分歧理论和突变理论-动力系统-V_PDF下载[131MB-百度云]阿诺德

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Both bifurcatiotheory and catastrophe theory are studies of smooth systems，tbcusing oproperties that seem manifestly non-smooth. Bifurcations are suddechanges that occur ia system as one or more parameters are varied.Catastrophe theory is accurately described as singularity theory and its applications.
　　These two theories are important tools ithe study of differential equations and of related physical systems.Analyzing the bifurcations or singularities of a system provides useful qualitative informatioabout its behaviour. The authors have writtethis book with reffeshing clarity.Theexpositiois masterful，with penetrating insights.

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PrefaceChapter 1． Bifurcations of Equilibria1． Families and Deformations1．1． Families of Vector Fields1．2． The Space of Jets1．3． Sard’s Lemma and Transversality Theorems1．4． Simplest Applications： Singular Points of Generic Vector Fields1．5． Topologically Versal Deformations1．6． The ReductioTheorem1．7． Generic and Principal Families2． Bifurcations of Singular Points iGeneric One-Parameter Families2．1 Typical Germs and Principal Families2．2． Soft and Hard Loss of Stability3． Bifurcations of Singular Points iGeneric Multi-Parameter Families with Simply Degenerate Linear Parts3．1． Principal Families3．2． BifurcatioDiagrams of the Principal Families （3- ） iTable 13．3． BifurcatioDiagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families （4- ） iTable 14． Bifurcations of Singular Points of Vector Fields with a Doubly-Degenerate Linear Part4．1． A List of Degeneracies4．2． Two Zero Eigenvalues4．3． Reductions to Two-Dimensional Systems4．4． One Zero and a Pair of Purely Imaginary Eigenvalues4．5． Two Purely Imaginary Pairs4．6． Principal Deformations of Equations of Difficult Type iProblems with Two Pairs of Purely Imaginary Eigenvalues （Following Zolitdek）5． The Exponents of Soft and Hard Loss of Stability5．1． Definitions5．2． Table of ExponentsChapter 2． Bifurcations of Limit Cycles1． Bifurcations of Limit Cycles iGeneric One-Parameter Families1．1． Multiplier I1．2． Multiplier-1 and Period-Doubling Bifurcations1．3． A Pair of Complex Conjugate Multipliers1．4． Nonlocal Bifurcations iOne-Parameter Families of Diffeomorphisms1．5． Nonlocal Bifurcations of Periodic Solutions1．6． Bifurcations Resulting iDestructions of Invariant Tori2． Bifurcations of Cycles iGeneric Two-Parameter Families with anAdditional Simple Degeneracy2．1． A List of Degeneracies2．2． A Multiplier 1or-1 with Additional Degeneracy ithe Nonlinear Terms2．3． A Pair of Multipliers othe Unit Circle with Additional Degeneracy ithe Nonlinear Terms3． Bifurcations of Cycles iGeneric Two-Parameter Families with Strong Resonances of Orders q≠43．1． The Normal Form ithe Case of Unipotent JordaBlocks3．2． Averaging ithe Seifert and the M6bius Foliations3．3． Principal Vector Fields and their Deformations3．4． Versality of Principal Deformations3．5． Bifurcations of Stationary Solutions of Periodic Differential Equations with Strong Resonances of Orders q≠44． Bifurcations of Limit Cycles for a Pair of Multipliers Crossing theUnit Circle at±i4．1． Degenerate Families4．2． Degenerate Families Found Analytically4．3． Degenerate Families Found Numerically4．4． Bifurcations iNondegenerate Families4．5． Limit Cycles of Systems with a Fourth Order Symmetry5． Finitely-Smooth Normal Forms of Local Families5．1． A Synopsis of Results5．2． Definitions and Examples5．3． General Theorems and Deformations of Nonresonant Germs5．4． Reductioto Linear Normal Form5．5． Deformations of Germs of Diffeomorphisms of Poincare Type5．6． Deformations of Simply Resonant Hyperbolic Germs5．7． Deformations of Germs of Vector Fields with One Zero Eigenvalue at a Singular Point5．8． Functional Invariants of Diffeomorphisms of the Line5．9． Functional Invariants of Local Families of Diffeomorphisms5．10． Functional Invariants of Families of Vector Fields5．11． Functional Invariants of Topological Classifications of Local Families of Diffeomorphisms of the Line6． Feigenbaum Universality for Diffeomorphisms and Flows6．1． Period-Doubling Cascades6．2． Perestroikas of Fixed Points6．3． Cascades of n-fold Increases of Period6．4． Doubling iHamiltoniaSystems6．5． The Period-Doubling Operator for One-Dimensional Mappings6．6． The Universal Period-Doubling Mechanism for DiffeomorphismsChapter 3． Nonlocal Bifurcations1． Degeneracies of Codimensio1． Summary of Results1．1． Local and Nonlocal Bifurcations1．2． Nonhyperbolic Singular Points1．3． Nonhyperbolic Cycles1．4． Nontransversal Intersections of Manifolds1．5． Contours1．6． BifurcatioSurfaces1．7． Characteristics of Bifurcations1．