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Title:
Spontaneously stochastic solutions in dynamical systems with singularities
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Abstract:
We consider a class of dynamical systems described by ordinary differential equations with an isolated singularity, where the singularity is characterized by the lack of Lipschitz continuity. Singularities are common in applications both for ODEs (e.g., particle collisions) and PDEs (e.g., finite-time blowup in fluid models). The fundamental obstacle is that solutions cannot be continued past the singularity uniquely: typically, there are infinitely many solutions. The conventional way to proceed is to define a regularization limit, such as vanishing viscosity or noise. It turns out that there are structurally stable situations when such a limit is not sensitive to a particular form of regularization. This is explained by the analysis of attractors for the rescaled (non-singular) dynamical system and their ergodic properties. What is even more surprising is that solutions in this limit may become probabilistic (spontaneously stochastic) with the unique probability measure past the singularity. We will present rigorous results and discuss applications of this phenomenon. This is joint work with Alexei A. Mailybaev (IMPA, Rio de Janeiro) and Artem Raibekas (UFF, Rio de Janeiro).
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