By David John Warwick Simpson
Real-world platforms that contain a few non-smooth switch are frequently well-modeled via piecewise-smooth platforms. even though there nonetheless stay many gaps within the mathematical idea of such structures. This doctoral thesis offers new effects relating to bifurcations of piecewise-smooth, non-stop, self sustaining platforms of standard differential equations and maps. a number of codimension-two, discontinuity brought on 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 entire heritage part is comfortably supplied for people with very little adventure in piecewise-smooth platforms.
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Extra resources for Bifurcations in Piecewise-smooth Continuous Systems (World Scientific Series on Nonlinear Science Series a)
2004); Coombes and Osbaldestin (2000)]. 9 Smooth Approximations Since piecewise-smooth systems may exhibit exotic, complicated bifurcations, many of which are not fully understood, many authors consider smooth approximations. By mollifying the nonsmooth components of a piecewise-smooth, continuous vector field, V , one obtains a smooth vector field, Vsmooth , and the norm, ||V − Vsmooth ||∞ , may be set as small as one likes. This can be useful when smoothness is a desired property. For instance in [L´azaro et al.
2(c). The point y ∗ is a fixed point of P if h(y ∗ ) ≡ P(y ∗ ) − y ∗ = 0. We have h(0) > 0 and h(PL (c1 )) < 0 thus by the intermediate value theorem there exists y ∗ with h(y ∗ ) = y ∗ . 1). 2 and is attracting. November 26, 2009 15:34 World Scientific Book - 9in x 6in bifurcations Discontinuous Bifurcations in Planar Systems 39 PR (y0 ) y 0 y0 yˆ c1 (b) γ1 0 P(y) x PL (c1 ) c1 γ2 yˆ c2 0 y∗ z ∗(R) (a) y (c) ∗(L) Fig. 1) when z is a repelling focus, z ∗(R) is an attracting node and µ < 0. Panels (b) and (c) show sketches of PR and P, respectively.
The associated eigenvalues of the equilibrium are complex-valued and jump from one side of the imaginary axis to the other at the bifurcation. Consequently, the discontinuous bifurcation is akin to a Hopf bifurcation. In codimension-one situations a periodic orbit is generated at the bifurcation. 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.
Bifurcations in Piecewise-smooth Continuous Systems (World Scientific Series on Nonlinear Science Series a) by David John Warwick Simpson