Download Bifurcations and chaos in piecewise-smooth dynamical systems by Zhusubaliyev T., Mosekilde E. PDF

By Zhusubaliyev T., Mosekilde E.

Technical difficulties frequently result in differential equations with piecewise-smooth right-hand aspects. difficulties in mechanical engineering, for example, violate the necessities of smoothness in the event that they contain collisions, finite clearances, or stick-slip phenomena. platforms of this sort can exhibit a wide number of complex bifurcation situations that also lack an in depth description. This publication offers the various attention-grabbing new phenomena that you'll discover in piecewise-smooth dynamical platforms. the sensible value of those phenomena is confirmed via a chain of well-documented and practical functions to switching energy converters, relay platforms, and kinds of pulse-width modulated keep watch over structures. different examples are derived from mechanical engineering, electronic electronics, and monetary business-cycle concept. the themes thought of within the booklet contain abrupt transitions linked to converted period-doubling, saddle-node and Hopf bifurcations, the interaction among classical bifurcations and border-collision bifurcations, truncated bifurcation eventualities, period-tripling and -quadrupling bifurcations, multiple-choice bifurcations, new forms of direct transitions to chaos, and torus destruction in nonsmooth platforms. inspite of its orientation in the direction of engineering difficulties, the booklet addresses theoretical and numerical difficulties in adequate aspect to be of curiosity to nonlinear scientists typically.

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Extra info for Bifurcations and chaos in piecewise-smooth dynamical systems

Example text

The separation can be written: In setting v = u, we obtain a = — , and therefore: Hence — v* e 3F2(w) and u* + v* E dFt(u). Whence we have the desired decomposition, u* = u^ + «*, with uf = u* + v* and u% = —v*. Finally, after considering the subdifferential of a sum of functions, let us examine the subdilferential of a composite function. s. V-+ Y with transpose A*: Y* -> K*. Let Fe F(F); the function Fo A: V-+R belong to F(F). 7. Lef f/zere fe a point Au where F is continuous and finite.

Examples Let s$ be a subset of V and x^ its indicator function. c. and positively homogeneous on V*, termed the support function ofs/. We have seen in Section 3 that y*f= Xso^- I*1 particular, J2/ and co s£ will have the same support function. Here is an extremely useful example of dual convex functions. || the norm of V and by || . ||# the norm of V*. Then V and V* are in duality and we supply them with the weak topologies a(V, V*) and a(V*, V). We take an even function

* its conjugate convex function which also belongs to r0(R).

14) which requires that the family um is bounded in V. 3. 2). 3. This has been given by Brezis [1] and exploits an idea of G. J. Minty [1]. 1. 3. 19) is called pseudo-monotone; c/H. Brezis[l],J. L. Lions [3]. 2. We find in Lions [3] many existence theorems for more general variational inequalities than the foregoing. Special cases. 1 we can deduce several special cases. 1. 5). For allfof V*, there isauetf such that Proof. 1, with

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