Methods of qualitative theory in nonlinear dynamics. Part 2 / Leonid P. Shilnikov [and others].
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 betw...
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|Series:||World Scientific series on nonlinear science. Monographs and treatises ;
v. 5. |
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- 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.