Multiple time scale dynamics.

*(English)*Zbl 1335.34001
Applied Mathematical Sciences 191. Cham: Springer (ISBN 978-3-319-12315-8/hbk; 978-3-319-12316-5/ebook). xiii, 814 p. (2015).

This interesting monograph is a self-contained, coherent overview of the backgrounds and progress of the dynamical systems with multiple time scales. Focusing on techniques, tools, and advances in numerical algorithms, the author develops material which, up until now, has been scattered in scientific journals and conference proceedings.

Requiring only basic knowledge of dynamical systems theory, the book is accessible to a broad audience of researchers and graduate students. It may be used for self-study or as a reference; portions of the text can be used in the advanced graduate courses and seminars.

The book is structured into 20 chapters. Chapter 1 is of a preparatory nature and outlines the broad field of study with a practical guide to how the book is structured and should be read, including notational conventions and some basic terminology for the systems with two time scales (fast and slow) \[ \begin{matrix} \varepsilon\dot x&=&f(x,y,\varepsilon) \\ \dot y&=&g(x,y,\varepsilon),\end{matrix} \] where \((x,y)\in\mathbb{R}^m\times\mathbb{R}^n\) and \(0<\varepsilon\ll 1\) is a small parameter representing the ratio of time scales. This chapter is short and aims to orient the readers and grasp their attention.

Chapter 2, General Fenichel Theory, and Chapter 3, Geometric Singular Perturbation Theory, discuss the fundamental and key ideas of geometric singular perturbation theory (GSPT) which provides a suitable mathematical framework for description and analysis of dynamical systems with multiscale behavior. In general, the singular perturbation theory concerns the study of the problems that depend on a parameter (or parameters) in such a way that solutions behave nonuniformly as the parameter tends toward some limiting value of interest, in the present book \(\varepsilon\rightarrow 0^+\). These chapters introduce some of core elements, definitions, and theorems of the GSPT, among others, the Fenichel’s theorem on the persistence and smoothness of normally hyperbolic invariant manifolds, a cornerstone theorem of the GSPT.

Chapter 4, Normal Forms, indicates some possibilities and transformations to bring a fast-slow system into the Fenichel Normal Form. The idea is introducing a local change of variables that “straightens” the normally hyperbolic invariant manifold and its stable and unstable manifolds and allows to write the dynamical system in a more convenient form.

Chapter 5, Direct Asymptotic Methods, presents asymptotic techniques centered on series expansions for regular and singular perturbation problems.

Chapter 6, Tracking Invariant Manifolds, analyses how the trajectories of fast-slow systems enter and leave the vicinity of a normally hyperbolic critical manifold. In this chapter, there is proved the Exchange lemma which describes accurately the behavior of the invariant objects near normally hyperbolic fast-to-slow and slow-to-fast transitions. The Exchange lemma is then applied to prove the existence of homoclinic orbits in the FitzHugh-Nagumo equation which models the qualitative behavior of the neurons in the brain of animals.

Chapter 7, The Blowup Method, deals in detail with the blowup method of Dumortier and Roussarie to study the regularization of singularities of nonhyperbolic equilibrium points.

Chapter 8, Singularities and Canards, using the Fenichel’s theorem and the blowup method for fast-slow system with a parameter \(\lambda\in\mathbb{R}\) explains the fast transition upon variation of a parameter \(\lambda\) from a small amplitude periodic orbits in the plane via canard cycles to a large amplitude canard periodic orbits.

Chapter 9, Advanced Asymptotic Methods, builds on the material of Chapter 5 on a more formal level. This chapter presents a collection of asymptotic and perturbation methods as, for example, matched asymptotic expansions, the boundary function method and WKB theory.

Chapter 10, Numerical Methods, is devoted to some numerical techniques for fast-slow dynamical systems. Here, we can find a brief introduction to boundary value problems.

Chapter 11, Computing Manifolds, discusses the basic techniques and algorithms as the CSP method or the ZDP method for obtaining an analytical or numerical representation of invariant manifolds for fast-slow system.

