# Bifurcations in piecewise-smooth continuous systems by Simpson D.J.W.

By Simpson D.J.W.

Real-world platforms that contain a few non-smooth swap are usually well-modeled through piecewise-smooth structures. despite the fact that there nonetheless stay many gaps within the mathematical idea of such platforms. This doctoral thesis provides new effects relating to bifurcations of piecewise-smooth, non-stop, independent structures of standard differential equations and maps. quite a few codimension-two, discontinuity prompted bifurcations are spread out 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 entire heritage part is with ease supplied for people with very little event in piecewise-smooth structures.

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**Example text**

Other invariants, such as periodic orbits, may be created at the bifurcation. This section discusses the the typical nature of discontinuous bifurcations. In general, a discontinuous bifurcation is either the analogue of a well-known, smooth bifurcation or novel and unique to piecewise-smooth systems. For smooth systems a local bifurcation occurs when one or more eigenvalues associated with an equilibrium cross the imaginary axis as a system parameter is continuously varied, Fig. 4(a). In contrast, as an equilibrium is followed through a discontinuous bifurcation, its associated eigenvalues may change discontinuously (panel (b)).

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). The y-axis is a switching manifold and the equilibrium crosses this manifold at the origin when µ = 0 and changes stability.

Then, if Λ < 0 there exists ε > 0 such that for all −ε < µ < 0 there is a stable periodic orbit whose radius is O(µ) away from z ∗ , and for 0 < µ < ε there are no periodic orbits near z ∗ . If on the other hand Λ > 0, there exists ε > 0 such that for all 0 < µ < ε there is an unstable periodic orbit whose radius is O(µ) away from z ∗ , and for all −ε < µ < 0 there are no periodic orbits near z ∗ . 1 is given in Appendix A. 11), as described in Sec. 2. Furthermore, as described in Sec. 34). See Sec.