Scientific literature provides several models describing the dynamics of different types of oscillators, both in deterministic and stochastic setting. The majority of the examples of stochastic oscillator are obtained by introducing a noisy ingredient in an underlying deterministic model. This noisy component may be an additive and/ or a multiplicative noise, a random frequency, a random damping, and so on (see [1] and references therein for a survey). A numerical issue tipically investigated in this setting is the attitude of classical numerical schemes for SDEs to preserve of long term properties of particular oscillators, see [2]. The variety of models inspires the construction of specific methods to solve particular stochastic equations, which describe oscillatory dynamics, see [3]. We present our results in terms of a study of conservation, see [4], and of providing targeted methods for very popular examples of oscillators, see [5, 6].

References

1. M. Gitterman. The noisy oscillator, The First Hundred Years, From Einstein Until Now. World Scientific, 2005.

2. K. Burrage, I. Lenane and G. Lythe. Numerical methods for second-order stochastic differential equations. SIAM Journal of Scientific Computing, 29, 245–264, 2007.

3. D. Cohen D. and M. Sigg. Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations, Numerische Mathematik, 121, pp. 1–29, 2012.

4. R. D’Ambrosio and C. Scalone. On the numerical structure preservation of nonlinear damped stochastic oscillators. Numerical Algorithms, 86, pp. 933-–952, 2021.

5. R. D’Ambrosio and C. Scalone. Filon quadrature for stochastic oscillators driven by time-varying forces. Applied Numerical Mathematics. https://doi.org/10.1016/j.apnum.2021.06.005, 2021.

6. R. D’Ambrosio and C. Scalone. A Magnus integrator for nonlinear stochastic oscillators with non-constant frequency, submitted.