Couple of new publications that have gone online during this summer: a preprint on Volterra-series approach to the Van der Pol oscillator driven by white noise, and a presentation on higher-order memory and inertia effects in the Onsager-Machlup fluctuation theory.

# Volterra-series approach to stochastic nonlinear dynamics: linear response of the Van der Pol oscillator driven by white noise

– R. Belousov, F. Berger, A.J. Hudspeth

http://arxiv.org/abs/1908.05313

The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further complicates their dynamics. Determination of the equation parameter values becomes then a difficult task. In a companion paper we have proposed an analytical approach to a similar problem for another classical nonlinear model—the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator’s time series.

# Higher-order memory and inertia effects in Onsager-Machlup theory of stochastic fluctuations

– R. Belousov, E. Roldan

In two seminal papers [1, 2] Onsager and Machlup introduced the celebrated theory of stochastic fluctuations for thermodynamic systems. Whereas their first contribution [1] proposes a variational principle that leads to a simple Langevin dynamics of first order in time, the second [2] extends this formalism to more complex systems that include inertia and memory effects via a second-order stochastic equation. I will present a higher-order generalization of the Onsager-Machlup theory and discuss its applications to equilibrium and nonequilibrium dynamics of currents.

[1] L. Onsager and S. Machlup, Phys. Rev. 91, pp. 1505-12, 1953.

[2] S. Machlup and L. Onsager, Phys. Rev. 91, pp. 1512-15, 1953.