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Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.



Homoclinic Orbits to Invariant Tori in Hamiltonian Systems

We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting distance between the perturbed whiskers, putting emphasis on the detection of their intersections, which give rise to homoclinic orbits to the perturbed tori. A geometric method is presented which takes into account the Lagrangian properties of the whiskers. In this way, the splitting distance is the gradient of a splitting potential. In the regular case (also known as a priori-unstable: The Lyapunov exponents of the whiskered tori remain fixed), the splitting potential is well-approximated by a Melnikov potential. This method is designed as a first step in the study of the singular case (also known as a priori-stable: The Lyapunov exponents of the whiskered tori approach to zero when the perturbation tends to zero).
Amadeu Delshams, Pere Gutiérrez

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.
Freddy Dumortier, Robert Roussarie

A Primer on the Exchange Lemma for Fast-Slow Systems

In this primer, we give a brief overview of the Exchange Lemma for fast-slow systems of ordinary differential equations. This Lemma has proven to be a useful tool for establishing the existence of homoclinic and heteroclinic orbits, especially those with many components or jumps, in a variety of traveling wave problems and perturbed near-integrable Hamiltonian systems. It has also been applied to models in which periodic orbits and solutions of boundary value problems are sought, including singularly perturbed two-point boundary value problems. The Exchange Lemma applies to fast-slow systems that have normally hyperbolic invariant manifolds (that are usually center manifolds). It enables one to track the dynamics of invariant manifolds and their tangent planes while orbits on them are in the neighborhood of a normally hyperbolic invariant manifold. The end result is a closeness estimate in the C 1 topology of the tracked manifold to a certain submanifold of the normally hyperbolic’s local unstable manifold. We review the general version of the Exchange Lemma due to Tin [24] that treats problems in which there is both fast and slow evolution on the center manifolds. The main normal form used in the neighborhoods of the invariant manifolds is obtained from the persistence theory for normally hyperbolic invariant manifolds due to Fenichel [5, 6, 7]. The works of Jones and Tin [11, 15, 24] form the basis for this work, and [15, 24] contain full presentations of all of the results stated here.
Tasso J. Kaper, Christopher K. R. T. Jones

Geometric Analysis of the Singularly Perturbed Planar Fold

The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, e.g. fold points or points of self-intersection of the critical manifold, the well developed geometric theory does not apply. We present a method based on blow-up techniques which leads to a rigorous geometric analysis of these problems. The blow-up method leads to problems which can be analysed by standard methods from the theory of invariant manifolds and global bifurcations. The presentation is in the context of a planar singularly perturbed fold. The blow-up used in the analysis is closely related to the rescalings used in the classical analysis based on matched asymptotic expansions. The relationship between these classical results and our geometric analysis is discussed.
M. Krupa, P. Szmolyan

Multiple Time Scales and Canards in a Chemical Oscillator

We present a geometric singular perturbation analysis of a chemical oscillator. Although the studied three-dimensional model is rather simple, its dynamics are quite complex. In the original scaling the problem has a folded critical manifold which additionally becomes tangent to the fast fibers in a region relevant to the dynamics. Thus normal hyperbolicity of the critical manifold is lost in two regions. The dynamics depends crucially on effects due to the loss of normal hyperbolicity. In particular, canard solutions play an essential role. We outline how rescalings and blow-up techniques can be used to prove the existence of canards in this problem and to explain other qualitative aspects of the dynamics.
Alexandra Milik, Peter Szmolyan

A Geometric Method for Periodic Orbits in Singularly-Perturbed Systems

In this work, we establish a modular geometric method to demonstrate the existence of periodic orbits in singularly perturbed systems of differential equations. These orbits have alternating fast and slow segments, reflecting the two time scales in the problems. The method involves converting the periodic orbit problem into a boundary value problem in an appropriately augmented system, and it employs several versions of the exchange lemmas due to Jones, Kopell, Kaper and Tin. It is applicable to models that arise in a wide variety of scientific disciplines, and applications are given to the FitzHugh-Nagumo, Hodgkin-Huxley, and Gray-Scott systems.
Cristina Soto-Treviño

The Phenomenon of Delayed Bifurcation and its Analyses

In this note, a general survey of the interesting phenomenon of delayed bifurcation is given. These delayed bifurcations arise in particular fast-slow systems under various situations, and their mathematical justification and application are discussed.
Jianzhong Su

Synchrony in Networks of Neuronal Oscillators

We review numerous recent results in which geometric singular perturbation methods have been used to analyze the population rhythms of neuronal networks. The neurons are modeled as relaxation oscillators and the coupling between neurons is modeled in a way that is motivated by properties of chemical synapses. The results give conditions for when excitatory or inhibitory synaptic coupling leads to either synchronized or desynchronized rhythms. Applications to models for sleep rhythms, image segmentation and wave propagation in inhibitory networks are also discussed.
D. Terman

Metastable dynamics and exponential asymptotics in multi-dimensional domains

Certain singularly perturbed partial differential equations exhibit a phenomenon known as dynamic metastability, whereby the solution evolves on an asymptotically exponentially long time interval as the singular perturbation parameter e tends to zero. This article illustrates a technique to analyze metastable behavior for a range of problems in multi-dimensional domains. The problems considered include the exit problem for diffusion in a potential well, models of interface propagation in materials science, an activator-inhibitor model in mathematical biology, and a flame-front problem. Many of these problems can be formulated in terms of non-local partial differential equations. This non-local feature is shown to be essential to the existence of metastable behavior.
Michael J. Ward


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