Chapter 12, Scaling And Delay, quantifies the scaling and delay effects in the fast-slow systems. In this context is analyzed delayed stability loss near a fast subsystem bifurcation points with a focus on Hopf bifurcation. The Newton polygon as a combinatorial tool is introduced which can be used as an auxiliary tool to compute critical manifolds, to find blowup coefficients, and to compute the scaling laws of delay times and slow manifolds.

Chapter 13, Oscillations, focuses on the examples and prototype of periodic oscillations where the fast-slow structure of systems plays a crucial role in the generating mechanism.

Chapter 14, Chaos In Fast-Slow Systems, gives the outline of some basic definitions and results from theory of chaotic dynamics adapted to fast-slow systems.

Chapter 15, Stochastic Systems, provides an introduction to the analysis of interaction between noise and multiscale dynamics. The basic techniques of stochastic fast-slow systems are developed and an analogue to the classical Fenichel’s theorem for stochastic system is formulated.

Chapter 16, Topological Methods, presents a different approach based on algebraic topology (the Conley index) to prove the existence of periodic orbits in fast-slow systems.

Chapter 17, Spatial Dynamics, covers the topic of traveling waves and their stability for time-dependent spatially extended systems in one space dimension.

Chapter 18, Infinite Dimensions, sketches possible directions of generalization the geometric theory of finite-dimensional fast-slow systems to infinite-dimensional dynamical systems in Banach space based on the semigroup theory.

Chapter 19, Other Topics, collects various topics that did not fit immediately within the main flow of the book – differential-algebraic equations, non-smooth dynamical systems, hysteresis, Hamiltonian systems, etc.

Chapter 20, Applications, centers on several application areas in which time scale separation arises naturally – control engineering (a feedback system in the case of a high-gain amplifier), neuroscience (the activation of sodium channels is a much faster process than activation of potassium and leak currents), fluid dynamics (case of high viscosity), etc.

The book contains excellent mathematics and is a well-written and unique source of information on the multiple time scale dynamics. I highly recommend it to all researchers and graduate students who would like to understand the geometric singular perturbation theory.

Requiring only basic knowledge of dynamical systems theory, the book is accessible to a broad audience of researchers and graduate students. It may be used for self-study or as a reference; portions of the text can be used in the advanced graduate courses and seminars.

The book is structured into 20 chapters. Chapter 1 is of a preparatory nature and outlines the broad field of study with a practical guide to how the book is structured and should be read, including notational conventions and some basic terminology for the systems with two time scales (fast and slow) \[ \begin{matrix} \varepsilon\dot x&=&f(x,y,\varepsilon) \\ \dot y&=&g(x,y,\varepsilon),\end{matrix} \] where \((x,y)\in\mathbb{R}^m\times\mathbb{R}^n\) and \(0<\varepsilon\ll 1\) is a small parameter representing the ratio of time scales. This chapter is short and aims to orient the readers and grasp their attention.

Chapter 2, General Fenichel Theory, and Chapter 3, Geometric Singular Perturbation Theory, discuss the fundamental and key ideas of geometric singular perturbation theory (GSPT) which provides a suitable mathematical framework for description and analysis of dynamical systems with multiscale behavior. In general, the singular perturbation theory concerns the study of the problems that depend on a parameter (or parameters) in such a way that solutions behave nonuniformly as the parameter tends toward some limiting value of interest, in the present book \(\varepsilon\rightarrow 0^+\). These chapters introduce some of core elements, definitions, and theorems of the GSPT, among others, the Fenichel’s theorem on the persistence and smoothness of normally hyperbolic invariant manifolds, a cornerstone theorem of the GSPT.

Chapter 4, Normal Forms, indicates some possibilities and transformations to bring a fast-slow system into the Fenichel Normal Form. The idea is introducing a local change of variables that “straightens” the normally hyperbolic invariant manifold and its stable and unstable manifolds and allows to write the dynamical system in a more convenient form.

Chapter 5, Direct Asymptotic Methods, presents asymptotic techniques centered on series expansions for regular and singular perturbation problems.

