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Regular version of the site
Master 2019/2020

Differential equations on manifolds

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Compulsory course (Mathematics)
Area of studies: Mathematics
When: 1 year, 1, 2 module
Mode of studies: offline
Instructors: Elena Gurevich
Master’s programme: Mathematics
Language: English
ECTS credits: 5

Course Syllabus

Abstract

Differential equations on manifolds is a half-year course for the first-year students of master degree in mathematics. It is based on courses of Mathematical Analysis, Linear Algebra and Topology. The course is devoted to studying of classical and modern methods of qualitative theory of dynamical systems in manifolds and prepares students for understanding further courses Ergodic Theory, Theory of Local Bifurcations, Modern Theory of Dynamical Systems.
Learning Objectives

Learning Objectives

  • 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.
Expected Learning Outcomes

Expected Learning Outcomes

  • 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.
Course Contents

Course Contents

  • 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 Systems
    Interrelation 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.
Assessment Elements

Assessment Elements

  • non-blocking контрольная работа
  • non-blocking экзамен
  • non-blocking экзамен
Interim Assessment

Interim Assessment

  • Interim assessment (1 module)
    0.3 * контрольная работа + 0.7 * экзамен
  • Interim assessment (2 module)
    0.3 * контрольная работа + 0.7 * экзамен
Bibliography

Bibliography

Recommended Core Bibliography

  • 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.

Recommended Additional Bibliography

  • Grines V., Medvedev Timur, Pochinka O. Dynamical Systems on 2- and 3-Manifolds. Switzerland : Springer, 2016.