Integrability and nonintegrability of dynamical systems / Alain Goriely.

Format:

Book

Author/Creator:

Goriely, Alain.

Series:

Advanced series in nonlinear dynamics ; v. 19.

Advanced series in nonlinear dynamics ; v. 19

Language:

English

Subjects (All):

Differentiable dynamical systems.

Differential equations, Nonlinear.

Physical Description:

1 online resource (435 p.)

Edition:

1st ed.

Place of Publication:

River Edge, NJ : World Scientific, c2001.

Language Note:

English

Summary:

This invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems)

Contents:

Contents ; Preface ; Chapter 1 Introduction ; 1.1 A planar system ; 1.1.1 A dynamical system approach ; 1.1.2 An algebraic approach ; 1.1.3 An analytic approach ; 1.1.4 Relevant questions ; 1.2 The Lorenz system ; 1.2.1 A dynamical system approach ; 1.2.2 An algebraic approach

1.2.3 An analytic approach 1.2.4 Relevant questions ; 1.3 Exercises ; Chapter 2 Integrability: an algebraic approach ; 2.1 First integrals ; 2.1.1 A canonical example: The rigid body motion ; 2.2 Classes of functions ; 2.2.1 Elementary first integrals ; 2.2.2 Differential fields

2.3 Homogeneous vector fields 2.3.1 Scale-invariant systems ; 2.3.2 Homogeneous and weight-homogeneous decompositions ; 2.3.3 Weight-homogeneous decompositions ; 2.4 Building first integrals ; 2.4.1 A simple algorithm for polynomial first integrals ; 2.5 Second integrals

2.5.1 Darboux polynomials 2.5.2 Darboux polynomials for planar vector fields ; 2.5.3 The Prelle-Singer Algorithm ; 2.6 Third integrals ; 2.7 Higher integrals ; 2.8 Class-reduction ; 2.9 First integrals for vector fields in R3: the compatibility analysis ; 2.10 Integrability

2.10.1 Local integrability 2.10.2 Liouville integrability ; 2.10.3 Algebraic integrability ; 2.11 Jacobi's last multiplier method ; 2.12 Lax pairs ; 2.12.1 General properties ; 2.12.2 Construction of Lax pairs ; 2.12.3 Completion of Lax pairs ; 2.12.4 Recycling integrable systems

2.12.5 More on Lax pairs

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 385-409) and index.

ISBN:

9786611960834

9781281960832

1281960837

9789812811943

981281194X

OCLC:

879023374