Research Seminar " Theory of Bifurcations of Multidimensional Systems"

2022/2023

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered at:

Department of Fundamental Mathematics (Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod))

Course type:

Compulsory course

When:

1 year, 3, 4 module

Instructor

Lerman, Lev

Course Syllabus

Abstract

This course aims to provide the student with a solid foundation in the theory of dynamical systems, the theory of bifurcations of multidimensional systems, and the necessary understanding of the approaches, methods, results, and terminology used in modern literature on applied mathematics. In fact, the course completed is enough to conduct a rather complicated bifurcation analysis of dynamic systems arising in applications. In this course, we try to provide the student with explicit procedures for applying general mathematical theorems to specific research problems.

Learning Objectives

• To lay down the basis in dynamical systems theory, theory of bifurcations and the necessary understanding of the approaches, methods, results, and terminology used in the modern applied mathematics literature
• Formation of the knowledge and skills applied to the study of main bifurcations using qualitative methods of dynamical systems
• Formation of sufficient knowledge to perform rather complex bifurcation analysis of dynamical systems arising in applications

Expected Learning Outcomes

• Be able to determine the equivalence of dynamic systems, conduct linearization, topological classification of General equilibria and fixed points. Know the Grobman-Hartman theorem. Be able to bring local bifurcations to topological normal forms
• Be able to find the fundamental matrix, the monodromy matrix. Know Floquet's theorem on the form of the fundamental matrix, Lyapunov's theorem on reducibility. existence of a matrix logarithm.
• Have an idea of the birth of the limit cycle from the saddle homoclinic loop on the plane, from the saddle-node homoclinic loop on the plane, from the saddle homoclinic loop in Rn, about the saddle value and stability of the cycle, orientable and non-orientable loops.
• Know the Central manifold theorems. To be able to compute the Central manifold.
• Know the definition of the dynamic system, orbits, and phase portraits. Be able to find invariant sets and limit sets. Be able to solve periodic differential equations, build Poincare maps and iterations of diffeomorphisms
• Know the simplest bifurcation conditions, the normal shape of the fold bifurcation. Have an idea of the General folded bifurcation, the normal form of flip bifurcation.
• Know the simplest bifurcation conditions, the normal shape of the fold bifurcation. Have an idea of the General folded bifurcation, the normal form of the Hopf bifurcation.
• To know the list of bifurcations of codim 2 equilibria. The bifurcation threshold. Bautin bifurcation (generalized Hopf). Bogdanov-Takens (double zero) bifurcation. Bifurcation of fold-Hopf (zero-pair). Hopf-Hopf Bifurcation.

Course Contents

• Introduction to Dynamical Systems
• Linear Periodic Differential Systems
• Codimension One Semi-Local Bifurcations
• Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems
• One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
• One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
• Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems
• A Review of Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

Assessment Elements

• Контрольная работа
• Экзамен

Interim Assessment

• 2022/2023 4th module
  0.2 * Контрольная работа + 0.6 * Экзамен

Bibliography

Recommended Core Bibliography

• Ma, Tian. Bifurcation Theory and Applications [Электронный ресурс ] / Tian Ma, Shouhong Wang; БД ebrary. - Koln: World Scientific Publishing Co, 2005.

Recommended Additional 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.