Master 2020/2021

Differential equations on manifolds

Category 'Best Course for New Knowledge and Skills'

Type: Compulsory course (Mathematics)

Area of studies: Mathematics

Delivered by: Department of Fundamental Mathematics

Where: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

When: 1 year, 1, 2 module

Mode of studies: offline

Instructors: Elena Gurevich

Master’s programme: Mathematics

Language: English

ECTS credits: 6

Contact hours: 84

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

• 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

• 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

• 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.
• Hyperbolic dynamics.
  Anosov diffeomorphisms. Smale Horse-shoe. Basic properties of hyperbolic sets.
• Morse-Smale Systems
  Interrelation between asymptotic behavior of trajectories and the topology of the ambient manifold. Gradient-like systems. Energy function. Topological classification.

Assessment Elements

• контрольная работа
• итоговый опрос по разделу
• итоговый опрос

Interim Assessment

• Interim assessment (1 module)
  0.7 * итоговый опрос по разделу + 0.3 * контрольная работа
• Interim assessment (2 module)
  0.7 * итоговый опрос + 0.3 * контрольная работа

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.