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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