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This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.



Chapter 1. Introduction

In this chapter, we begin in Section 1.1 with a practical guide to orient the reader to how the book is structured and how it can be utilized. Several notational conventions are introduced as well. Section 1.2 covers some basic terminology for systems with two time scales.
Christian Kuehn

Chapter 2. General Fenichel Theory

Important Remark: This chapter provides a proof of the perturbation of normally hyperbolic invariant manifolds due to Fenichel. Chapter 3 specializes the general case to fast–slow systems, so that the current chapter is necessary only for readers interested in the proof and its strategy. It is possible to skip this chapter at first reading, get an idea how the theorem is used, and then return to the proof.
Christian Kuehn

Chapter 3. Geometric Singular Perturbation Theory

This chapter introduces some of the core elements, definitions, and theorems of the theory of normally hyperbolic invariant manifold theory for fast–slow systems. Several other chapters build on this material.
Christian Kuehn

Chapter 4. Normal Forms

Having developed the main theorems of perturbations of invariant manifolds, we aim to bring a fast–slow system into normal form . As this book was written, there was no complete general theory for what a “normal form” for a fast–slow system should be.
Christian Kuehn

Chapter 5. Direct Asymptotic Methods

In this chapter, we shall just attempt to compute asymptotic expansions for fast–slow systems by more or less brute force. It is a very instructive technique: just substitute an asymptotic expansion for the solution and see what happens. In other words, where does such a substitution seem to give a good approximation, and where are modifications required?
Christian Kuehn

Chapter 6. Tracking Invariant Manifolds

The main goal of this chapter is to discuss the tracking of invariant manifolds when they transition from a fast to a slow motion and vice versa. That is, we would like to understand how trajectories or more general objects enter and leave the vicinity of a normally hyperbolic critical manifold. The main application is to show how the geometric theory of fast–slow systems can be used to prove the persistence of candidate orbits for \(0 <\varepsilon \ll 1\).
Christian Kuehn

Chapter 7. The Blowup Method

This chapter deals with geometric desingularization of nonhyperbolic equilibrium points using the so-called blowup method. The main insight, due to Dumortier and Roussarie, is that singularities at which fast and slow directions interact may be converted into partially hyperbolic problems using the blowup method. The method inserts a suitable manifold, e.g., a sphere, at the singularity.
Christian Kuehn

Chapter 8. Singularities and Canards

In this chapter, we put our previous tools such as Fenichel’s theorem and the blowup method to good use. The main goal is to delve even further into the analysis of singularities where normal hyperbolicity is lost and to track trajectories through a region near the singularity.
Christian Kuehn

Chapter 9. Advanced Asymptotic Methods

Asymptotic analysis is a key ingredient in capturing multiscale dynamics. In this chapter, a collection of asymptotic and perturbation methods is presented. The focus is on the basic principles of methods and key examples to understand their application. All methods can be applied in many other circumstances, and although the algebraic manipulations change, the principles of the methods tend to carry over.
Christian Kuehn

Chapter 10. Numerical Methods

For the analysis of many nonlinear dynamical systems, numerical methods are indispensable. Fast–slow systems are no exception. In fact, multiscale differential equations provide a big challenge for efficient numerics.
Christian Kuehn

Chapter 11. Computing Manifolds

We have extensively discussed the properties of invariant manifolds and their relevance for fast–slow systems in previous chapters. However, we usually used explicit algebraic expressions or asymptotic expansions to deal with critical and slow manifolds. For a general multiple time scale system, there are several complications. They may not be in standard form, and even if they are, then calculating a slow manifold analytically may be intractable. This chapter deals with algorithms to find and compute invariant manifolds for fast–slow systems numerically.
Christian Kuehn

Chapter 12. Scaling and Delay

This chapter has two major goals. The first is to analyze delayed loss of stability near fast subsystem bifurcation points with a focus on Hopf bifurcation. The second goal is to introduce several algebraic-combinatorial flavored tools, which turn out to be very helpful for multiscale systems.
Christian Kuehn

Chapter 13. Oscillations

Many multiple time scale systems are capable of generating intricate patterns. In this chapter, we are going to focus on periodic oscillations where the fast–slow structure plays a crucial role in the generating mechanism. Let us point out that we do not aim at a complete classification. The focus is on examples and prototype mechanisms. There are two main keywords associated with this area that we want to explore: mixed-mode oscillations (MMOs) and bursting.
Christian Kuehn

Chapter 14. Chaos in Fast-Slow Systems

Similar to Chapter 13, the current chapter uses the theory previously discussed in parts of this book to gain substantial insight into nonlinear dynamics.
Christian Kuehn

Chapter 15. Stochastic Systems

The interaction between noise and multiscale dynamics is already a large area, and it is still a field of intensive research. This chapter aims to provide a number of diverse and interlinked techniques that reflect some recent developments.
Christian Kuehn

Chapter 16. Topological Methods

All methods to understand multiple time scale systems we have presented so far needed some kind of mathematical analysis. In particular, for any geometric construction, asymptotic calculation, or numerical method, we needed tools such as transversality arguments, asymptotic comparison, and error estimates. But what if we are primarily interested in existence statements such as, does a given fast–slow system have a periodic orbit? In this chapter, we shall use a different approach, based on (algebraic) topology, to answer existence questions using only a minimal amount of analytic information.
Christian Kuehn

Chapter 17. Spatial Dynamics

In this chapter, the main topic is traveling waves for time-dependent spatially extended systems in one space dimension. Note that we have already extensively discussed various techniques to prove the existence of waves for partial differential equations (PDEs); see, e.g., Chapter 6 Hence, we focus here on further topics beyond the existence of waves in PDEs.
Christian Kuehn

Chapter 18. Infinite Dimensions

Generalizing the geometric viewpoint of fast–slow ODEs to truly infinite- dimensional dynamical systems is notoriously difficult. However, there has been quite a bit of progress in recent years.
Christian Kuehn

Chapter 19. Other Topics

This chapter collects various topics that did not fit immediately within the main flow of the book. Nevertheless, they have been included here due to their general importance and interaction with fast–slow systems.
Christian Kuehn

Chapter 20. Applications

In this chapter we touch on several application areas in which time scale separation arises naturally. As you can guess from the rather diverse set of section headings, it is quite reasonable to conjecture that most quantitative sciences that employ mathematical modeling may eventually encounter various multiscale problems. Each section centers on one or two key examples in which one can clearly identify the time scale separation parameter as well as apply many of the methods discussed in this book.
Christian Kuehn


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