# Bifurcations in Piecewise-smooth Continuous Systems (World by David John Warwick Simpson

By David John Warwick Simpson

Real-world platforms that contain a few non-smooth switch are usually well-modeled via piecewise-smooth platforms. notwithstanding there nonetheless stay many gaps within the mathematical thought of such platforms. This doctoral thesis provides new effects relating to bifurcations of piecewise-smooth, non-stop, self reliant structures of normal differential equations and maps. a number of codimension-two, discontinuity triggered bifurcations are opened up in a rigorous demeanour. numerous 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; particularly, 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 complete history part is comfortably supplied for people with very little adventure in piecewise-smooth platforms.

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Example text

2002); di Bernardo et al. (2008a)]. (For a different approach see [Bemporad et al. ) The word canonical instead of normal is used because in smooth bifurcation theory, normal forms have a precise meaning [Kuznetsov (2004); Arnold (1988)]. This section explains why the canonical form may only be obtained when the linearizations of the left and right half-systems have no eigenspace tangent to the switching manifold at the bifurcation. Call an N ×N matrix whose ith column is ei−1 for all i = 1, a companion matrix.

2004)]. Also, a sequence of nested spirals has been observed in a three-dimensional, piecewise-linear, continuous map that models an electrical circuit [Bonatto and Gallas (2008)]. For basic border-collision bifurcations in three-dimensional maps, see [Roy and Roy (2008)]. Particularly when CL and CR have complex-valued multipliers, it is November 26, 2009 24 15:34 World Scientific Book - 9in x 6in Bifurcations in Piecewise-Smooth, Continuous Systems of interest to compute regions in parameter space within which periodic solutions of a particular period or rotation number exist and are attracting.

As shown in [di Bernardo et al. − − (1999)], the 2-cycle is admissible for one sign of µ if and only if σL + σR − is odd, where σi denotes the number of real multipliers of Ai (0) less than -1. 14) that exists for small values of µ, the 2-cycle shrinks linearly to a point as µ → 0. Consequently, the border-collision bifurcation may resemble a period-doubling bifurcation. Effective period-doubling via a border-collision bifurcation has recently been suggested to play an important role in cardiac dynamics [Zhao et al.