to Rice's formula) and probability of serviceability.