On the limit of regularized piecewise-smooth dynamical systems

Ernst Hairer (Université de Genève), Nicola Guglielmi

This work deals with piecewise-smooth dynamical systems and with regularizations, where the jump discontinuities of the vector field are smoothed out in an \(\varepsilon\)-neighbourhood by using a continuous transition function. It addresses the following questions:

• does the solution of the regularization, for \(\varepsilon\to 0\), converge to a Filippov solution of the discontinuous problem?

• under which condition is the limit for \(\varepsilon\to 0\) of the regularized solution independent of the transition function?

Emphasis is put on the situation, where there is non-uniqueness of solutions for the discontinuous problem. The results are complemented by numerical simulations.

This work is a continuation of the results in the publications

[1] N. Guglielmi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems. SIAM J. Appl. Dyn. Syst. 14(3) (2015) 1454–1477.

[2] N. Guglielmi and E. Hairer, Solutions leaving a codimension-2 sliding. Nonlinear Dynamics 88(2) (2017) 1427-1439

which can be downloaded from http://www.unige.ch/~hairer/preprints.html.