Overview

The subject of differential and difference equations is an old and much-honored chapter in science, one which germinated in applied fields such as celestial mechanics, nonlinear oscillations, and fluid dynamics. In recent years, due primarily to the proliferation of computers, dynamical systems has once more turned to its roots in applications with perhaps a more mature look. Many of the available books and expository narratives either require extensive mathematical preparation, or are not designed to be used as textbooks. The authors have filled this void with the present book.

Full Product Details

Author:   Jack K. Hale ,  H. Buttanri ,  Hüseyin Kocak
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1991
Volume:   3
Dimensions:   Width: 15.50cm , Height: 3.00cm , Length: 23.50cm
Weight:   0.885kg
ISBN:  

9781461287650

ISBN 10:   1461287650
Pages:   574
Publication Date:   19 November 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

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 Poincaré 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. Poincaré-Andronov-Hopf Bifurcation.- 11.3. Computing Bifurcation Curves.- 12. Periodic Orbits.- 12.1. Poincaré-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. Poincaré-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.

Reviews

J.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples. -MATHEMATIKA

J.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples. MATHEMATIKA

"J.K. Hale, H. Kocak, and H. Buttanri Dynamics and Bifurcations ""This book takes the reader step by step through the vast subject of dynamical systems. Proceeding from 1 to 2 dimensions and onto higher dimensions in separate self-contained sections, the text is mathematically rigorous yet devoid of excess formalism. A refreshing balance is further achieved by the use of many excellent illustrations and a wealth of worked and unworked examples.""—MATHEMATIKA"

Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions