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Author V.I.Arnol'd

Press Science Press

Isbn 7030234936/9787030234933

Publication Date Jan 2009

Pages 271

Size 16K

Edition First

Cover Hardcover

Language English

Page material gelatine plate paper

Product Description:

Both bifurcation theory and catastrophe theory are studies of smooth systems，tbcusing on properties that seem manifestly non-smooth. Bifurcations are sudden changes that occur in a 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 in the study of differential equations and of related physical systems.Analyzing the bifurcations or singularities of a system provides useful qualitative information about its behaviour. The authors have written this book with reffeshing clarity.Theexposition is masterful，with penetrating insights.

Content:

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 Reduction Theorem

1.7. Generic and Principal Families

2. Bifurcations of Singular Points in Generic One-Parameter Families

2.1 Typical Germs and Principal Families

2.2. Soft and Hard Loss of Stability

3. Bifurcations of Singular Points in Generic Multi-Parameter Families with Simply Degenerate Linear Parts

3.1. Principal Families

3.2. Bifurcation Diagrams of the Principal Families (3-+) in Table 1

3.3. Bifurcation Diagrams with Respect to Weak Equivalence and Phase Portraits of the Principal Families (4-+) in Table 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 in Problems 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 in Generic 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 in One-Parameter Families of Diffeomorphisms

1.5. Nonlocal Bifurcations of Periodic Solutions

1.6. Bifurcations Resulting in Destructions of Invariant Tori

2. Bifurcations of Cycles in Generic Two-Parameter Families with an

Additional Simple Degeneracy

2.1. A List of Degeneracies

2.2. A Multiplier+1or-1 with Additional Degeneracy in the Nonlinear Terms

2.3. A Pair of Multipliers on the Unit Circle with Additional Degeneracy in the Nonlinear Terms

3. Bifurcations of Cycles in Generic Two-Parameter Families with Strong Resonances of Orders q≠4

3.1. The Normal Form in the Case of Unipotent Jordan Blocks

3.2. Averaging in the 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 in Nondegenerate 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. Reduction to 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 in Hamiltonian Systems

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 Codimension 1. 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. Bifurcation Surfaces

1.7. Characteristics of Bifurcations

1.8. Summary of Results

2. Nonlocal Bifurcations of Flows on Two-Dimensional Surfaces

2.1. Semilocal Bifurcations of Flows on Surfaces

2.2. Nonlocal Bifurcations on a Sphere: The One-Parameter Case .

2.3. Generic Families of Vector Fields

2.4. Conditions for Genericity

2.5. One-Parameter Families on Surfaces different from the Sphere

2.6. Global Bifurcations of Systems with a Global Transversal Section on a Torus

2.7. Some Global Bifurcations on a Klein bottle

2.8. Bifurcations on a Two-Dimensional Sphere: The Multi-Parameter Case

2.9. Some Open Questions

3. Bifurcations of Trajectories Homoclinic to a Nonhyperbolic Singular Point

3.1. A Node in its Hyperbolic Variables

3.2. A Saddle in its Hyperbolic Variables: One Homoclinic Trajectory

3.3. The Topological Bernoulli Automorphism

3.4. A Saddle in its 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. Creation of a Smooth Two-Dimensional Attractor

4.4. Creation of Complex Invariant Sets (The Noncritical Case) ...

4.5. The Critical Case

4.6. A Two-Step Transition from 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 Open Questions

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 on the Boundary of the Set of Morse-Smale Systems

5.3. Requirements for Genericity

5, 4, Principal Families in R3 and their Properties

5.5. Versality of the Principal Families

5.6. A Saddle with Complex Leading Direction in R3

5.7. An Addition: Bifurcations of Homoclinic Loops Outside the Boundary of a Set of Morse-Smale Systems

5.8. An Addition: Creation of a Strange Attractor upon Bifurcation of 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 on Inaccessibility

6.3. Moduli

6.4. Systems with Contours

6.5. Diffeomorphisms with Nontrivial Basic Sets

6.6, Vector Fields in R3 with Trajectories Homoclinic to a Cycle

6.7. Symbolic Dynamics

6.8. Bifurcations of Smale Horseshoes

6.9. Vector Fields on a Bifurcation Surface

6.10. Diffeomorphisms with an Infinite Set of Stable Periodic Trajectories

7. Infinite Nonwandering Sets

7.1. Vector Fields on the 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 in Function Space

7.7. Birth of Tori in a 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. Relaxation Oscillations

1. Fundamental Concepts

1.1. An Example: van der Pors Equation

1.2. Fast and Slow Motions

1.3. The Slow Surface and Slow Equations

1.4. The Slow Motion as an Approximation to the Perturbed Motion

1.5. The Phenomenon of 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 Motion for Systems with One Slow Variable

2.4. The Slow Motion for Systems with Two Slow Variables

2.5. Normal Forms of Phase Curves of the Slow Motion

2.6. Connection with the Theory of Implicit Differential Equations

2.7. Degeneration of the Contact Structure

3. The Asymptotics of Relaxation Oscillations

3.1. Degenerate Systems

3.2. Systems of First Approximation

3.3. Normalizations of Fast-Slow Systems with Two Slow Variables for

3.4. Derivation of the Systems of First Approximation

3.5. Investigation of the Systems of First Approximation

3.6. Funnels

3.7. Periodic Relaxation Oscillations in the 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 in Analytic Systems of Type 2

4.4. Hysteresis

4.5. The Mechanism of Delay

4.6. Computation of the Moment of Jumping in Analytic Systems

4.7. Delay Upon Loss of Stability by a Cycle

4.8. Delayed Loss of Stability and "Ducks" .

5. Duck Solutions

5.1. An Example: A Singular Point on the Fold of the Slow Surface

5.2. Existence of Duck Solutions

5.3. The Evolution of Simple Degenerate Ducks

5.4. A Semi-local Phenomenon: Ducks with Relaxation

5.5. Ducks in R3 and Rn

Recommended Literature

References

Additional References

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