Methods of Qualitative Theory in Nonlinear Dynamics, Part 2

by Shilnikov, Leonid P.; Shilnikov, Andrey L.; Turaev, Dmitry V.; Chua, Leon O.; Shilnikov, Leonid P.

• ISBN13: 9789810240721
• ISBN10: 9810240724
• Format: Hardcover
• Copyright: 2001-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

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 Poincare, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in the book have been developed only recently and have not yet appeared in textbook form.In keeping with the self-contained nature of the book, all the topics are developed with introductory background and complete mathematical rigor. Generously illustrated and written at a high level of exposition, this invaluable book will appeal to both the beginner and the advanced,student of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject.

Table of Contents

Introduction to Part II	v
Structurally Stable Systems	393	(36)
Rough systems on a plane. Andronov--Pontryagin theorem	394	(5)
The set of center motions	399	(5)
General classification of center motions	404	(5)
Remarks on roughness of high-order dynamical systems	409	(3)
Morse--Smale systems	412	(7)
Some properties of Morse--Smale systems	419	(10)
Bifurcations of Dynamical Systems	429	(22)
Systems of first degree of non-roughness	430	(7)
Remarks on bifurcations of multi-dimensional systems	437	(3)
Structurally unstable homoclinic and heteroclinic orbits. Moduli of topological equivalence	440	(4)
Bifurcations in finite-parameter families of systems. Andronov's setup	444	(7)
The Behavior of Dynamical Systems on Stability Boundaries of Equilibrium States	451	(24)
The reduction theorems. The Lyapunov functions	452	(6)
The first critical case	458	(7)
The second critical case	465	(10)
The Behavior of Dynamical Systems on Stability Boundaries of Periodic Trajectories	475	(56)
The reduction of the Poincare map. Lyapunov functions	475	(5)
The first critical case	480	(9)
The second critical case	489	(4)
The third critical case. Weak resonances	493	(7)
Strong resonances	500	(15)
Passage through strong resonance on stability boundary	515	(12)
Additional remarks on resonances	527	(4)
Local Bifurcations on the Route over Stability Boundaries	531	(106)
Bifurcation surface and transverse families	531	(6)
Bifurcation of an equilibrium state with one zero exponent	537	(22)
Bifurcation of periodic orbits with multiplier +1	559	(19)
Bifurcation of periodic orbits with multiplier -1	578	(20)
Andronov-Hopf bifurcation	598	(13)
Birth of invariant torus	611	(12)
Bifurcations of resonant periodic orbits accompanying the birth of invariant torus	623	(14)
Global Bifurcations at the Disappearance of Saddle-Node Equilibrium States and Periodic Orbits	637	(50)
Bifurcations of a homoclinic loop to a saddle-node equilibrium state	638	(11)
Creation of an invariant torus	649	(17)
The formation of a Klein bottle	666	(4)
The blue sky catastrophe	670	(11)
On embedding into the flow	681	(6)
Bifurcations of Homoclinic Loops of Saddle Equilibrium States	687	(114)
Stability of a separatrix loop on the plane	688	(12)
Bifurcation of a limit cycle from a separatrix loop of a saddle with non-zero saddle value	700	(12)
Bifurcations of a separatrix loop with zero saddle value	712	(8)
Birth of periodic orbits from a homoclinic loop (the case dim Wu = 1)	720	(25)
Behavior of trajectories near a homoclinic loop in the case dim Wu > 1	745	(3)
Codimension-two bifurcations of homoclinic loops	748	(17)
Bifurcations of the homoclinic-8 and heteroclinic cycles	765	(24)
Estimates of the behavior of trajectories near a saddle equilibrium state	789	(12)
Safe and Dangerous Boundaries	801	(18)
Main stability boundaries of equilibrium states and periodic orbits	802	(2)
Classification of codimension-one boundaries of stability regions	804	(9)
Dynamically definite and indefinite boundaries of stability regions	813	(6)
Appendix C: Examples, Problems & Exercises	819	(108)
Bibliography	927	(16)
Index---Parts I & II	943