分歧理论和突变理论-动力系统-V 本书特色
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.
分歧理论和突变理论-动力系统-V 内容简介
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分歧理论和突变理论-动力系统-V 目录
Preface
Chapter 1． Bifurcations of Equilibria
1． Families and Deformations
1．1． Families of Vector Fields
1．2． The Space of Jets
1．3． Sard's Lemma and Transversality Theorems
1．4． Simplest Applications： Singular Points of Generic Vector Fields
1．5． Topologically Versal Deformations
1．6． The ReductioTheorem
1．7． Generic and Principal Families
2． Bifurcations of Singular Points iGeneric One-Parameter Families
2．1 Typical Germs and Principal Families
2．2． Soft and Hard Loss of Stability
3． Bifurcations of Singular Points iGeneric Multi-Parameter Families with Simply Degenerate Linear Parts
3．1． Principal Families
3．2． BifurcatioDiagrams of the Principal Families （3- ） iTable 1
3．3． BifurcatioDiagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families （4- ） iTable 1
4． Bifurcations of Singular Points of Vector Fields with a Doubly-Degenerate Linear Part
4．1． A List of Degeneracies
4．2． Two Zero Eigenvalues
4．3． Reductions to Two-Dimensional Systems
4．4． One Zero and a Pair of Purely Imaginary Eigenvalues
4．5． Two Purely Imaginary Pairs
4．6． 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 Stability
5．1． Definitions
5．2． Table of Exponents
Chapter 2． Bifurcations of Limit Cycles
1． Bifurcations of Limit Cycles iGeneric One-Parameter Families
1．1． Multiplier I
1．2． Multiplier-1 and Period-Doubling Bifurcations
1．3． A Pair of Complex Conjugate Multipliers
1．4． Nonlocal Bifurcations iOne-Parameter Families of Diffeomorphisms
1．5． Nonlocal Bifurcations of Periodic Solutions
1．6． Bifurcations Resulting iDestructions of Invariant Tori
2． Bifurcations of Cycles iGeneric Two-Parameter Families with an
Additional Simple Degeneracy
2．1． A List of Degeneracies
2．2． A Multiplier 1or-1 with Additional Degeneracy ithe Nonlinear Terms
2．3． A Pair of Multipliers othe Unit Circle with Additional Degeneracy ithe Nonlinear Terms
3． Bifurcations of Cycles iGeneric Two-Parameter Families with Strong Resonances of Orders q≠4
3．1． The Normal Form ithe Case of Unipotent JordaBlocks
3．2． Averaging ithe Seifert and the M6bius Foliations
3．3． Principal Vector Fields and their Deformations
3．4． Versality of Principal Deformations
3．5． Bifurcations of Stationary Solutions of Periodic Differential Equations with Strong Resonances of Orders q≠4
4． Bifurcations of Limit Cycles for a Pair of Multipliers Crossing the
Unit Circle at±i
4．1． Degenerate Families
4．2． Degenerate Families Found Analytically
4．3． Degenerate Families Found Numerically
4．4． Bifurcations iNondegenerate Families
4．5． Limit Cycles of Systems with a Fourth Order Symmetry
5． Finitely-Smooth Normal Forms of Local Families
5．1． A Synopsis of Results
5．2． Definitions and Examples
5．3． General Theorems and Deformations of Nonresonant Germs
5．4． Reductioto Linear Normal Form
5．5． Deformations of Germs of Diffeomorphisms of Poincare Type
5．6． Deformations of Simply Resonant Hyperbolic Germs
5．7． Deformations of Germs of Vector Fields with One Zero Eigenvalue at a Singular Point
5．8． Functional Invariants of Diffeomorphisms of the Line
5．9． Functional Invariants of Local Families of Diffeomorphisms
5．10． Functional Invariants of Families of Vector Fields
5．11． Functional Invariants of Topological Classifications of Local Families of Diffeomorphisms of the Line
6． Feigenbaum Universality for Diffeomorphisms and Flows
6．1． Period-Doubling Cascades
6．2． Perestroikas of Fixed Points
6．3． Cascades of n-fold Increases of Period
6．4． Doubling iHamiltoniaSystems
6．5． The Period-Doubling Operator for One-Dimensional Mappings
6．6． The Universal Period-Doubling Mechanism for Diffeomorphisms
Chapter 3． Nonlocal Bifurcations
1． Degeneracies of Codimensio1． Summary of Results
1．1． Local and Nonlocal Bifurcations
1．2． Nonhyperbolic Singular Points
1．3． Nonhyperbolic Cycles
1．4． Nontransversal Intersections of Manifolds
1．5． Contours
1．6． BifurcatioSurfaces
1．7． Characteristics of Bifurcations
1．8． Summary of Results
2． Nonlocal Bifurcations of Flows oTwo-Dimensional Surfaces
2．1． Semilocal Bifurcations of Flows oSurfaces
2．2． Nonlocal Bifurcations oa Sphere： The One-Parameter Case ．
2．3． Generic Families of Vector Fields
2．4． Conditions for Genericity
2．5． One-Parameter Families oSurfaces different from the Sphere
2．6． Global Bifurcations of Systems with a Global Transversal Sectiooa Torus