8． Summary of Results2． Nonlocal Bifurcations of Flows oTwo-Dimensional Surfaces2．1． Semilocal Bifurcations of Flows oSurfaces2．2． Nonlocal Bifurcations oa Sphere： The One-Parameter Case ．2．3． Generic Families of Vector Fields2．4． Conditions for Genericity2．5． One-Parameter Families oSurfaces different from the Sphere2．6． Global Bifurcations of Systems with a Global Transversal Sectiooa Torus2．7． Some Global Bifurcations oa Kleibottle2．8． Bifurcations oa Two-Dimensional Sphere： The Multi-Parameter Case2．9． Some OpeQuestions3． Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Point3．1． A Node iits Hyperbolic Variables3．2． A Saddle iits Hyperbolic Variables： One Homoclinic Trajectory3．3． The Topological Bernoulli Automorphism3．4． A Saddle iits Hyperbolic Variables： Several Homoclinic Trajectories3．5． Principal Families4． Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Cycle4．1． The Structure of a Family of Homoclinic Trajectories4．2． Critical and Noncritical Cycles4．3． Creatioof a Smooth Two-Dimensional Attractor4．4． Creatioof Complex Invariant Sets （The Noncritical Case）．．．4．5． The Critical Case4．6． A Two-Step Transitiofrom Stability to Turbulence4．7． A Noncompact Set of Homoclinic Trajectories4．8． Intermittency4．9． Accessibility and Nonaccessibility4．10． Stability of Families of Diffeomorphisms4．11． Some OpeQuestions5． Hyperbolic Singular Points with Homoclinic Trajectories5．1． Preliminary Notions： Leading Directions and Saddle Numbers5．2． Bifurcations of Homoclinic Trajectories of a Saddle that Take Place othe Boundary of the Set of Morse-Smale Systems5．3． Requirements for Genericity5，4， Principal Families iR3 and their Properties5．5． Versality of the Principal Families5．6． A Saddle with Complex Leading DirectioiR35．7． AAddition： Bifurcations of Homoclinic Loops Outside the Boundary of a Set of Morse-Smale Systems5．8． AAddition： Creatioof a Strange Attractor upoBifurcatioof a Trajectory Homoclinic to a Saddle6． Bifurcations Related to Nontransversal Intersections6．1． Vector Fields with No Contours and No Homoclinic Trajectories6．2． A Theorem oInaccessibility6．3． Moduli6．4． Systems with Contours6．5． Diffeomorphisms with Nontrivial Basic Sets6．6， Vector Fields iR3 with Trajectories Homoclinic to a Cycle6．7． Symbolic Dynamics6．8． Bifurcations of Smale Horseshoes6．9． Vector Fields oa BifurcatioSurface6．10． Diffeomorphisms with aInfinite Set of Stable Periodic Trajectories7． Infinite Nonwandering Sets7．1． Vector Fields othe Two-Dimensional Torus7．2． Bifurcations of Systems with Two Homoclinic Curves of a Saddle7．3． Systems with Feigenbaum Attractors7．4． Birth of Nonwandering Sets7．5． Persistence and Smoothness of Invariant Manifolds7．6． The Degenerate Family and Its Neighborhood iFunctioSpace7．7． Birth of Tori ia Three-Dimensional Phase Space8． Attractors and their Bifurcations8．1． The Likely Limit Set According to Milnor （1985）8．2． Statistical Limit Sets8．3． Internal Bifurcations and Crises of Attractors8．4． Internal Bifurcations and Crises of Equilibria and Cycles8．5． Bifurcations of the Two-Dimensional TorusChapter 4． RelaxatioOscillations1． Fundamental Concepts1．1． AExample： vader Pors Equation1．2． Fast and Slow Motions1．3． The Slow Surface and Slow Equations1．4． The Slow Motioas aApproximatioto the Perturbed Motion1．5． The Phenomenoof Jumping2． Singularities of the Fast and Slow Motions2．1． Singularities of Fast Motions at Jump Points of Systems with One Fast Variable2．2． Singularities of Projections of the Slow Surface2．3． The Slow Motiofor Systems with One Slow Variable2．4． The Slow Motiofor Systems with Two Slow Variables2．5． Normal Forms of Phase Curves of the Slow Motion2．6． Connectiowith the Theory of Implicit Differential Equations2．7． Degeneratioof the Contact Structure3． The Asymptotics of RelaxatioOscillations3．1． Degenerate Systems3．2． Systems of First Approximation3．3． Normalizations of Fast-Slow Systems with Two Slow Variables for3．4． Derivatioof the Systems of First Approximation3．5． Investigatioof the Systems of First Approximation3．6． Funnels3．7． Periodic RelaxatioOscillations ithe Plane4． Delayed Loss of Stability as a Pair of Eigenvalues Cross the Imaginary Axis4．1． Generic Systems4．2． Delayed Loss of Stability4．3． Hard Loss of Stability iAnalytic Systems of Type 24．4． Hysteresis4．5． The Mechanism of Delay4．6． Computatioof the Moment of Jumping iAnalytic Systems4．7． Delay UpoLoss of Stability by a Cycle4．8． Delayed Loss of Stability and “Ducks” ．5． Duck Solutions5．1． AExample： A Singular Point othe Fold of the Slow Surface5．2． Existence of Duck Solutions5．3． The Evolutioof Simple Degenerate Ducks5．4． A Semi-local Phenomenon： Ducks with Relaxation5．5． Ducks iR3 and RnRecommended LiteratureReferencesAdditional References