Differential equations on manifolds
- Studying of basic methods of qualitative theory and important classes od dynamical systems on manifolds: local and global analysis, hyperbolic points and sets, energy function, Morse-Smale systems, hyperbolic systems.
- A student knows motivation and basic technics of the topic, is able to apply his knowledge to solution of textbook problems and confidently use a terminology of the subject.
- Topology of manifolds and vector fields on manifolds.Smooth manifolds. Differential equations and dynamical systems on manifolds. Transversality. Structural stability.
- Local analysis.Linear vector fields and maps. Behavior of orbits near regular point. Singular points. Index of singular point. Local classification of hyperbolic singular points. Invariant manifolds. Lambda-lemma.
- Morse-Smale SystemsInterrelation between asymptotic behavior of trajectories and the topology of the ambient manifold. Gradient-like systems. Energy function. Topological classification.
- Hyperbolic dynamics.Anosov diffeomorphisms. Smale Horse-shoe. Basic properties of hyperbolic sets.
- Interim assessment (1 module)0.3 * контрольная работа + 0.7 * экзамен
- Interim assessment (2 module)0.3 * контрольная работа + 0.7 * экзамен
- Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods Of Qualitative Theory In Nonlinear Dynamics (Part II). World Sci //Singapore, New Jersey, London, Hong Kong. – 2001.
- Grines V., Medvedev Timur, Pochinka O. Dynamical Systems on 2- and 3-Manifolds. Switzerland : Springer, 2016.