Chapter 6, Tracking Invariant Manifolds, analyses how the trajectories of fast-slow systems enter and leave the vicinity of a normally hyperbolic critical manifold. In this chapter, there is proved the Exchange lemma which describes accurately the behavior of the invariant objects near normally hyperbolic fast-to-slow and slow-to-fast transitions. The Exchange lemma is then applied to prove the existence of homoclinic orbits in the FitzHugh-Nagumo equation which models the qualitative behavior of the neurons in the brain of animals.

Chapter 7, The Blowup Method, deals in detail with the blowup method of Dumortier and Roussarie to study the regularization of singularities of nonhyperbolic equilibrium points.

Chapter 8, Singularities and Canards, using the Fenichel’s theorem and the blowup method for fast-slow system with a parameter \(\lambda\in\mathbb{R}\) explains the fast transition upon variation of a parameter \(\lambda\) from a small amplitude periodic orbits in the plane via canard cycles to a large amplitude canard periodic orbits.

Chapter 9, Advanced Asymptotic Methods, builds on the material of Chapter 5 on a more formal level. This chapter presents a collection of asymptotic and perturbation methods as, for example, matched asymptotic expansions, the boundary function method and WKB theory.

Chapter 10, Numerical Methods, is devoted to some numerical techniques for fast-slow dynamical systems. Here, we can find a brief introduction to boundary value problems.

Chapter 11, Computing Manifolds, discusses the basic techniques and algorithms as the CSP method or the ZDP method for obtaining an analytical or numerical representation of invariant manifolds for fast-slow system.

Chapter 12, Scaling And Delay, quantifies the scaling and delay effects in the fast-slow systems. In this context is analyzed delayed stability loss near a fast subsystem bifurcation points with a focus on Hopf bifurcation. The Newton polygon as a combinatorial tool is introduced which can be used as an auxiliary tool to compute critical manifolds, to find blowup coefficients, and to compute the scaling laws of delay times and slow manifolds.

Chapter 13, Oscillations, focuses on the examples and prototype of periodic oscillations where the fast-slow structure of systems plays a crucial role in the generating mechanism.

Chapter 14, Chaos In Fast-Slow Systems, gives the outline of some basic definitions and results from theory of chaotic dynamics adapted to fast-slow systems.

Chapter 15, Stochastic Systems, provides an introduction to the analysis of interaction between noise and multiscale dynamics. The basic techniques of stochastic fast-slow systems are developed and an analogue to the classical Fenichel’s theorem for stochastic system is formulated.

Chapter 16, Topological Methods, presents a different approach based on algebraic topology (the Conley index) to prove the existence of periodic orbits in fast-slow systems.

Chapter 17, Spatial Dynamics, covers the topic of traveling waves and their stability for time-dependent spatially extended systems in one space dimension.

Chapter 18, Infinite Dimensions, sketches possible directions of generalization the geometric theory of finite-dimensional fast-slow systems to infinite-dimensional dynamical systems in Banach space based on the semigroup theory.

Chapter 19, Other Topics, collects various topics that did not fit immediately within the main flow of the book – differential-algebraic equations, non-smooth dynamical systems, hysteresis, Hamiltonian systems, etc.

Chapter 20, Applications, centers on several application areas in which time scale separation arises naturally – control engineering (a feedback system in the case of a high-gain amplifier), neuroscience (the activation of sodium channels is a much faster process than activation of potassium and leak currents), fluid dynamics (case of high viscosity), etc.

The book contains excellent mathematics and is a well-written and unique source of information on the multiple time scale dynamics. I highly recommend it to all researchers and graduate students who would like to understand the geometric singular perturbation theory.

Reviewer: Robert Vrabel (Trnava)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34E13 | Multiple scale methods for ordinary differential equations |

34D15 | Singular perturbations of ordinary differential equations |

34C45 | Invariant manifolds for ordinary differential equations |

34E17 | Canard solutions to ordinary differential equations |

34C26 | Relaxation oscillations for ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

37D10 | Invariant manifold theory for dynamical systems |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |

34D45 | Attractors of solutions to ordinary differential equations |