A geometric singular perturbation theory approach to constrained differential equations

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.

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 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 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.

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

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.

We consider wavefronts that arise in a mathematical modelfor high Lewis number combustion processes. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possesses essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis numbers to the spectral stability of the front in the case of infinite Lewis number. We discuss the implication for nonlinear stability of fronts with high Lewis numbers.This work builds on the ideas developed by Gardner and Jones [12] and generalized in the papers by Bates, Fife, Gardner and Jones [3, 4].

Background and Significance of the topic: Complexity of many advanced models practically precludes its robust analysis. Fortunately, in many cases the models involve multiple time or size scales and thus yield themselves to asymptotic analysis that allows for significant simplifications of them without losing essential features of their dynamics. Methodology: We apply various methods of asymptotic analysis, such as the Tikhonov-Vasilieva theory, geometric singular perturbation theory, asymptotic expansions, or the degenerate convergence theory. Application/Relevance to systems analysis: The presented theory allows for significant simplifications of multiscale complex systems in a way that preserves the main features of their dynamics. Policy and/or practice implications: In many cases a smart aggregation of the equations of a model leads to a complete solution of the problem at a lower computational cost. Discussion and conclusion: We presented two models and two ways of their asymptotic analysis. It is important to note that not always does the naive approach work–applicability of the asymptotic theories often requires a subtle mathematical analysis. Sometimes only the approximation of macroscopic variables is available. Whenever possible, the analysis of complex multiscale systems should be preceded by assessing the possibility of their simplification through the aggregation of equations and variables. This often requires application of advanced analytic methods but, if successful, leads to significantly simpler systems that can be solved at a lower computational cost.

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

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.

The paper is devoted to the order reduction for critical traveling wave problems for a reaction-diffusion type systems. The mathematical apparatus is based on the geometric theory of singular perturbations and the canards technique. The use of the method of invariant manifolds of singularly perturbed systems allows us to replace the study of traveling waves of the original PDE system by analyzing their profiles in a ODE system of a lower order.