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In this paper, we revisit the classical linear turning point problem for the second order differential equation $\epsilon^2 x'' +\mu(t)x=0$ with $\mu(0)=0,\,\mu'(0)\ne 0$ for $0<\epsilon\ll 1$. Written as a first order system, $t=0$ therefore corresponds to a turning point connecting hyperbolic and elliptic regimes. Our main result is that we provide an alternative approach to WBK that is based upon dynamical systems theory, including GSPT and blowup, and we bridge -- perhaps for the first time -- hyperbolic and elliptic theories of slow-fast systems. As an advantage, we only require finite smoothness of $\mu$. The approach we develop will be useful in other singular perturbation problems with hyperbolic-to-elliptic turning points., Comment: Updated. We have added extra references and improved our results (basically, we have solved the issue raised in Remark 4.12 of the previous version). The improvement of our result leads to improved results in our subsequent work on the eigenvalue problem

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as $\epsilon\rightarrow 0$. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called $VI_3$). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.

In this work, we study the dynamics of piecewise smooth systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be extended to such intersections, but this approach does not provide a unique sliding vector and, as opposed to the classical sliding vector-field on codimension-1 discontinuity manifolds, there is no agreed notion of stability in the codimension-2 context. In this paper, we perform a regularization of the piecewise smooth system, introducing two regularization functions and a small perturbation parameter. Then, based on singular perturbation theory, we define sliding and stability of sliding through a critical manifold of the singularly perturbed, regularized system. We show that this notion of sliding vector-field coincides with the Filippov one. The regularized system gives a parameterized surface (the canopy) independent of the regularization functions. This surface serves as our natural basis to derive new and simple geometric criteria on the existence, multiplicity and stability of the sliding flow, depending only on the smooth vector fields around the intersection. Interestingly, we are able to show that if there exist two sliding vector-fields then one is a saddle and the other is of focus/node/center type. This means that there is at most one stable sliding vector-field. We then investigate the effect of the choice of the regularization functions, and, using a blowup approach, we demonstrate the mechanisms through which sliding behavior can appear or disappear on the intersection and describe what consequences this has on the dynamics on the adjacent codimension-1 discontinuity sets. Finally, we show the existence of canard explosions of regularizations of PWS systems in $\mathbb R^3$ that depend on a single unfolding parameter.

In this paper we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge-Knopoff model with Ruina rate-and-state friction law. In the relevant parameter regime, the system is slow-fast with two slow variables and one fast. The oscillation is special for several reasons: Firstly, its singular limit is unbounded, the amplitude of the cycle growing like $\log \epsilon^{-1}$ as $\epsilon\rightarrow 0$. As a consequence of this estimate, the unboundedness of the cycle cannot be captured by a simple $\epsilon$-dependent scaling of the variables, see e.g. \cite{Gucwa2009783}. We therefore obtain its limit on the Poincar\'e sphere. Here we find that the singular limit consists of a slow part on an attracting critical manifold, and a fast part on the equator (i.e. at $\infty$) of the Poincar\'e sphere, which includes motion along a center manifold. The reduced flow on this center manifold runs out along the manifold's boundary, in a special way, leading to a complex return to the slow manifold. We prove the existence of the limit cycle by showing that a return map is a contraction. The main technical difficulty in this part is due to the fact that the critical manifold loses hyperbolicity at an exponential rate at infinity. We therefore use the method in \cite{kristiansen2017a}, applying the standard blowup technique in an extended phase space. In this way we identify a singular cycle, consisting of $12$ pieces, all with desirable hyperbolicity properties, that enables the perturbation into an actual limit cycle for $0<\epsilon\ll 1$., Comment: This is a first draft. Comments are welcome

We consider the problem of a slender rod slipping along a rough surface. Painlev\'e \cite{Painleve1895, Painleve1905a,Painleve1905b} showed that the governing rigid body equations for this problem can exhibit multiple solutions (the {\it indeterminate} case) or no solutions at all (the {\it inconsistent} case), provided the coefficient of friction $\mu$ exceeds a certain critical value $\mu_P$. Subsequently G\'enot and Brogliato \cite{GenotBrogliato1999} proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead there is a special solution for $\mu>\mu_C>\mu_P$, with $\mu_C$ a new critical value of the coefficient of friction, where the rod continues to slip until it reaches a singular `$0/0$' point $P$. Even though the rigid body equations can not describe what happens to the rod beyond the singular point $P$, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a {\it canard} \cite{Benoit81}. To overcome the inadequacy of the rigid body equations beyond $P$, the rigid body assumption is relaxed in the neighbourhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point $P$ and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced $4D$ slow-fast phase space, connecting a $2D$ focus-type slow manifold with the stable manifold of a $2D$ saddle-type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup., Comment: We have modified the interpretation of Proposition 1 on p. 10

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and indeterminate Painlev\'e paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies., Comment: Compared to previous version of the paper, we have: (a) Added a new theorem 2, (b) added a new discussion section with numerical computations, and (c) changed the overall exposition of the manuscript

We use blowup to study the regularization of codimension one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudo-equlibria and limit cycles that can occur in the PWS system. Using the regularization of Sotomayor and Teixeira \cite{Sotomayor96}, we show rigorously how singular canards can persist and how the bifurcation of pseudo-equilibria is related to bifurcations of equilibria in the regularized system. We also show that PWS limit cycles are connected to Hopf bifurcations of the regularization. In addition, we show how regularization can create another type of limit cycle that does not appear to be present in the original PWS system. For both types of limit cycle, we show that the criticality of the Hopf bifurcation that gives rise to periodic orbits is strongly dependent on the precise form of the regularization. Finally, we analyse the limit cycles as locally unique families of periodic orbits of the regularization and connect them, when possible, to limit cycles of the PWS system. We illustrate our analysis with numerical simulations and show how the regularized system can undergo a canard explosion phenomenon., Comment: To appear in SIAM Journal on Applied Dynamical Systems

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance., Comment: To appear in SIAM Journal of Applied Dynamical Systems