非线性动力系统和混沌应用导论

series prefacepreface to the second editionintroduction1 equilibrium solutions， stability， and linearized stability1.1 equilibria of vector fields1.2 stability of trajectories1.3 maps1.4 some terminology associated with fixed points1.5 application to the unforced duffing oscillator1.6 exercises2 liapunov functions2.1 exercises3 invariant manifolds： linear and nonlinear systems3.1 stable， unstable， and center subspaces of linear， autonomous vector fields3.2 stable， unstable， and center manifolds for fixed points of nonlinear， autonomous vector fields3.3 maps3.4 some examples3.5 existence of invariant manifolds： the main methods of proof， and how they work3.6 time-dependent hyperbolic trajectories and their stable and unstable manifolds3.7 invariant manifolds in a broader context3.8 exercises4 periodic orbits4.1 nonexistence of periodic orbits for two-dimensional， autonomous vector fields4.2 further remarks on periodic orbits4.3 exercises5 vector fields possessing an integral5.1vector fields on two-manifolds having an integral5.2 two degree-of-freedom hamiltonian systems and geometry5.3 exercises6 index theory6.1exercises7 some general properties of vector fields：existence， uniqueness， differentiability， and flows7.1 existence， uniqueness， differentiability with respect to initial conditions7.2 continuation of solutions7.3 differentiability with respect to parameters7.4 autonomous vector fields7.5 nonautonomous vector fields7.6 liouville's theorem7.7 exercises8 asymptotic behavior8.1 the asymptotic behavior of trajectories8.2 attracting sets， attractors， and basins of attraction8.3 the lasalle invariance principle8.4 attraction in nonautonomous systems8.5 exercises9 the poincare-bendixson theorem9.1 exercises10 poincare maps10.1 case 1：poincar6 map near a periodic orbit10.2 case 2：the poincare map of a time-periodic ordinary differential equation10.3 case 3：the poincare map near a homoclinic orbit10.4 case 4：poincar6 map associated with a two degree-of-freedom hamiltonian system10.5 exercises11 conjugacies of maps， and varying the cross-section11.1 case 1：poincar6 map near a periodic orbit： variation of the cross-section11.2 case 2：the poincare map of a time-periodic ordinary differential equation： variation of the cross-section12 structural stability， genericity， and transversality12.1 definitions of structural stability and genericity12.2 transversality12.3 exercises13 1 agrange's equations13.1 generalized coordinates13.2 derivation of lagrange's equations13.3 the energy integral13.4 momentum integrals13.5 hamilton's equations13.6 cyclic coordinates， routh's equations， and reduction of the number of equations13.7 variational methods13.8 the hamilton-jacobi equation13.9 exercises14 harniltonian vector fields14.1 symplectic forms14.2 poisson brackets14.3 symplectic or canonical transformations14.4 transformation of hamilton's equations under symplectic transformations14.5 completely integrable hamiltonian systems14.6 dynamics of completely integrable hamiltonian systems in action-angle coordinates14.7 perturbations of completely integrable hamiltonian systems in action-angle coordinates14.8 stability of elliptic equilibria14.9 discrete-time hamiltonian dynamical systems： iteration of symplectic maps14.10 generic properties of hamiltonian dynamical systems14.11 exercises15 gradient vector fields15.1 exercises16 reversible dynamical systems16.1 the definition of reversible dynamical systems16.2 examples of reversible dynamical systems16.3 linearization of reversible dynamical systems16.4 additional properties of reversible dynamical systems16.5 exercises17 asymptotically autonomous vector fields17.1 exercises18 center manifolds18.1 center manifolds for vector fields18.2 center manifolds