By David John Warwick Simpson

Real-world structures that contain a few non-smooth switch are usually well-modeled through piecewise-smooth structures. even though there nonetheless stay many gaps within the mathematical concept of such structures. This doctoral thesis provides new effects concerning bifurcations of piecewise-smooth, non-stop, independent structures of normal differential equations and maps. a variety of codimension-two, discontinuity precipitated bifurcations are spread out 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; 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 historical past part is very easily supplied for people with very little adventure in piecewise-smooth structures.

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Real-world structures that contain a few non-smooth switch are usually well-modeled via piecewise-smooth structures. in spite of the fact that there nonetheless stay many gaps within the mathematical conception of such platforms. This doctoral thesis offers new effects relating to bifurcations of piecewise-smooth, non-stop, self sustaining structures of standard differential equations and maps.

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4: Sketches of eigenvalues under variation of a system parameter near a local bifurcation in a smooth system, panel (a), and a piecewise-smooth, continuous system, panel (b). called the eigenvalue path. Let n ˆ denote the number of values of s for which (1 − s)AL + sAR has one or more eigenvalues on the imaginary axis. The idea is that n ˆ represents the possible degree of complexity of the discontinuous bifurcation. It is conjectured that if n ˆ = 0 then no bifurcation occurs at µ = 0. This has been verified for many examples but to the author’s knowledge has yet to be proved [Leine (2006); Leine and Nijmeijer (2004)].

In order to calculate the period for our situation, we must first understand the shape of the Hopf cycle. As the Hopf cycle shrinks to a point, it becomes better approximated by the periodic orbit in the corresponding piecewise-linear system. 16)), shrinks to zero in a self-similar manner due to the inherent scaling symmetry. Thus an arbitrarily small Hopf cycle generated in this discontinuous bifurcation takes the shape of the periodic orbit in the corresponding piecewise-linear system, hence the orbit consists of two spiral segments, see Fig.

Here the discontinuity maps will be continuous. Trajectories that enter the attracting November 26, 2009 15:34 World Scientific Book - 9in x 6in 28 bifurcations Bifurcations in Piecewise-Smooth, Continuous Systems Σ Σ (a1) (a2) (b1) p0 p (b2) Σ (a3) (a4) g(p0 ) g(p0 ) g(p0 ) g(p0 ) p Σ p0 p (b3) p0 p p0 (b4) Fig. 8: The top panels show phase portraits for Filippov systems with four different grazing scenarios: panel (a1) - grazing-sliding, panel (a2) - regular grazing, panel (a3) - continuous grazing, panel (a4) - grazing at a corner.