Research Seminar " Theory of Bifurcations of Multidimensional Systems"

Master 2020/2021

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, 1-4 module

Mode of studies: offline

Instructors: Lev Lerman

Master’s programme: Mathematics

Language: English

ECTS credits: 5

Contact hours: 72

Full Syllabus Ask Question

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

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
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.
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
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.
Know the Central manifold theorems. To be able to compute the Central manifold.
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.
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 simplest bifurcation conditions, the normal shape of the fold bifurcation. Have an idea of the General folded bifurcation, the normal form of flip bifurcation.

Course Contents

Introduction to Dynamical Systems
Autonomous differential equations, their flow. Definition of a dynamical system. Orbits and phase portraits. Invariant sets and limit sets. Periodic differential equations. Poincare maps and iterations of diffeomorphisms.
Linear Periodic Differential Systems
Fundamental matrix, monodromy matrix and Poincare map, multipliers. Floquet theorem on the form of fundamental matrix. Lyapunov’s reducibility theorem. Existence of matrix logarithm.
Codimension One Semi-Local Bifurcations
The birth of a limit cycle from a saddle homoclinic loop on the plane. The birth of a limit cycle from a saddle-node homoclinic loop on the plane. The birth of a limit cycle from a saddle homoclinic loop in Rn, saddle-value and stability of the cycle, orientable and non-orientable loops. The birth of a stable (unstable) limit cycle from a saddle-node homoclinic loop in Rn. The birth of a saddle limit cycle from a saddle-saddle homoclinic loop in Rn, the case of several loops, complicated dynamics. Shilnikov’s saddle-focus loop, complicated dynamics. Smale’s horseshoe.
Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems
Equivalence of dynamical systems. Linearization, topological classification of generic equilibria and fixed points, Grobman-Hartman theorem. Local bifurcations and bifurcation diagrams. Topological normal forms for local bifurcations. Structural stability
One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
Simplest bifurcation conditions. The normal form of the fold bifurcation. Generic fold bifurcation. The normal form of the Hopf bifurcation
One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
Simplest bifurcation conditions. The normal form of the fold bifurcation. Generic fold bifurcation. The normal form of the flip bifurcation. Generic flip bifurcation. The "normal form" of the Neimark-Sacker bifurcation.
Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems
Center manifold theorems and reduction. Center manifolds in parameter-dependent systems. Computation of center manifolds.
A Review of Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
List of codim 2 bifurcations of equilibria. Cusp bifurcation. Bautin (generalized Hopf) bifurcation. Bogdanov-Takens (double-zero) bifurcation. Fold-Hopf (zero-pair) bifurcation. Hopf-Hopf bifurcation.

Assessment Elements

реферат
итоговый устный опрос

Interim Assessment

Interim assessment (2 module)
0.7 * итоговый устный опрос + 0.3 * реферат
Interim assessment (4 module)
0.7 * итоговый устный опрос + 0.3 * реферат

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.

Course Syllabus