Contents -- Preface to the fourth edition -- 1 Second-order differential equations in the phase plane -- 1.1 Phase diagram for the pendulum equation -- 1.2 Autonomous equations in the phase plane -- 1.3 Mechanical analogy for the conservative system x=f(x) -- 1.4 The damped linear oscillator -- 1.5 Nonlinear damping: limit cycles -- 1.6 Some applications -- 1.7 Parameter-dependent conservative systems -- 1.8 Graphical representation of solutions -- Problems -- 2 Plane autonomous systems and linearization -- 2.1 The general phase plane -- 2.2 Some population models -- 2.3 Linear approximation at equilibrium points -- 2.4 The general solution of linear autonomous plane systems -- 2.5 The phase paths of linear autonomous plane systems -- 2.6 Scaling in the phase diagram for a linear autonomous system -- 2.7 Constructing a phase diagram -- 2.8 Hamiltonian systems -- Problems -- 3 Geometrical aspects of plane autonomous systems -- 3.1 The index of a point -- 3.2 The index at infinity -- 3.3 The phase diagram at infinity -- 3.4 Limit cycles and other closed paths -- 3.5 Computation of the phase diagram -- 3.6 Homoclinic and heteroclinic paths -- Problems -- 4 Periodic solutions -- averaging methods -- 4.1 An energy-balance method for limit cycles -- 4.2 Amplitude and frequency estimates: polar coordinates -- 4.3 An averaging method for spiral phase paths -- 4.4 Periodic solutions: harmonic balance -- 4.5 The equivalent linear equation by harmonic balance -- Problems -- 5 Perturbation methods -- 5.1 Nonautonomous systems: forced oscillations -- 5.2 The direct perturbation method for the undamped Duffing's equation -- 5.3 Forced oscillations far from resonance -- 5.4 Forced oscillations near resonance with weak excitation -- 5.5 The amplitude equation for the undamped pendulum -- 5.6 The amplitude equation for a damped pendulum.

5.7 Soft and hard springs -- 5.8 Amplitude-phase perturbation for the pendulum equation -- 5.9 Periodic solutions of autonomous equations (Lindstedt's method) -- 5.10 Forced oscillation of a self-excited equation -- 5.11 The perturbation method and Fourier series -- 5.12 Homoclinic bifurcation: an example -- Problems -- 6 Singular perturbation methods -- 6.1 Non-uniform approximations to functions on an interval -- 6.2 Coordinate perturbation -- 6.3 Lighthill's method -- 6.4 Time-scaling for series solutions of autonomous equations -- 6.5 The multiple-scale technique applied to saddle points and nodes -- 6.6 Matching approximations on an interval -- 6.7 A matching technique for differential equations -- Problems -- 7 Forced oscillations: harmonic and subharmonic response, stability, and entrainment -- 7.1 General forced periodic solutions -- 7.2 Harmonic solutions, transients, and stability for Duffing's equation -- 7.3 The jump phenomenon -- 7.4 Harmonic oscillations, stability, and transients for the forced van der Pol equation -- 7.5 Frequency entrainment for the van der Pol equation -- 7.6 Subharmonics of Duffing's equation by perturbation -- 7.7 Stability and transients for subharmonics of Duffing's equation -- Problems -- 8 Stability -- 8.1 Poincaré stability (stability of paths) -- 8.2 Paths and solution curves for general systems -- 8.3 Stability of time solutions: Liapunov stability -- 8.4 Liapunov stability of plane autonomous linear systems -- 8.5 Structure of the solutions of n-dimensional linear systems -- 8.6 Structure of n-dimensional inhomogeneous linear systems -- 8.7 Stability and boundedness for linear systems -- 8.8 Stability of linear systems with constant coefficients -- 8.9 Linear approximation at equilibrium points for first-order systems in n variables.

8.10 Stability of a class of non-autonomous linear systems in n dimensions -- 8.11 Stability of the zero solutions of nearly linear systems -- Problems -- 9 Stability by solution perturbation: Mathieu's equation -- 9.1 The stability of forced oscillations by solution perturbation -- 9.2 Equations with periodic coefficients (Floquet theory) -- 9.3 Mathieu's equation arising from a Duffing equation -- 9.4 Transition curves for Mathieu's equation by perturbation -- 9.5 Mathieu's damped equation arising from a Duffing equation -- Problems -- 10 Liapunov methods for determining stability of the zero solution -- 10.1 Introducing the Liapunov method -- 10.2 Topographic systems and the Poincaré-Bendixson theorem -- 10.3 Liapunov stability of the zero solution -- 10.4 Asymptotic stability of the zero solution -- 10.5 Extending weak Liapunov functions to asymptotic stability -- 10.6 A more general theory for autonomous systems -- 10.7 A test for instability of the zero solution: n dimensions -- 10.8 Stability and the linear approximation in two dimensions -- 10.9 Exponential function of a matrix -- 10.10 Stability and the linear approximation for nth order autonomous systems -- 10.11 Special systems -- Problems -- 11 The existence of periodic solutions -- 11.1 The Poincaré-Bendixson theorem and periodic solutions -- 11.2 A theorem on the existence of a centre -- 11.3 A theorem on the existence of a limit cycle -- 11.4 Van der Pol's equation with large parameter -- Problems -- 12 Bifurcations and manifolds -- 12.1 Examples of simple bifurcations -- 12.2 The fold and the cusp -- 12.3 Further types of bifurcation -- 12.4 Hopf bifurcations -- 12.5 Higher-order systems: manifolds -- 12.6 Linear approximation: centre manifolds -- Problems -- 13 Poincaré sequences, homoclinic bifurcation, and chaos -- 13.1 Poincaré sequences.

13.2 Poincaré sections for nonautonomous systems -- 13.3 Subharmonics and period doubling -- 13.4 Homoclinic paths, strange attractors and chaos -- 13.5 The Duffing oscillator -- 13.6 A discrete system: the logistic difference equation -- 13.7 Liapunov exponents and difference equations -- 13.8 Homoclinic bifurcation for forced systems -- 13.9 The horseshoe map -- 13.10 Melnikov's method for detecting homoclinic bifurcation -- 13.11 Liapunov exponents and differential equations -- 13.12 Power spectra -- 13.13 Some further features of chaotic oscillations -- Problems -- Answers to the exercises -- Appendices -- A: Existence and uniqueness theorems -- B: Topographic systems -- C: Norms for vectors and matrices -- D: A contour integral -- E: Useful results -- References and further reading -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.