series preface preface to the second edition introduction 1 equilibrium solutions， stability， and linearized stability 1.1 equilibria of vector fields 1.2 stability of trajectories 1.3 maps 1.4 some terminology associated with fixed points 1.5 application to the unforced duffing oscillator 1.6 exercises 2 liapunov functions 2.1 exercises 3 invariant manifolds： linear and nonlinear systems 3.1 stable， unstable， and center subspaces of linear， autonomous vector fields 3.2 stable， unstable， and center manifolds for fixed points of nonlinear， autonomous vector fields 3.3 maps 3.4 some examples 3.5 existence of invariant manifolds： the main methods of proof， and how they work 3.6 time-dependent hyperbolic trajectories and their stable and unstable manifolds 3.7 invariant manifolds in a broader context 3.8 exercises 4 periodic orbits 4.1 nonexistence of periodic orbits for two-dimensional， autonomous vector fields 4.2 further remarks on periodic orbits 4.3 exercises 5 vector fields possessing an integral 5.1vector fields on two-manifolds having an integral 5.2 two degree-of-freedom hamiltonian systems and geometry 5.3 exercises 6 index theory 6.1exercises 7 some general properties of vector fields：existence， uniqueness， differentiability， and flows 7.1 existence， uniqueness， differentiability with respect to initial conditions 7.2 continuation of solutions 7.3 differentiability with respect to parameters 7.4 autonomous vector fields 7.5 nonautonomous vector fields 7.6 liouville's theorem 7.7 exercises 8 asymptotic behavior 8.1 the asymptotic behavior of trajectories 8.2 attracting sets， attractors， and basins of attraction 8.3 the lasalle invariance principle 8.4 attraction in nonautonomous systems 8.5 exercises 9 the poincare-bendixson theorem 9.1 exercises 10 poincare maps 10.1 case 1：poincar6 map near a periodic orbit 10.2 case 2：the poincare map of a time-periodic ordinary differential equation 10.3 case 3：the poincare map near a homoclinic orbit 10.4 case 4：poincar6 map associated with a two degree-of-freedom hamiltonian system 10.5 exercises 11 conjugacies of maps， and varying the cross-section 11.1 case 1：poincar6 map near a periodic orbit： variation of the cross-section 11.2 case 2：the poincare map of a time-periodic ordinary differential equation： variation of the cross-section 12 structural stability， genericity， and transversality 12.1 definitions of structural stability and genericity 12.2 transversality 12.3 exercises 13 1 agrange's equations 13.1 generalized coordinates 13.2 derivation of lagrange's equations 13.3 the energy integral 13.4 momentum integrals 13.5 hamilton's equations 13.6 cyclic coordinates， routh's equations， and reduction of the number of equations 13.7 variational methods 13.8 the hamilton-jacobi equation 13.9 exercises 14 harniltonian vector fields 14.1 symplectic forms 14.2 poisson brackets 14.3 symplectic or canonical transformations 14.4 transformation of hamilton's equations under symplectic transformations 14.5 completely integrable hamiltonian systems 14.6 dynamics of completely integrable hamiltonian systems in action-angle coordinates 14.7 perturbations of completely integrable hamiltonian systems in action-angle coordinates 14.8 stability of elliptic equilibria 14.9 discrete-time hamiltonian dynamical systems： iteration of symplectic maps 14.10 generic properties of hamiltonian dynamical systems 14.11 exercises 15 gradient vector fields 15.1 exercises 16 reversible dynamical systems 16.1 the definition of reversible dynamical systems 16.2 examples of reversible dynamical systems 16.3 linearization of reversible dynamical systems 16.4 additional properties of reversible dynamical systems 16.5 exercises 17 asymptotically autonomous vector fields 17.1 exercises 18 center manifolds 18.1 center manifolds for vector fields 18.2 center manifolds depending on parameters. 18.3 the inclusion of linearly unstable directions 18.4 center manifolds for maps 18.5 properties of center manifolds 18.6 final remarks on center manifolds 18.7 exercises 19 normal forms 19.1 normal forms for vector fields 19.2 normal forms for vector fields with parameters 19.3 normal forms for maps 19.4 exercises 19.5 the elphick-tirapegui-brachet-coullet-iooss 19.6 exercises 19.7 lie groups， lie group actions， and symmetries 19.8 exercises 19.9 normal form coefficients 19.10 hamiltonian normal forms 19.11 exercises 19.12 conjugacies and equivalences of vector fields 19.13 final remarks on normal forms 20 bifurcation of fixed points of vector fields 20.1 a zero eigenvalue 20.2 a pure imaginary pair of eigenvalues： the poincare-andronov-hopf bifurcation 20.3 stability of bifurcations under perturbations 20.4 the idea of the codimension of a bifurcation 20.5 versal deformations of families of matrices 20.6 the double-zero eigenvalue： the takens-bogdanov bifurcation 20.7 a zero and a pure imaginary pair of eigenvalues： the hopf-steady state bifurcation 20.8 versal deformations of linear hamiltonian systems 20.9 elementary hamiltonian bifurcations 21 bifurcations of fixed points of maps 21.1 an eigenvalue of i 21.2 an eigenvalue of -1： period doubling 21.3 a pair of eigenvalues of 1viodulus 1： the naimark-sacker bifurcation 21.4 the codimension of local bifurcations of maps 21.5 exercises 21.6 maps of the circle 22 on the interpretation and application of bifurcation diagrams： a word of caution 23 the smale horseshoe 23.1 definition of the smale horseshoe map 23.2 construction of the invariant set 23.3 symbolic dynamics 23.4 the dynamics on the invariant set 23.5 chaos 23.6 final remarks and observations 24 symbolic dynamics 24.1 the structure of the space of symbol sequences 24.2 the shift map 24.3 exercises 25 the conley-moser conditions， or “how to prove that a dynamical system is chaotic” 25.1 the main theorem 25.2 sector bundles 25.3 exercises 26 dynamics near homoclinic points of two-dimensional maps 26.1 heteroclinic cycles 26.2 exercises 27 orbits homoclinic to hyperbolic fixed points in three-dimensional autonomous vector fields 27.1 the technique of analysis 27.2 orbits homoclinic to a saddle-point with purely real eigenvalues 27.3 orbits homoclinic to a saddle-focus 27.4 exercises 28 melnikov's method for homoclinic orbits in two-dimensional， time-periodic vector fields 28.1 the general theory 28.2 poincare maps and the geometry of the melnikov function 28.3 some properties of the melnikov function 28.4 homoclinic bifurcations 28.5 application to the damped， forced duffing oscillator 28.6 exercises 29 liapunov exponents 29.1 liapunov exponents of a trajectory 29.2 examples 29.3 numerical computation of liapunov exponents 29.4 exercises 30 chaos and strange attractors 30.1 exercises 31 hyperbolic invariant sets： a chaotic saddle 31.1 hyperbolicity of the invariant cantor set a constructed in chapter 25 31.2 hyperbolic invariant sets in r“ 31.3 a consequence of hyperbolicity： the shadowing lemma 31.4 exercises 32 long period sinks in dissipative systems and elliptic islands in conservative systems 32.1 homoclinic bifurcations 32.2 newhouse sinks in dissipative systems 32.3 islands of stability in conservative systems 32.4 exercises 33 global bifurcations arising from local codimension——two bifurcations 33.1 the double-zero eigenvalue 33.2 a zero and a pure imaginary pair of eigenvalues 33.3 exercises 34 glossary of frequently used terms bibliography index
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