2．7． Some Global Bifurcations oa Kleibottle
2．8． Bifurcations oa Two-Dimensional Sphere： The Multi-Parameter Case
2．9． Some OpeQuestions
3． Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Point
3．1． A Node iits Hyperbolic Variables
3．2． A Saddle iits Hyperbolic Variables： One Homoclinic Trajectory
3．3． The Topological Bernoulli Automorphism
3．4． A Saddle iits Hyperbolic Variables： Several Homoclinic Trajectories
3．5． Principal Families
4． Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Cycle
4．1． The Structure of a Family of Homoclinic Trajectories
4．2． Critical and Noncritical Cycles
4．3． Creatioof a Smooth Two-Dimensional Attractor
4．4． Creatioof Complex Invariant Sets （The Noncritical Case） ．．．
4．5． The Critical Case
4．6． A Two-Step Transitiofrom Stability to Turbulence
4．7． A Noncompact Set of Homoclinic Trajectories
4．8． Intermittency
4．9． Accessibility and Nonaccessibility
4．10． Stability of Families of Diffeomorphisms
4．11． Some OpeQuestions
5． Hyperbolic Singular Points with Homoclinic Trajectories
5．1． Preliminary Notions： Leading Directions and Saddle Numbers
5．2． Bifurcations of Homoclinic Trajectories of a Saddle that Take Place othe Boundary of the Set of Morse-Smale Systems
5．3． Requirements for Genericity
5，4， Principal Families iR3 and their Properties
5．5． Versality of the Principal Families
5．6． A Saddle with Complex Leading DirectioiR3
5．7． AAddition： Bifurcations of Homoclinic Loops Outside the Boundary of a Set of Morse-Smale Systems
5．8． AAddition： Creatioof a Strange Attractor upoBifurcatioof a Trajectory Homoclinic to a Saddle
6． Bifurcations Related to Nontransversal Intersections
6．1． Vector Fields with No Contours and No Homoclinic Trajectories
6．2． A Theorem oInaccessibility
6．3． Moduli
6．4． Systems with Contours
6．5． Diffeomorphisms with Nontrivial Basic Sets
6．6， Vector Fields iR3 with Trajectories Homoclinic to a Cycle
6．7． Symbolic Dynamics
6．8． Bifurcations of Smale Horseshoes
6．9． Vector Fields oa BifurcatioSurface
6．10． Diffeomorphisms with aInfinite Set of Stable Periodic Trajectories
7． Infinite Nonwandering Sets
7．1． Vector Fields othe Two-Dimensional Torus
7．2． Bifurcations of Systems with Two Homoclinic Curves of a Saddle
7．3． Systems with Feigenbaum Attractors
7．4． Birth of Nonwandering Sets
7．5． Persistence and Smoothness of Invariant Manifolds
7．6． The Degenerate Family and Its Neighborhood iFunctioSpace
7．7． Birth of Tori ia Three-Dimensional Phase Space
8． Attractors and their Bifurcations
8．1． The Likely Limit Set According to Milnor （1985）
8．2． Statistical Limit Sets
8．3． Internal Bifurcations and Crises of Attractors
8．4． Internal Bifurcations and Crises of Equilibria and Cycles
8．5． Bifurcations of the Two-Dimensional Torus
Chapter 4． RelaxatioOscillations
1． Fundamental Concepts
1．1． AExample： vader Pors Equation
1．2． Fast and Slow Motions
1．3． The Slow Surface and Slow Equations
1．4． The Slow Motioas aApproximatioto the Perturbed Motion
1．5． The Phenomenoof Jumping
2． Singularities of the Fast and Slow Motions
2．1． Singularities of Fast Motions at Jump Points of Systems with One Fast Variable
2．2． Singularities of Projections of the Slow Surface
2．3． The Slow Motiofor Systems with One Slow Variable
2．4． The Slow Motiofor Systems with Two Slow Variables
2．5． Normal Forms of Phase Curves of the Slow Motion
2．6． Connectiowith the Theory of Implicit Differential Equations
2．7． Degeneratioof the Contact Structure
3． The Asymptotics of RelaxatioOscillations
3．1． Degenerate Systems
3．2． Systems of First Approximation
3．3． Normalizations of Fast-Slow Systems with Two Slow Variables for
3．4． Derivatioof the Systems of First Approximation
3．5． Investigatioof the Systems of First Approximation
3．6． Funnels
3．7． Periodic RelaxatioOscillations ithe Plane
4． Delayed Loss of Stability as a Pair of Eigenvalues Cross the Imaginary Axis
4．1． Generic Systems
4．2． Delayed Loss of Stability
4．3． Hard Loss of Stability iAnalytic Systems of Type 2
4．4． Hysteresis
4．5． The Mechanism of Delay
4．6． Computatioof the Moment of Jumping iAnalytic Systems
4．7． Delay UpoLoss of Stability by a Cycle
4．8． Delayed Loss of Stability and “Ducks” ．
5． Duck Solutions
5．1． AExample： A Singular Point othe Fold of the Slow Surface
5．2． Existence of Duck Solutions
5．3． The Evolutioof Simple Degenerate Ducks
5．4． A Semi-local Phenomenon： Ducks with Relaxation
5．5． Ducks iR3 and Rn
Recommended Literature
References
Additional References
