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 -- 2.13 Exercises -- Chapter 3 Integrability: an analytic approach -- 3.1 Singularities of functions -- 3.2 Solutions of differential equations -- 3.3 Singularities of linear differential equations -- 3.3.1 Fundamental solutions -- 3.3.2 Regular singular points -- 3.4 Singularities of nonlinear differential equations -- 3.4.1 Fixed and movable singularities.

3.5 The Painleve property -- 3.5.1 Historical digression I -- 3.5.2 Painleve's a-method -- 3.5.3 The isomonodromy deformation problem -- 3.5.4 Applications -- 3.6 Painleve equations and integrable PDEs -- 3.6.1 The theory of solitons and the Inverse Scattering Transform -- 3.6.2 The Ablowitz-Ramani-Segur conjecture -- 3.7 The PDE Painleve test -- 3.7.1 Integrability of ODEs -- 3.8 Singularity analysis -- 3.8.1 Step 1: The dominant behavior -- 3.8.2 Step 2: Kovalevskaya exponents -- 3.8.3 Step 3: The local solution -- 3.8.4 Formal existence of local solutions -- 3.8.5 Companion systems -- 3.8.6 Convergence of local solutions -- 3.8.7 A short list of singularity analyses -- 3.9 The Painleve tests -- 3.9.1 Painleve test #1: The Hoyer-Kovalevskaya method -- 3.9.2 Painleve test #2: The Gambier-ARS algorithm -- 3.9.3 Painleve test #3: The Painleve-CFP algorithm -- 3.9.4 Painleve property and normal forms -- 3.10 The weak-Painleve conjecture -- 3.11 Patterns of singularities for nonintegrable systems -- 3.11.1 Kovalevskaya fractals -- 3.11.2 Singularity clustering -- 3.12 Finite time blow-up -- 3.13 Exercises -- Chapter 4 Polynomial and quasi-polynomial vector fields -- 4.1 The quasimonomial systems -- 4.2 The quasimonomial transformations -- 4.3 New-time transformations -- 4.4 Canonical forms -- 4.5 The Newton polyhedron -- 4.6 Transformation of the Newton polyhedron -- 4.7 Historical digression: a new-old formalism -- 4.8 Algebraic Degeneracy -- 4.8.1 Degeneracy of matrix A -- 4.8.2 Degeneracy of matrix B -- 4.9 Transformation of first integrals -- 4.10 An algorithm for polynomial first integrals -- 4.11 Jacobi's last multiplier for quasimonomial systems -- 4.12 Application: semi-simple normal forms -- 4.13 Quasimonomial transformation and the Painleve property.

4.13.1 The transformations group of the Riemann sphere -- 4.14 Painleve tests and quasimonomial transformations -- 4.14.1 The dominant balances -- 4.14.2 Dominant balance and Newton's polyhedron -- 4.14.3 The Kovalevskaya exponents -- 4.14.4 Quasimonomial transformation of local series -- 4.15 The Painleve test for the Lotka-Volterra form -- 4.15.1 Step 1: dominant balances -- 4.15.2 Step 2: Kovalevskaya exponents -- 4.15.3 Step 3: compatibility conditions -- 4.16 Transformation of singularities -- 4.16.1 New-time transformation of local series -- 4.16.2 The weak-Painleve conjecture -- 4.16.3 New integrable systems -- 4.17 Exercises -- Chapter 5 Nonintegrability -- 5.1 The general approach: the variational equation -- 5.1.1 Nonintegrability of linear systems -- 5.2 First integrals and linear eigenvalues -- 5.3 First integrals and Kovalevskaya exponents -- 5.3.1 Yoshida's analysis -- 5.3.2 Resonances between Kovalevskaya exponents -- 5.3.3 Kovalevskaya exponents and Darboux polynomials -- 5.3.4 Kovalevskaya exponents for Hamiltonian systems -- 5.4 Complete integrability and resonances -- 5.5 Complete integrability and logarithmic branch points -- 5.6 Multivalued first integral and local solutions -- 5.7 Partial integrability -- 5.7.1 A natural arbitrary parameter -- 5.7.2 Necessary conditions for partial integrability -- 5.8 Exercises -- Chapter 6 Hamiltonian systems -- 6.1 Hamiltonian systems -- 6.1.1 First integrals -- 6.2 Complete integrability -- 6.2.1 Liouville integrability -- 6.2.2 Arnold-Liouville integrability -- 6.3 Algebraic integrability -- 6.4 Ziglin's theory of nonintegrability -- 6.4.1 Hamiltonian systems with two degrees of freedom -- 6.4.2 Ziglin's theorem in n dimensions -- 6.4.3 More on Ziglin's theory -- 6.4.4 The Morales-Ruiz and Ramis theorem -- 6.5 Exercises.

Chapter 7 Nearly integrable dynamical systems -- 7.0.1 An introductory example -- 7.1 General setup -- 7.1.1 General assumptions -- 7.1.2 Comments on the assumptions -- 7.2 A perturbative singularity analysis -- 7.2.1 The Painleve test -- 7.2.2 The w-series -- 7.2.3 Epsilon-expansion for the w-series -- 7.3 The Melnikov vector in n dimensions -- 7.3.1 The variational equation -- 7.3.2 The Melnikov vector -- 7.3.3 The method of residues -- 7.4 Singularity analysis and the Melnikov vector -- 7.4.1 The fundamental local solution -- 7.4.2 Residues and local solutions -- 7.4.3 The compatibility conditions and the residues -- 7.5 The algorithmic procedure -- 7.5.1 The computation of the Melnikov vector: a non-algorithmic procedure -- 7.5.2 The algorithmic procedure -- 7.6 Some illustrative examples -- 7.7 Exercises -- Chapter 8 Open problems -- Glossary -- Bibliography -- Index.