The goal of these lectures is an exposition of the geometric approach to singular
perturbation problems. Singularly perturbed equations gain their special structure from the …

In this book we will study equations of the following form x= f (x, t; µ),(0.0. 1) and x↦→ g (x;
µ),(0.0. 2) with x∈ U⊂ Rn, t∈ R1, and µ∈ V⊂ Rp where U and V are open sets in Rn and …

Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have
the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled …

In this paper we develop new global perturbation techniques for detecting homoclinic and
heteroclinic orbits in a class of four dimensional ordinary differential equations that are …

In this note we present an energy equation concerning the H2 norm of a forced one-
dimensional nonlinear Schrödinger equation. As an application we are able to corroborate …

A growing list of diverse biological systems and their equally diverse functionalities provides
realizations of a paradigm of emergent behavior. In each of these biological systems …

In this paper we develop methods to show the existence of orbits homoclinic or heteroclinic
to periodic orbits, hyperbolic fixed points or combinations of hyperbolic fixed points and/or …

In this paper we prove the existence of multi-pulse orbits homoclinic to a slow manifold in a
two-mode truncation of the damped-forced nonlinear Schrödinger equation (first suggested …

Driven damped coupled oscillators exhibit complex spatiotemporal dynamics. An archetype
model is the driven damped sine-Gordon equation, which can describe several physical …

This paper presents a new unified theory of orbits homoclinic to resonance bands in a class
of near-integrable dissipative systems. It describes three sets of conditions, each of which …