# 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 through piecewise-smooth platforms. although there nonetheless stay many gaps within the mathematical conception of such platforms. This doctoral thesis provides new effects relating to bifurcations of piecewise-smooth, non-stop, self sufficient platforms of standard differential equations and maps. quite a few codimension-two, discontinuity precipitated 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; 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 complete history part is with ease 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 platforms. notwithstanding there nonetheless stay many gaps within the mathematical idea of such structures. This doctoral thesis provides new effects relating to bifurcations of piecewise-smooth, non-stop, self reliant platforms of normal differential equations and maps.

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2. 18) has at most one limit cycle. See [Freire et al. (1998)] for a proof. Recall, a limit cycle is a periodic orbit that has a nearby trajectory that limits on the periodic orbit as either t → ∞ or t → −∞. Freire et al. 1) and observing that multiple limit cycles never coexist. 2 as fact prior to computing periodic orbits. Consequently the mathematical arguments below are much simpler than those in [Freire et al. (1998)], but still instructive and insightful. 8) denote subsets of the switching manifold.

If the orbit is stable, it encircles the repelling focus; this is known as the supercritical case. Conversely if the orbit is unstable, it encircles the attracting focus and this is known as the subcritical case. This section derives a condition governing the criticality of the bifurcation, extending the result of Freire et al. [Freire et al. (1997)] to piecewise-smooth systems. As an example, consider the piecewise-linear, continuous system: x˙ = −x − |x| + y , y˙ = −3x + y + µ . 16) has a unique equilibrium, namely an attracting √ focus at (µ, 2µ) when µ > 0 (with eigenvalues, − 21 ± 23 i) and a repelling √ focus at ( µ3 , 0) when µ < 0 (with eigenvalues, 21 ± 211 i).

This section restates the result in Sec. 14) in terms of critical multipliers as originally derived by Feigin. Known phenomena relating to border-collision bifurcations is then overviewed. 25) it is seen that for small values of µ, the fixed point, x∗(i) , exists exactly when det(I − Ai (0)) is nonsingular. If x∗(i) exists and T map (0)b(0) = 0, then the sign of µ for which the fixed point is admis- November 26, 2009 15:34 World Scientific Book - 9in x 6in Fundamentals of Piecewise-Smooth, Continuous Systems bifurcations 21 sible is determined by the sign of det(I − Ai (0)).