Nonlinear differential equations and dynamical systems

TLDR: In this article, the Poincare-Bendixson theorem is applied to the analysis of two-dimensional linear systems with first integrals and integral manifolds, and the Lagrange standard form is used.

Abstract: 1 Introduction.- 1.1 Definitions and notation.- 1.2 Existence and uniqueness.- 1.3 Gronwall's inequality.- 2 Autonomous equations.- 2.1 Phase-space, orbits.- 2.2 Critical points and linearisation.- 2.3 Periodic solutions.- 2.4 First integrals and integral manifolds.- 2.5 Evolution of a volume element, Liouville's theorem.- 2.6 Exercises.- 3 Critical points.- 3.1 Two-dimensional linear systems.- 3.2 Remarks on three-dimensional linear systems.- 3.3 Critical points of nonlinear equations.- 3.4 Exercises.- 4 Periodic solutions.- 4.1 Bendixson's criterion.- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem.- 4.3 The Poincare-Bendixson theorem.- 4.4 Applications of the Poincare-Bendixson theorem.- 4.5 Periodic solutions in ?n.- 4.6 Exercises.- 5 Introduction to the theory of stability.- 5.1 Simple examples.- 5.2 Stability of equilibrium solutions.- 5.3 Stability of periodic solutions.- 5.4 Linearisation.- 5.5 Exercises.- 6 Linear Equations.- 6.1 Equations with constant coefficients.- 6.2 Equations with coefficients which have a limit.- 6.3 Equations with periodic coefficients.- 6.4 Exercises.- 7 Stability by linearisation.- 7.1 Asymptotic stability of the trivial solution.- 7.2 Instability of the trivial solution.- 7.3 Stability of periodic solutions of autonomous equations.- 7.4 Exercises.- 8 Stability analysis by the direct method.- 8.1 Introduction.- 8.2 Lyapunov functions.- 8.3 Hamiltonian systems and systems with first integrals.- 8.4 Applications and examples.- 8.5 Exercises.- 9 Introduction to perturbation theory.- 9.1 Background and elementary examples.- 9.2 Basic material.- 9.3 Naive expansion.- 9.4 The Poincare expansion theorem.- 9.5 Exercises.- 10 The Poincare-Lindstedt method.- 10.1 Periodic solutions of autonomous second-order equations.- 10.2 Approximation of periodic solutions on arbitrary long time-scales.- 10.3 Periodic solutions of equations with forcing terms.- 10.4 The existence of periodic solutions.- 10.5 Exercises.- 11 The method of averaging.- 11.1 Introduction.- 11.2 The Lagrange standard form.- 11.3 Averaging in the periodic case.- 11.4 Averaging in the general case.- 11.5 Adiabatic invariants.- 11.6 Averaging over one angle, resonance manifolds.- 11.7 Averaging over more than one angle, an introduction.- 11.8 Periodic solutions.- 11.9 Exercises.- 12 Relaxation Oscillations.- 12.1 Introduction.- 12.2 Mechanical systems with large friction.- 12.3 The van der Pol-equation.- 12.4 The Volterra-Lotka equations.- 12.5 Exercises.- 13 Bifurcation Theory.- 13.1 Introduction.- 13.2 Normalisation.- 13.3 Averaging and normalisation.- 13.4 Centre manifolds.- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation.- 13.6 Exercises.- 14 Chaos.- 14.1 Introduction and historical context.- 14.2 The Lorenz-equations.- 14.3 Maps associated with the Lorenz-equations.- 14.4 One-dimensional dynamics.- 14.5 One-dimensional chaos: the quadratic map.- 14.6 One-dimensional chaos: the tent map.- 14.7 Fractal sets.- 14.8 Dynamical characterisations of fractal sets.- 14.9 Lyapunov exponents.- 14.10 Ideas and references to the literature.- 15 Hamiltonian systems.- 15.1 Introduction.- 15.2 A nonlinear example with two degrees of freedom.- 15.3 Birkhoff-normalisation.- 15.4 The phenomenon of recurrence.- 15.5 Periodic solutions.- 15.6 Invariant tori and chaos.- 15.7 The KAM theorem.- 15.8 Exercises.- Appendix 1: The Morse lemma.- Appendix 2: Linear periodic equations with a small parameter.- Appendix 3: Trigonometric formulas and averages.- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3.- Appendix 5: Bifurcations of self-excited oscillations.- Appendix 6: Normal forms of Hamiltonian systems near equilibria.- Answers and hints to the exercises.- References.