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. (c) 2005 Elsevier B.V. All rights reserved.