Noise and topology in driven systems -- an application to interface dynamics (Q2888617)

The authors start with one of the models from the interface dynamics, describing the interaction between the phase \(\phi\) and the position \(r\) of the wall. The study of such a system reveals that it has a saddle-node bifurcation at \(f=f_s\) (where \(f\) is a constant depinning, or tilt, force) and, furthermore, homoclinic orbits (leaving an unstable fixed point and returning to it). The presence of noise is well studied. In earlier papers the discrete dynamical system of the form \(x_{i+1}=x_i-\varepsilon+x_i^2+\xi_i+h(x_i)\) was considered, where \(x_i\in \mathbb R\), \(\varepsilon\in \mathbb R\) is the bifurcation parameter, \(\xi_i\) is the noise, and \(h(x)\) is a function vanishing in the neighborhood of \(x=0\) but is such that orbits return to a neighborhood of 0. In this article, the authors introduce a parametrization \(f=1-\varepsilon^2\). There are several easier cases. First, the authors consider the problem of depinning from a periodic potential without the phase \(\phi\), then the case when the temperature is zero, i.e., the noise terms are absent. The phase space of the corresponding equation is the torus \((r,\phi)\in [0,2\pi)\times [0,\pi)\). The study of the general case of the problem seems to be new. Its description and the results for the one- and two-dimensional cases contain in Section 5. The intermittent behavior of the system is explained by the saddle-node methods with taking into account global topological aspects. It is shown also that the mean velocity of the system is a non-monotone function.