By Simpson D.J.W.
Real-world platforms that contain a few non-smooth swap are frequently well-modeled via piecewise-smooth structures. even if there nonetheless stay many gaps within the mathematical thought of such platforms. This doctoral thesis offers new effects relating to bifurcations of piecewise-smooth, non-stop, independent platforms of normal differential equations and maps. numerous codimension-two, discontinuity prompted 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 accomplished heritage part is very easily supplied for people with very little event in piecewise-smooth platforms.
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28), and a unique companion matrix, C, with the same characteristic polynomial. With the idea of applying a similarity transformation, one might like to know when a given matrix A is similar to its companion C. The answer is that A and C are similar if and only if there exists no monic polynomial, p∗ , of degree less than N with p∗ (A) = 0, [Dummit and Foote (2004); November 26, 2009 15:34 World Scientific Book - 9in x 6in Fundamentals of Piecewise-Smooth, Continuous Systems bifurcations 15 Hartley and Hawkes (1970); Turnbull and Aitken (1932)].
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.
Consequently, the attracting nature of the attracting equilibrium must be stronger than the repelling nature of the repelling equilibrium. Since the time to move 180◦ is π/ωj , this requirement is equivalent to −νR /ωR > νL /ωL , hence Λ < 0, in agreement with the theorem. In smooth systems, Hopf cycles grow as ellipses and the period of an arbitrarily small Hopf cycle is easy to calculate [Kuznetsov (2004); Wiggins (2003)]. In order to calculate the period for our situation, we must first understand the shape of the Hopf cycle.