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We consider smooth systems limiting as → 0 to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with 0 < 1 using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an −dependent domain which shrinks to zero as → 0, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the −dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.

Two-stroke relaxation oscillations consist of two distinct phases per cycle -one slow and one fast -which distinguishes them from the well-known van der Pol-type 'four-stroke' relaxation oscillations. These type of oscillations can be found in singular perturbation problems in non-standard form where the slow-fast timescale splitting is not necessarily reflected in a slow-fast variable splitting. We provide a framework for the application of geometric singular perturbation theory to problems of this kind, and apply it to prove the existence, uniqueness and stability of the observed relaxation oscillations. The analysis of such two-stroke oscillations is motivated by applications which arise in the dynamics of nonlinear transistors, and models for mechanical oscillators with friction.

Singular exponential nonlinearities of the form e h(x) −1 with > 0 small occur in many different applications. These terms have essential singularities for = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in [20], and prove -for both systems -existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

Boundary equilibria bifurcation (BEB) arises in piecewise-smooth (PWS) systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some → 0 to PWS systems which undergo a BEB. This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory and a method of geometric desingularization known as blow-up. After deriving a local normal form capable of generating all 12 singularly perturbed BEBs, we describe the unfolding in each case. Detailed quantitative results on saddle-node, Andronov-Hopf, homoclinic and codimension-2 Bogdanov-Takens bifurcations involved in the unfoldings and classification are presented. Each bifurcation is singular in the sense that it occurs within a domain which shrinks to zero as → 0 at a rate determined by the rate at which the system loses smoothness. Detailed asymptotics for a distinguished homoclinic connection which forms the boundary between two singularly perturbed BEBs in parameter space are also given. Finally, we describe the explosive onset of oscillations arising in the unfolding of a particular singularly perturbed