8.1 Introduction

Many works have been devoted to the study of the global behaviour of smooth nonlinear oscillators (Duffing oscillator [Ueda (1979)], shallow arches [Lamarque and Malasoma (1992)], [Malasoma et. al. (1994)], [Szempli?ska-Stupnicka (1969)], Lorenz's system [Lorenz (1963)], [Sparrow (1982)] or smooth maps (logistic map [Coullet and Tresser (1984)], [Feigenbaum (1978)], etc.): Both periodic, quasi-periodic or chaotic behaviour have been investigated, bifurcations, transitions, universal behaviour have been studied. At the same time a number of papers deal with "simpler, integrable" systems which seem to be "paradigms": unimodal maps [Collet and Eckmann (1980)], [Li and Yorke (1975)], Lozi's attractor [Lozi (1978)], Chua's double scroll circuit [Madan (1993)], etc. Indeed the latter are concerned with piecewise linear dynamics exhibiting chaotic behaviour via Poincar maps [Chua et. al. (1986a)], [Komuro et. al. (1991)], [Lozi (1978)], that have been build analytically.

But few studies deal with the bifurcations and the global behaviour of unsmooth systems, i.e. systems with mathematical difficulties such as discontinuities or/and multivalued differential equations. Such models are interesting from the point of view of applications: impacts, friction, shocks and constitutive laws provide unsmooth models of that type [Awrejcewicz and Delfs (1990a)], [Awrejcewicz and Delfs (1990b)], [Capecchi and Vestroni (1995)], [Deimling (1992)], [Dowell and Schwartz (1983a)], [Dowell and Schwartz (1983b)], [Ferri and Bindemann (1995)], [Mahla and Badan Palhares (1993)], [Monteiro Marques (1994)], [Moreau (1988)], [Paoli (1993)], [Paoli et. al. (1992)], [Pfeiffer (1988a)], [Popp and Stelter (1990)], [Shaw and Shaw (1989)], [Whiston (1987)].

Mathematical results are sometimes available for...