Dynamics and bifurcations
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Bibliographic Information
Dynamics and bifurcations
（Texts in applied mathematics, 3）
Springer-Verlag, 1996
- : us
- : gw
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"Corrected third printing, 1996"--T.p. verso
Includes bibliographical references (p. [545]-558) and index
Description and Table of Contents
- Volume
-
: us ISBN 9780387971414
Description
In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.
Table of Contents
I: Dimension One.- 1. Scalar Autonomous Equations.- 1.1. Existence and Uniqueness.- 1.2. Geometry of Flows.- 1.3. Stability of Equilibria.- 1.4. Equations on a Circle.- 2. Elementary Bifurcations.- 2.1. Dependence on Parameters - Examples.- 2.2. The Implicit Function Theorem.- 2.3. Local Perturbations Near Equilibria.- 2.4. An Example on a Circle.- 2.5. Computing Bifurcation Diagrams.- 2.6. Equivalence of Flows.- 3. Scalar Maps.- 3.1. Euler's Algorithm and Maps.- 3.2. Geometry of Scalar Maps.- 3.3. Bifurcations of Monotone Maps.- 3.4. Period-doubling Bifurcation.- 3.5. An Example: The Logistic Map.- II: Dimension One and One Half.- 4. Scalar Nonautonomous Equations.- 4.1. General Properties of Solutions.- 4.2. Geometry of Periodic Equations.- 4.3. Periodic Equations on a Cylinder.- 4.4. Examples of Periodic Equations.- 4.5. Stability of Periodic Solutions.- 5. Bifurcation of Periodic Equations.- 5.1. Bifurcations of Poincare Maps.- 5.2. Stability of Nonhyperbolic Periodic Solutions.- 5.3. Perturbations of Vector Fields.- 6. On Tori and Circles.- 6.1. Differential Equations on a Torus.- 6.2. Rotation Number.- 6.3. An Example: The Standard Circle Map.- III: Dimension Two.- 7. Planar Autonomous Systems.- 7.1. "Natural" Examples of Planar Systems.- 7.2. General Properties and Geometry.- 7.3. Product Systems.- 7.4. First Integrals and Conservative Systems.- 7.5. Examples of Elementary Bifurcations.- 8. Linear Systems.- 8.1. Properties of Solutions of Linear Systems.- 8.2. Reduction to Canonical Forms.- 8.3. Qualitative Equivalence in Linear Systems.- 8.4. Bifurcations in Linear Systems.- 8.5. Nonhomogeneous Linear Systems.- 8.6. Linear Systems with 1-periodic Coefficients.- 9. Near Equilibria.- 9.1. Asymptotic Stability from Linearization.- 9.2. Instability from Linearization.- 9.3. Liapunov Functions.- 9.4. An Invariance Principle.- 9.5. Preservation of a Saddle.- 9.6. Flow Equivalence Near Hyperbolic Equilibria.- 9.7. Saddle Connections.- 10. In the Presence of a Zero Eigenvalue.- 10.1. Stability.- 10.2. Bifurcations.- 10.3. Center Manifolds.- 11. In the Presence of Purely Imaginary Eigenvalues.- 11.1. Stability.- 11.2. Poincare-Andronov-Hopf Bifurcation.- 11.3. Computing Bifurcation Curves.- 12. Periodic Orbits.- 12.1. Poincare-Bendixson Theorem.- 12.2. Stability of Periodic Orbits.- 12.3. Local Bifurcations of Periodic Orbits.- 12.4. A Homoclinic Bifurcation.- 13. All Planar Things Considered.- 13.1. Structurally Stable Vector Fields.- 13.2. Dissipative Systems.- 13.3. One-parameter Generic Bifurcations.- 13.4. Bifurcations in the Presence of Symmetry.- 13.5. Local Two-parameter Bifurcations.- 14- Conservative and Gradient Systems.- 14.1. Second-order Conservative Systems.- 14.2. Bifurcations in Conservative Systems.- 14.3. Gradient Vector Fields.- 15. Planar Maps.- 15.1. Linear Maps.- 15.2. Near Fixed Points.- 15.3. Numerical Algorithms and Maps.- 15.4. Saddle Node and Period Doubling.- 15.5. Poincare-Andronov-Hopf Bifurcation.- 15.6. Area-preserving Maps.- IV: Higher Dimensions.- 16. Dimension Two and One Half.- 16.1. Forced Van der Pol.- 16.2. Forced Duffing.- 16.3. Near a Transversal Homoclinic Point.- 16.4. Forced and Damped Duffing.- 17. Dimension Three.- 17.1. Period Doubling.- 17.2. Bifurcation to Invariant Torus.- 17.3. Silnikov Orbits.- 17.4. The Lorenz Equations.- 18. Dimension Four.- 18.1. Integrable Hamiltonians.- 18.2. A Nonintegrable Hamiltonian.- Farewell.- APPENDIX: A Catalogue of Fundamental Theorems.- References.
- Volume
-
: gw ISBN 9783540971412
Description
This study presents ideas and examples about the geometry of dynamics and bifurcations of ordinary differential equations. The subject of differential and difference equations is an old and much-honoured chapter in science. In recent years, due primarily to the proliferation of computers, dynamical systems have returned to their roots in applications. It is the aim of this book to provide a modest foundation of knowledge for undergraduate and beginning graduate students in mathematics or science and engineering. Equations in dimension one and two constitute the majority of the text. It is demonstrated in particular that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Proceeding, the authors investigate the dynamics of planar autonomous equations where new dynamical behaviour, such as periodic and homoclinic orbits, appears.
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