Methods of qualitative theory in nonlinear dynamics. Part 2 /

Methods of qualitative theory in nonlinear dynamics. Part 2 /

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Bibliographic Details
Imprint:New Jersey : World Scientific, ©2001.
Description:1 online resource (1 volume) : illustrations
Language:English
Series:World scientific series on nonlinear science. Series A, Monographs and treatises ; v. 5
World Scientific series on nonlinear science. Series A, Monographs and treatises ; v. 5.
Subject:
Differentiable dynamical systems.
Nonlinear theories.
Nonlinear mechanics.
Differentiable dynamical systems.
Nonlinear mechanics.
Nonlinear theories.
Electronic books.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/11178915
Hidden Bibliographic Details
Other authors / contributors:Shilʹnikov, L. P.
ISBN:9789812798558
9812798552
Notes:Includes bibliographical references (pages 927-942) and index.
Print version record.
Summary:Bifurcation and chaos has dominated research in nonlinear dynamics for over two decades, and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book has been written to serve that unfulfilled need. Following the footsteps of Poincaré, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many o.
Other form:Print version: Methods of qualitative theory in nonlinear dynamics. Part 2. New Jersey : World Scientific, ©2001 9810240724 9789810240721
Table of Contents:
  • Introduction to Part II; Contents; Chapter 7. STRUCTURALLY STABLE SYSTEMS; 7.1. Rough systems on a plane. Andronov-Pontryagin theorem; 7.2. The set of center motions; 7.3. General classification of center motions; 7.4. Remarks on roughness of high-order dynamical systems; 7.5. Morse-Smale systems; 7.6. Some properties of Morse-Smale systems; Chapter 8. BIFURCATIONS OF DYNAMICAL SYSTEMS; 8.1. Systems of first degree of non-roughness; 8.2. Remarks on bifurcations of multi-dimensional systems; 8.3. Structurally unstable homoclinic and heteroclinic orbits. Moduli of topological equivalence.
  • 8.4. Bifurcations in finite-parameter families of systems. Andronov's setupChapter 9. THE BEHAVIOR OF DYNAMICAL SYSTEMS ON STABILITY BOUNDARIES OF EQUILIBRIUM STATES; 9.1. The reduction theorems. The Lyapunov functions; 9.2. The first critical case; 9.3. The second critical case; Chapter 10. THE BEHAVIOR OF DYNAMICAL SYSTEMS ON STABILITY BOUNDARIES OF PERIODIC TRAJECTORIES; 10.1. The reduction of the Poincare map. Lyapunov functions; 10.2. The first critical case; 10.3. The second critical case; 10.4. The third critical case. Weak resonances; 10.5. Strong resonances.
  • 10.6. Passage through strong resonance on stability boundary10.7. Additional remarks on resonances; Chapter 11. LOCAL BIFURCATIONS ON THE ROUTE OVER STABILITY BOUNDARIES; 11.1. Bifurcation surface and transverse families; 11.2. Bifurcation of an equilibrium state with one zero exponent; 11.3. Bifurcation of periodic orbits with multiplier +1; 11.4. Bifurcation of periodic orbits with multiplier -1; 11.5. Andronov-Hopf bifurcation; 11.6. Birth of invariant torus; 11.7. Bifurcations of resonant periodic orbits accompanying the birth of invariant torus.
  • Chapter 12. GLOBAL BIFURCATIONS AT THE DISAPPEARANCE OF SADDLE-NODE EQUILIBRIUM STATES AND PERIODIC ORBITS12.1. Bifurcations of a homoclinic loop to a saddle-node equilibrium state; 12.2. Creation of an invariant torus; 12.3. The formation of a Klein bottle; 12.4. The blue sky catastrophe; 12.5. On embedding into the flow; Chapter 13. BIFURCATIONS OF HOMOCLINIC LOOPS OF SADDLE EQUILIBRIUM STATES; 13.1. Stability of a separatrix loop on the plane; 13.2. Bifurcation of a limit cycle from a separatrix loop of a saddle with non-zero saddle value.
  • 13.3. Bifurcations of a separatrix loop with zero saddle value13.4. Birth of periodic orbits from a homoclinic loop (the case dim Wu = 1); 13.5. Behavior of trajectories near a homoclinic loop in the case dim Wu> 1; 13.6. Codimension-two bifurcations of homoclinic loops; 13.7. Bifurcations of the homoclinic-8 and heteroclinic cycles; 13.8. Estimates of the behavior of trajectories near a saddle equilibrium state; Chapter 14. SAFE AND DANGEROUS BOUNDARIES; 14.1. Main stability boundaries of equilibrium states and periodic orbits.

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