This paper treats bifurcations of periodic solutions in discontinuous systems of the Filippov type. Furthermore, bifurcations of fixed points in non-smooth continuous systems are addressed. Filippov's theory for the definition of solutions of discontinuous systems is surveyed and jumps in fundamental solution matrices are discussed. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. The Floquet multipliers can jump through the unit circle causing discontinuous bifurcations. Numerical examples are treated which show various discontinuous bifurcations. Also infinitely unstable periodic solutions are addressed.

/pdf/bifurcations-in-nonlinear-discontinuous-systems-2t0ff1lg78.pdf

Bifurcations in Nonlinear Discontinuous Systems

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/pdf/bifurcations-in-discontinuous-mechanical-systems-of-the-49z66wz9rc.pdf

Bifurcations in discontinuous mechanical systems of the Fillippov-type

http://www.mate.tue.nl/mate/pdfs/4576.pdf

Energy dissipation of a friction damper

Two-degree-of-freedom autonomous system with friction is analyzed numerically. The friction coefficient has been smoothened using arc tan function. The standard, but slightly modified chaos identification tools have been applied for the analyzed discontinuous system. Some interesting examples of stick-slip regular and chaotic dynamics have been illustrated and discussed.

Stick-slip dynamics of a two-degree-of-freedom system

The purpose of these lectures is to show how the topological degree theory for (non-convex) multivalued mappings can be usefully applied to differential inclusions. Namely, we shall apply it to get a topological characterization of the set of solutions and periodic solutions for some differential inclusions. We discuss these problems in the case when the considered differential inclusions are defined on Banach spaces or on proximate retracts. Recall that a proximate retract is a compact subset A of the Euclidean space ℝn such that there exists an open neighbourhood U of A in ℝn and a metric retraction r: U → A. It is well known that any compact convex subset A of ℝn or any compact C 2-manifold M ⊂ ℝn is a proximate retract. Moreover, a topological degree method for implicit differential equations and differential inclusions is presented.

Topological approach to differential inclusions

Dry friction problems lead to discontinuous differential equations, e.g. to

$$x'' + xx' + \mu \operatorname{sgn} x' + \beta ^2 x = \varphi (t),$$

where sgn γ=γγ for γ ≠ 0 and sgn (0)=[ − 1. 1]. We study existence of ω-periodic solutions of (1) in case ϕ is ω-periodic. Results forx> 0 are given in the book “Multivalued Differential Equations” (K. Deimling: De Gruyter 1992). and preliminary ones forx=0 are contained in K. Deimling “Multivalued differential equations and dry friction problems” (Proc. Conf. Differential & Delay Equations. World Sci. Publ. 1992). Based on the latter and considerable additional analysis, we give a complete description of the resonant casex=0.β=1. ϕ(t)=sint. In particular, it turned out that for μ e (π 4. 1) there is a unique globally asymptotically stable 2π-periodic solutionxgm, which necessarily has deadzones (i.e.xμ (t) ≡c in certain intervals). In addition, the nonresonant case is solved by means of degree theory for multivalued maps, since in this situation a priori bounds can be found easily.

Periodic solutions of dry friction problems

Let X be a Banach space, 2x\⊘ the nonempty subsets of X,J = [o,a]⊂R and F:J×X→2x\⊘ a multivalued map. We consider U′ ∊ F(t,u) a.e. on J, u(o) = Xp ∊ X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X0 + ∫0 t u′(s)ds on J and u′(t)∊F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ⊘ in CX (J), the space of continuous v : J → X with ∣v∣0 = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact Rδ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.

Klaus Deimling

Papers

Periodic solutions of dry friction problems

On solution sets of multivalued differential equations