Synchronization and Non-Smooth Dynamical Systems

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.

In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb law of friction. In many cases, these models are perfectly adequate for engineering purposes. Formally, however, since the Fundamental Theorem of Existence and Uniqueness does not apply to these situations, one generally expects that these models possess forward nonuniqueness of solutions. Consequently, numerical computations of such systems might possibly unknowingly discard certain solutions. In this paper, we try to shed further light on this issue by studying solutions of a simple friction oscillator subject to stiction friction. The stiction law is a simple nonsmooth model of friction that is a modification of Coulomb based on the fundamental observation that the dynamic friction force, when the mass is in motion, is smaller than the static friction force during stick. The resulting piecewise smooth vector field of this discontinuous model does not follow the classical Filippov convention, and the concept of a Filippov solution cannot be used. Furthermore, some Carathéodory solutions, i.e., absolutely continuous solutions satisfying the differential equation in a weaker sense, are nonphysical. 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 understanding of the nonunique dynamics, we then introduce a general regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. Using geometric singular perturbation theory, we identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. Most interestingly, we find that these new solutions do not correspond to stiction solutions in the piecewise-smooth limit, and are therefore seemingly nonphysical, yet they are robust and appear generically in the class of regularizations we consider. Finally, 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 are otherwise disconnected.

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.

The following overview on non-smooth dynamical systems is given from an application point of view. The aim is to show that real world problems often lead to non-smooth systems and, thus, there is a need to deal with such problems. Moreover, further work shall be stimulated by demonstrating some examples of practical relevance. At present, non-smooth systems are a challenge to engineers and mathematicians as well.

The coupling of different scales in nonlinear systems may lead to some special dynamical phenomena, which always behaves in the combination between large-amplitude oscillations and small-amplitude oscillations, namely bursting oscillations. Up to now, most of therelevant reports have focused on the smooth dynamical systems. However, the coupling of different scales in non-smooth systems may lead to more complicated forms of bursting oscillations because of the existences of different types of non-conventional bifurcations in non-smooth systems. The main purpose of the paper is to explore the coupling effects of multiple scales in non-smooth dynamical systems with non-conventional bifurcations which may occur at the non-smooth boundaries. According to the typical generalized Chua's electrical circuit which contains two non-smooth boundaries, we establish a four-dimensional piecewise-linear dynamical model with different scales in frequency domain. In the model, we introduce a periodically changed current source as well as a capacity for controlling. We select suitable parameter values such that an order gap exists between the exciting frequency and the natural frequency. The state space is divided into several regions in which different types of equilibrium points of the fast sub-system can be observed. By employing the generalized Clarke derivative, different forms of non-smooth bifurcations as well as the conditions are derived when the trajectory passes across the non-smooth boundaries. The case of codimension-1 non-conventional bifurcation is taken for example to investigate the effects of multiple scales on the dynamics of the system. Periodic bursting oscillations can be observed in which codimension-1 bifurcation causes the transitions between the quiescent states and the spiking states. The structure analysis of the attractor points out that the trajectory can be divided into three segments located in different regions. The theoretical period of the movement as well as the amplitudes of the spiking oscillations is derived accordingly, which agrees well with the numerical result. Based on the envelope analysis, the mechanism of the bursting oscillations is presented, which reveals the characteristics of the quiescent states and the repetitive spiking oscillations. Furthermore, unlike the fold bifurcations which may lead to jumping phenomena between two different equilibrium points of the system, the non-smooth fold bifurcation may cause the jumping phenomenon between two equilibrium points located in two regions divided by the non-smooth boundaries. When the trajectory of the system passes across the non-smooth boundaries, non-smooth fold bifurcations may cause the system to tend to different equilibrium points, corresponding to the transitions between quiescent states and spiking states, which may lead to the bursting oscillations.

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.

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

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 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.

Bifurcation phenomena in non-smooth dynamical systems

This paper provides a mathematical analysis of a virus-marine bacteria interaction model. The model is a simplified case of the model published and used by Middelboe (Middelboe, M. 2000 Microb. Ecol. 40, 114-124). It takes account of the virus, the susceptible bacteria, the infected bacteria and the substrate in a chemostat. We show that the numerical values of the parameters given by Middelboe allow two different time scales to be considered. We then use the geometrical singular perturbation theory to study the model. We show that there are two invariant submanifolds of dimension two in the four-dimensional phase space and that these manifolds cross themselves on the boundary of the domain of biological relevance. We then perform a rescaling to understand the dynamics in the vicinity of the intersection of the manifolds. Our results are discussed in the marine ecological context.

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. This type of oscillation 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. The existing literature on such non-standard problems has developed primarily through applications—we complement this by illustrating the suitability of a more general framework for geometric singular perturbation theory to prove existence and uniqueness for a general class of two-stroke relaxation oscillators. While this result can be derived from a more general result in de Maesschalck et al (2011 Indagationes Math. 22 165–206), our methods emphasise the scope, simplicity and applicability of this non-standard approach. We apply this non-standard geometric singular perturbation toolbox to a collection of examples arising in the dynamics of nonlinear transistors and models for mechanical oscillators with friction.

Analyzing the backbone curve of piecewise-linear non-smooth systems using a generalized bilinear frequency approximation method