Symmetries and Regularization in Planar Piecewise Smooth Systems

In this paper, we investigate whether symmetries and reversing symmetries of planar piecewise smooth differential systems are preserved under regularization. Although regularization method is well studied in the context of nonsmooth dynamical systems, the impact of this procedure on system symmetries has remained unexplored. We establish general conditions under which a symmetry (respectively, a reversing symmetry) of the piecewise smooth system is also a symmetry (respectively, a reversing symmetry) of the regularized system. The analysis includes both linear and nonlinear regularizations. We illustrate the theory with examples including normal forms of symmetric fold-fold singularities and a discontinuous Liénard-type oscillator arising from structural mechanics.