By David John Warwick Simpson
Real-world platforms that contain a few non-smooth switch are frequently well-modeled through piecewise-smooth platforms. in spite of the fact that there nonetheless stay many gaps within the mathematical concept of such structures. This doctoral thesis offers new effects relating to bifurcations of piecewise-smooth, non-stop, self sufficient structures of normal differential equations and maps. a variety of codimension-two, discontinuity brought on bifurcations are opened up in a rigorous demeanour. a number of of those unfoldings are utilized to a mathematical version of the expansion of Saccharomyces cerevisiae (a universal yeast). the character of resonance close to border-collision bifurcations is defined; specifically, the curious geometry of resonance tongues in piecewise-smooth non-stop maps is defined intimately. Neimark-Sacker-like border-collision bifurcations are either numerically and theoretically investigated. A entire history part is comfortably supplied for people with very little adventure in piecewise-smooth structures.
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Extra resources for Bifurcations in Piecewise-smooth Continuous Systems (World Scientific Series on Nonlinear Science Series a)
Determining the existence and admissibility of cycles of higher period in terms of the multipliers of AL and AR is a far more difficult task, see Chapter 6. For an extension of Feigin’s results to piecewise-smooth, discontinuous maps, see [Dutta et al. (2008)]. Under smooth parameter variation, periodic cycles may collide with a switching manifold. e. 11). This is shown in Chapter 6. Hence Feigin’s results may also be used to determine the admissibility of n and 2n-cycles local to this border-collision bifurcation [di Bernardo et al.
Then PL (c1 ) = eT 2 ϕT (0, c1 ; µ) where T is the L transition time obtained by solving eT 1 ϕT (0, c1 ; µ) = 0. Unfortunately this X last equation is transcendental (like e cos(X) = 2). Consequently qµ has no known explicit form. This exemplifies a recurring theme regarding discontinuous bifurcations: local dynamics may be determined by global properties of a piecewise-linear system. 21), are attracting and repelling foci for arbitrarily small values of µ. In this scenario, near the discontinuous bifurcation there exists a unique admissible equilibrium.
Now suppose there are no sliding regions in a neighborhood of the grazing point. When the grazing trajectory, Γ, is admissible, as in Fig. 8(a2), the grazing point is known as a regular grazing point. The criterion of November 26, 2009 15:34 World Scientific Book - 9in x 6in Fundamentals of Piecewise-Smooth, Continuous Systems bifurcations 29 no sliding generally corresponds to adding a codimension to the bifurcation, however in practice underlying physical assumptions often eliminate the possibility for sliding motion.