Two-parameter degenerate sliding bifurcations in Filippov systems

This paper is concerned with the extension to the case of codimension-2 degenerate sliding bifurcations of the theory of sliding bifurcations in Filippov systems presented in [M. di Bernardo, P. Kowalczyk, A. Nordmark, Bifurcations of dynamical systems with sliding: derivation of normal form mappings, Physica D, 170 (2002) 175-205]. These bifurcations were detected in experimental systems such as the dry-friction oscillator and turn out to be organising centres for branches of codimension-1 sliding bifurcations. The analysis is carried out for generic n-dimensional piecewise smooth systems. The possible degenerate scenarios are classified. It is shown that several branches of codimension-1 sliding bifurcations originate from the degenerate codimension-2 points. Such branches are appropriately classified in the degenerate crossing-sliding case. A friction oscillator is used as a representative example to illustrate and confirm the theoretical derivations. The importance is discussed of the unfolding of the degenerate sliding bifurcations for the development of continuation techniques.