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

By David John Warwick Simpson

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 thought of such platforms. This doctoral thesis provides new effects relating to bifurcations of piecewise-smooth, non-stop, independent platforms of normal differential equations and maps. numerous codimension-two, discontinuity triggered 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 finished history part is very easily supplied for people with very little event in piecewise-smooth platforms.

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14). A detailed analysis of lens-chain structures is presented in Chapter 6. 7 Poincar´ e Maps and Discontinuity Maps Let Π be an (N − 1)-dimensional manifold, contained in the phase space of an N -dimensional, ODE system. The Poincar´e map for Π, is a function g : Π → Π, where for any p ∈ Π, g(p) is the next intersection of the trajectory of the ODE system passing through p, if such a point exists. Poincar´e maps are often ill-defined globally [Lee et al. (2008)]. It is usual to study a Poincar´e map locally, say in a neighborhood of a periodic orbit (where g(p) = p) and choose the Poincar´e section, Π, such that intersections are transversal.

2. e. equilibria and periodic orbits, on both sides of the bifurcation. Determination of equilibria is straightforward; this was accomplished in Sec. 3. 21). The computation of periodic orbits, however, is more difficult. 18). For a treatment periodic orbits in piecewise-linear systems with multiple switching manifolds, the reader is referred to [Mitrovski and Kocarev (2001); Gon¸calves (2005)]. The remainder of this chapter is organized as follows. In Sec. 18), in two 33 bifurcations November 26, 2009 34 15:34 World Scientific Book - 9in x 6in bifurcations Bifurcations in Piecewise-Smooth, Continuous Systems dimensions.

There are two cases. If det(AL (0)) and det(AR (0)) have the same sign then x∗(L) and x∗(R) are admissible for different signs of µ. 10) has a unique equilibrium. Alternatively if det(AL (0)) and det(AR (0)) have opposite sign, x∗(L) and x∗(R) are admissible for the same sign of µ and collide and annihilate at the origin as µ → 0. The first case is known as persistence, the second is called a nonsmooth fold. 14). 4 T = eT 1 adj(I − AL (µ)) = e1 adj(I − AR (µ)) . 26) The Observer Canonical Form To analyze any local bifurcation from a general point of view it is often a good idea to first obtain a normal form.

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