A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularization Using Geometric Singular Perturbation Theory

We study the solutions of a friction oscillator subject to stiction. The vector field of this discontinuous model does not follow the Filippov convention, and the concept of Filippov solution cannot be used. Furthermore, some Carathéodory solutions are unphysical. Therefore, we introduce the concept of stiction solutions: these are the Carathéodory solutions that are physically relevant, i.e., the ones that follow the stiction law. However, we find that some of the stiction solutions are forward nonunique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the nonunique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. We show that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, which were otherwise disconnected.

We investigate the macroscopic description of a dilute, gas-like system of particles, which interact through binary collisions that conserve momentum and mass, but which dissipate energy, as in the case of granular media with inelastic collisions. Our starting point is on the mesoscopic level, through the Boltzmann equation. We deduce hydrodynamic equations for the macroscopic description that would reduce to the compressible Navier–Stokes equations if there were no energy dissipation. We do this in a regime where both the Knudsen number (the ratio of mesoscopic to macroscopic length scales) and the restitution deficit (which measures the inelasticity) are small but non-zero. In this regime, we show that for small values of the Knudsen number and small inelasticity it is possible to relate the actual dynamics to a reduced dynamics on a ‘slow manifold’, which in the limit of zero inelasticity and zero Knudsen number is simply the ‘manifold’ of local Maxwellians. Instead of expanding the Boltzmann equation itself, we expand this manifold in terms of these two small parameters. In this way, a number of ideas from the theory of dynamical systems, and especially geometric singular perturbation theory, enter our analysis. We discuss the resulting hydrodynamic equations, and compare them with those obtained by other researchers using other methods (suited to other regimes). As we explain here, the particular regime we investigate is especially interesting in the context of pattern formation in driven granular media.

This paper is concerned with a geometric study of ()‐parameter families of constrained differential systems, where . Our main results say that the dynamics of such a family close to the impasse set is equivalent to the dynamics of a multiple time scale singular perturbation problem (that is a singularly perturbed system containing several small parameters). This enables us to use a geometric theory for multiscale systems in order to describe the behaviour of such a family close to the impasse set. We think that a systematic program towards a combination between geometric singular perturbation theory and constrained systems and problems involving persistence of typical minimal sets are currently emergent. Some illustrations and applications of the main results are provided.

Two-fold singularities in a piecewise smooth (PWS) dynamical system in $$\mathbb {R}^3$$ have long been the subject of intensive investigation. The interest stems from the fact that trajectories which enter the two-fold are associated with forward non-uniqueness. The key questions are: how do we continue orbits forward in time? Are there orbits that are distinguished among all the candidates? We address these questions by regularizing the PWS dynamical system for the case of the visible–invisible two-fold. Within this framework, we consider a regularization function outside the class of Sotomayor and Teixeira. We then undertake a rigorous investigation, using geometric singular perturbation theory and blowup. We show that there is indeed a forward orbit U that is distinguished amongst all the possible forward orbits leaving the two-fold. Working with a normal form of the visible–invisible two-fold, we show that attracting limit cycles can be obtained (due to the contraction towards U), upon composition with a global return mechanism. We provide some illustrative examples.

Kinks and rotations are studied in long Josephson junctions for small and large surface losses. Geometric singular perturbation theory is used to prove existence for small surface losses, while numerical continuation is necessary to handle large surface losses. A survey of the system behaviour in terms of dissipation parameters and bias current is given. Linear orbital stability for kinks is proved for small surface losses by calculating the spectrum of the linearized problem. The spectrum is split into essential spectrum and discrete spectrum. For the determination of the discrete spectrum, robustness of exponential dichotomies is used. Puiseux series together with perturbation theory for linear operators are an essential tool. In a final step, a smooth Evans function together with geometric singular perturbation theory is used to count eigenvalues. For kinks, non‐linear orbital stability is shown. For this purpose, the asymptotic behaviour of a semigroup is given and the theory of centre and stable manifolds is applied. Copyright © 2001 John Wiley & Sons, Ltd.

This paper investigates traveling wave solutions for the spatial diffusion predator-prey model with Allee effects. From the perspective of singular perturbations, we demonstrate the existence of the traveling fronts connecting periodic orbits and the positive equilibrium. On the one hand, we use geometric singular perturbation theory and exchange lemma to study the existence of periodic wave trains, including the degenerate case in which the periodic orbits pass through the small neighbourhoods of two folds on the critical manifold. We utilize the Fenichel normal form and central manifold theory to analyze the flows near the folds where normal hyperbolicity fails. Additionally, we derive the period functions of these periodic orbits. On the other hand, by invariant manifold theory, we prove the transversal intersection between the stable manifolds of positive equilibrium and the unstable manifolds of the periodic orbit. In particular, we identify two types of traveling fronts: pulled and pushed fronts connecting periodic orbits to the positive equilibrium with the steepest decay. Finally, numerical simulations illustrate these theoretical results.

This paper is concerned with the traveling wave solutions of a singularly perturbed system, which arises from the coupled arrays of Chua's circuit. By the geometric singular perturbation theory and invariant manifold theory, we prove that there exists a heteroclinic cycle consisting of the traveling front and back waves with the same wave speed. In particular, the expression of corresponding wave speed is also obtained. Furthermore, we show that the chaotic behavior induced by this heteroclinic cycle is hyperchaos.

This paper focuses on the persistence of solitary waves and periodic waves of a singularly perturbed generalized Drinfel’d–Sokolov system. Geometric singular perturbation theory is first employed to reduce the higher-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we are then able to find some sufficient conditions about the wave speed to guarantee the existence of homoclinic orbits and periodic orbits, which indicates the existence of solitary waves and periodic waves. Furthermore, we find the lower and upper bounds of the limit wave speed.

Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the breakdown of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multipulse solutions.

Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model

Traveling waves in a quintic BBM equation under both distributed delay and weak backward diffusion

In this article we establish an interaction between non-smooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems (l = 3) and consider the case known as a T-singularity.

We consider the dynamics of the general Maxwell–Bloch equations modified by a linear restoring term. We use the slowly varying amplitude approximation to eliminate second-order derivatives in the polarization, but we keep the second-order derivatives in the electric field. This model governs a two-level laser in a ring cavity. For some class of lasers, this system of equations has two separated time scales and forms a singularly perturbed, hyperbolic system with three distinct characteristics. We extend Menon–Haller’s infinite-dimensional geometric singular perturbation theory [G. Menon and G. Haller, Siam J. Math. Anal. Society for Industrial and Applied Mathematics Vol. 33, No. 2, pp. 315–346 c (2001)] to the general Maxwell–Bloch equations by proving the persistence of an arbitrarily smooth, infinite-dimensional, slow manifold, under an unbounded perturbation. The attractor of the system is contained in this manifold. We use a modified graph transform method to prove the existence of this manifold and a cone condition to define the graph transform. The fact that the Maxwell–Bloch and the modified general Maxwell–Bloch equations have the same asymptotic behavior suggests a justification of the slowly varying amplitude approximation.KeywordsMaxwell–Bloch systemInvariant manifolds theoryGeometric singular perturbation

Maximal canards in a slow–fast Rosenzweig–MacArthur model with intraspecific competition among predators

In geometric singular perturbation theory, Fenichel manifolds are typically only finitely smooth. In this paper, we prove better local smoothness properties in the analytic setting, under the condition that no singularities in the slow flow are present. We also investigate cases where the slow flow has a node or focus, where summability results are obtained. Various techniques are being employed like formal power series methods, majorant equations, Gevrey-asymptotics, and studies in the Borel plane.