In this survey we discuss current directions of research in the dynamics of nonsmooth systems, with emphasis on bifurcation theory. An introduction to the state-of-the-art (also for non-specialists) is complemented by a presentation of main open problems. We illustrate the theory by means of elementary examples. The main focus is on piecewise smooth systems, that have recently attracted a lot of attention, but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities. This extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context. A dedicated section is devoted to concrete applications that stimulate the development of the field. This survey is concluded by an extensive bibliography.

Abstract. In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for the Lipschitz systemẋ = εg (t, x, ε). Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct, for small ε, the curves of dependence of the amplitude of asymptotically stable 2π-periodic solutions of the nonsmooth van der Pol oscillatorü+ε (|u| − 1)u+(1+aε)u = ελ sin t, on the detuning parameter a and the amplitude of the perturbation λ. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillatorü + ε`u 2 − 1´u + (1 + aε)u = ελ sin t, which were first constructed by Andronov and Witt.Key words. Periodic solution, asymptotic stability, averaging theory, nonsmooth differential system, nonsmooth van der Pol oscillator. AMS subject classifications. 34C29, 34C25, 47H11.1. Introduction. In the present paper we study the existence, uniqueness and asymptotic stability of the T -periodic solutions for the systeṁwhere ε > 0 is a small parameter and the function gis T -periodic in the first variable and locally Lipschitz with respect to the second one. As usual a key role will be played by the averaging functionand we shall look for those periodic solutions that starts near some v 0 ∈ g −1 0 (0). In the case that g is of class C 1 , we remind the periodic case of the second Bogolyubov's theorem ([6], Ch. 1, § 5, Theorem II) which represents a part of the averaging principle: det (g 0 ) ′ (v 0 ) = 0 assures the existence and uniqueness, for ε > 0 small, of a T -periodic solution of system (1.1) in a neighborhood of v 0 , while the fact that all the eigenvalues of the Jacobian matrix (g 0 ) ′ (v 0 ) have negative real part, provides also its asymptotic stability. This theorem has a long history and it includes results by Fatou Second Bogolubov's theorem gave a theoretical justification of resonance phenomenons in many real physical systems. The most significant example is the classical lamp oscillator whose scheme is drawn at Fig. 1.1 and whose current u is described by the following second order differential equation Fig. 3-5).where R = εR 0 , M = εM 0 , ω 2 = 1 + εb, F (t) = ελ sin t, ε > 0 is assumed to be small and the lamp characteristic is drawn at Fig 1.2a. The analysis of bifurcation

This paper develops an analytic framework to design both stress-controlled and displacement-controlled T -periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t->(e(t),p(t)), where ei(t) and pi(t) are the elastic and plastic deformations of spring i, that satisfies the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the elastic component t->e(t) always converges to a T-periodic function. The achievement of the present paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. In this way we obtain an analogue of the Frederick-Armstrong theorem from continuum mechanics.

The paper deals with a T -periodically perturbed autonomous system in R n of the forṁwith ε > 0 small. The main goal of the paper is to provide conditions ensuring the existence of T -periodic solutions to (PS) belonging to a given open setThis problem is considered in the case when the boundary ∂W of W contains at most a finite number of nondegenerate T -periodic solutions of the autonomous systemẋ = ψ(x). The starting point of our approach is the following property due to Malkin: if for any T -periodic limit cycle x0 ofẋ = ψ(x) belonging to ∂W the so-called bifurcation function fx 0 (θ), θ ∈ [0, T ], associated to x0, see (1.11), satisfies the condition fx 0 (0) = 0 then the integral operatordoes not have fixed points on ∂W for all ε > 0 sufficiently small. By means of the Malkin's bifurcation function we then establish a formula to evaluate the Leray-Schauder topological degree of I − Qε on W. This formula permits to state existence results that generalize or improve several results of the existing literature. In particular, we extend a continuation principle due to Capietto, Mawhin and Zanolin where it is assumed that ∂W does not contain any T -periodic solutions of the unperturbed system. Moreover, we obtain generalizations or improvements of some existence results due to Malkin and Loud.

By a nondegenerate k-parameterized family K of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on K system equals to k. Bifurcation of asymptotically stable periodic solutions from K is well studied in the literature and different conditions have been proposed depending on whether the algebraic multiplicity of +1 is k or not (by Malkin, Loud, Melnikov, Yagasaki). In this paper we assume that the later is unknown. Asymptotic stability can not be understood in this case, but we demonstrate that the information about uniqueness of periodic solutions is still available. Moreover, we show that differentiability of the right hand sides is not necessary for the results of this kind and our theorems are proven under a kind of Lipschitz continuity.1991 Mathematics Subject Classification. 34C29; 34C25; 58F22.