# Bifurcation and Chaos in Nonsmooth Mechanical Systems: v. 45 by J. Awrejcewicz, Claude-Henri Lamarque By J. Awrejcewicz, Claude-Henri Lamarque

This booklet provides the theoretical body for learning lumped nonsmooth dynamical platforms: the mathematical tools are recalled, and tailored numerical tools are brought (differential inclusions, maximal monotone operators, Filippov idea, Aizerman conception, etc.). instruments on hand for the research of classical gentle nonlinear dynamics (stability research, the Melnikov process, bifurcation eventualities, numerical integrators, solvers, etc.) are prolonged to the nonsmooth body. Many types and purposes bobbing up from mechanical engineering, electric circuits, fabric habit and civil engineering are investigated to demonstrate theoretical and computational advancements.

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A general solution to the Eq. (1-45) for arbitrary initial conditions AQ, VQ = y'o > 0 has the form t/o = —sina 0 * + (A —^cosao* T —o, (1-49) which is valid in the interval 0 < t < t\, where y'o > 0. 51) in (1-49) we define the initial conditions in the next time interval t\ < t < t 0, are as follows: A2 =2/1 (£2), v2 = 0.

4 An operator A on H is a maximal monotone operator iff i) A is monotone on H, ii) A is maximal in the set of all the monotone operators on H. Let us notice that "maximal" is associated to inclusion of graphs; AcB iff Vz € D(A), x € D(B) and Ax C Bx. 33) Now let us consider a maximal monotone operator A on H. 5 The section of Ax for x € H, denoted A°x, is defined by A°x = projAx(0). 34) This is the projection on Ax in the sense of the theorem of the projection on a convex set. Details, examples and properties of maximal monotone operators could be found in [Brezis (1973)] or [Brezis (1992)].

66) Thus, for given F, v, b, h, XQ and io, we seek functions x and from [0, T] to R such that the Eqs. 60). 66) and the following equation is obtained: x(t) + a{x(t) - v(t))h{t - b{t, x-v))£ F(t, x(t),x(t)). 66) is equivalent to the particular case of Coulomb model. 68) where y verifies the initial condition 1/(0) = x0. 69) F is continuous with respect to all its arguments and Lipschitzcontinuous with respect to its last two arguments x and x. 67). In the particular case of Coulomb's friction, x is unique.