depending on parameters.18.3 the inclusion of linearly unstable directions18.4 center manifolds for maps18.5 properties of center manifolds18.6 final remarks on center manifolds18.7 exercises19 normal forms19.1 normal forms for vector fields19.2 normal forms for vector fields with parameters19.3 normal forms for maps19.4 exercises19.5 the elphick-tirapegui-brachet-coullet-iooss19.6 exercises19.7 lie groups， lie group actions， and symmetries19.8 exercises19.9 normal form coefficients19.10 hamiltonian normal forms19.11 exercises19.12 conjugacies and equivalences of vector fields19.13 final remarks on normal forms20 bifurcation of fixed points of vector fields20.1 a zero eigenvalue20.2 a pure imaginary pair of eigenvalues： the poincare-andronov-hopf bifurcation20.3 stability of bifurcations under perturbations20.4 the idea of the codimension of a bifurcation20.5 versal deformations of families of matrices20.6 the double-zero eigenvalue： the takens-bogdanov bifurcation20.7 a zero and a pure imaginary pair of eigenvalues： the hopf-steady state bifurcation20.8 versal deformations of linear hamiltonian systems20.9 elementary hamiltonian bifurcations21 bifurcations of fixed points of maps21.1 an eigenvalue of i21.2 an eigenvalue of -1： period doubling21.3 a pair of eigenvalues of 1viodulus 1： the naimark-sacker bifurcation21.4 the codimension of local bifurcations of maps21.5 exercises21.6 maps of the circle22 on the interpretation and application of bifurcation diagrams： a word of caution23 the smale horseshoe23.1 definition of the smale horseshoe map23.2 construction of the invariant set23.3 symbolic dynamics23.4 the dynamics on the invariant set23.5 chaos23.6 final remarks and observations24 symbolic dynamics24.1 the structure of the space of symbol sequences24.2 the shift map24.3 exercises25 the conley-moser conditions， or “how to prove that a dynamical system is chaotic”25.1 the main theorem25.2 sector bundles25.3 exercises26 dynamics near homoclinic points of two-dimensional maps26.1 heteroclinic cycles26.2 exercises27 orbits homoclinic to hyperbolic fixed points in three-dimensional autonomous vector fields27.1 the technique of analysis27.2 orbits homoclinic to a saddle-point with purely real eigenvalues27.3 orbits homoclinic to a saddle-focus27.4 exercises28 melnikov's method for homoclinic orbits in two-dimensional， time-periodic vector fields28.1 the general theory28.2 poincare maps and the geometry of the melnikov function28.3 some properties of the melnikov function28.4 homoclinic bifurcations28.5 application to the damped， forced duffing oscillator28.6 exercises29 liapunov exponents29.1 liapunov exponents of a trajectory29.2 examples29.3 numerical computation of liapunov exponents29.4 exercises30 chaos and strange attractors30.1 exercises31 hyperbolic invariant sets： a chaotic saddle31.1 hyperbolicity of the invariant cantor set a constructed in chapter 2531.2 hyperbolic invariant sets in r“31.3 a consequence of hyperbolicity： the shadowing lemma31.4 exercises32 long period sinks in dissipative systems and elliptic islands in conservative systems 32.1 homoclinic bifurcations32.2 newhouse sinks in dissipative systems32.3 islands of stability in conservative systems32.4 exercises33 global bifurcations arising from local codimension——two bifurcations33.1 the double-zero eigenvalue33.2 a zero and a pure imaginary pair of eigenvalues33.3 exercises34 glossary of frequently used termsbibliographyindex

《非线性动力系统和混沌应用导论（第2版）》是一部高年级的本科生和研究生学生学习应用非线性动力学和混沌的入门教程。《非线性动力系统和混沌应用导论（第2版）》的重点讲述大量的技巧和观点，包括了深层次学习本科目的必备的核心知识，这些可以使学生能够学习特殊动力系统并获得学习这些系统大量信息。因此，像工程、物理、化学和生物专业读者不需要另外学习大量的预备知识。新的版本中包括了大量有关不变流形理论和规范模的新材料，拉格朗日、哈密尔顿、梯度和可逆动力系统的也有讨论，也包括了哈密尔顿分叉和环映射的基本性质。本书附了丰富的参考资料和详细的术语表，似的《非线性动力系统和混沌应用导论（第2版）》的可读性更加增大。