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Master 2019/2020

Research seminar "Theory of local bifurcations"

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
Master’s programme: Mathematics
Language: English
ECTS credits: 5

Course Syllabus

Abstract

This course aims to provide the student with both a solid basis in dynamical systems theory, theory of local bifurcations and the necessary understanding of the approaches, methods, results, and terminology used in the modern applied mathematics literature. A key theme is that of topological equivalence and codimension, or "what one may expect to occur in the dynamics with a given number of parameters allowed to vary". Actually, the course covered is sufficient to perform quite complex bifurcation analysis of dynamical systems arising in applications. The course examines the basic topics of bifurcation theory and could be used to compose in future a course on nonlinear dynamical systems or systems theory. Certain classical results, such as Andronov-Hopf and homoclinic bifurcation in two-dimensional systems, are presented in details, including self-contained proofs. For more complex topics of the theory, such as homoclinic bifurcations in more than two dimensions and two-parameter local bifurcations, we try to make clear the relevant geometrical ideas behind the proofs but only sketch them or, sometimes, discuss and illustrate the results but give only references of where to find the proofs. This approach, we hope, makes the course important for a wide audience and keeps it relatively short and able to be browsed. We also present several recent theoretical results concerning, in particular, bifurcations of homoclinic orbits to nonhyperbolic equilibria and one-parameter bifurcations of limit cycles in systems with reflectional symmetry. These results are hardly covered in standard graduate-level textbooks but seem to be important in applications. In this course we try to provide the student with explicit procedures for application of general mathematical theorems to particular research problems. Special attention is given to numerical implementation of the developed techniques. Several examples, mainly from mathematical biology, are used as illustrations.
Learning Objectives

Learning Objectives

  • • formation of the basis in dynamical systems theory, theory of local 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 method of dynamical systems; • formation of sufficient knowledge to perform quite complex bifurcation analysis of dynamical systems arising in applications; • formation of the knowledge on numerical implementation of the developed and presented techniques.
Expected Learning Outcomes

Expected Learning Outcomes

  • understand definitions proposed during the first topic of the course, remember some knowledge from previous courses that are necessary to continue to study the course
  • understand definitions proposed during the first topic of the course, know how to construct and analyze bifurcation diagrams
  • understand definitions proposed during the topic of the course, know how to analyze scenarios of one-parameter bifurcations of equilibria in continuous-time dynamical systems
  • understand definitions proposed during the topic of the course, know how to analyze scenarios of one-parameter bifurcations of fixed points in discrete-time dynamical systems
  • understand definitions proposed during the topic of the course, know how to analyze scenarios of one-parameter bifurcations of equilibria and periodic orbits in n-dimensional dynamical systems
  • understand definitions proposed during the topic of the course, know how to analyze scenarios of two-parameter bifurcations of equilibria in continuous-time dynamical systems
  • understand definitions proposed during the topic of the course, know how to analyze scenarios of two-parameter bifurcations of fixed points in discrete-time dynamical systems
  • understand definitions proposed during the topic of the course, know how to use numerical analysis in bifurcation theory
Course Contents

Course Contents

  • Introduction to Dynamical Systems
    1.1 Definition of a dynamical system 1.2 Orbits and phase portraits 1.3 Invariant sets 1.4 Differential equations and dynamical systems 1.5 Poincare maps
  • 2 Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems
    2.1 Equivalence of dynamical systems 2.2 Topological classification of generic equilibria and fixed points 2.3 Bifurcations and bifurcation diagrams 2.4 Topological normal forms for bifurcations 2.5 Structural stability
  • 3 One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
    3.1 Simplest bifurcation conditions 3.2 The normal form of the fold bifurcation 3.3 Generic fold bifurcation 3.4 The normal form of the Hopf bifurcation 3.5 Generic Hopf bifurcation
  • 4. One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
    4.1 Simplest bifurcation conditions 4.2 The normal form of the fold bifurcation 4.3 Generic fold bifurcation 4.4 The normal form of the flip bifurcation 4.5 Generic flip bifurcation 4.6 The "normal form" of the Neimark-Sacker bifurcation 4.7 Generic Neimark-Sacker bifurcation
  • 5. Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems
    5.1 Center manifold theorems 5.2 Center manifolds in parameter-dependent systems 5.3 Bifurcations of limit cycles 5.4 Computation of center manifolds
  • 6. Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems
    6.1 List of codim 2 bifurcations of equilibria 6.2 Cusp bifurcation 6.3 Bautin (generalized Hopf) bifurcation 6.4 Bogdanov-Takens (double-zero) bifurcation 6.5 Fold-Hopf (zero-pair) bifurcation 6.6 Hopf-Hopf bifurcation
  • 7. Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems
    7.1 List of codim 2 bifurcations of fixed points 7.2 Cusp bifurcation 7.3 Generalized flip bifurcation 7.4 Chenciner (generalized Neimark-Sacker) bifurcation 7.5 Strong resonances 7.6 Codim 2 bifurcations of limit cycles
  • 8. Numerical Analysis of Bifurcations
    8.1 Numerical analysis at fixed parameter values 8.2 One-parameter bifurcation analysis 8.3 Two-parameter bifurcation analysis 8.4 Continuation strategy 8.5 Convergence theorems for Newton methods 8.6 Detection of codim 2 homoclinic bifurcations
  • 9
Assessment Elements

Assessment Elements

  • non-blocking homework, independent work
  • non-blocking exam work
Interim Assessment

Interim Assessment

  • Interim assessment (1 module)
    0.5 * exam work + 0.5 * homework, independent work
  • Interim assessment (2 module)
    0.5 * exam work + 0.5 * homework, independent work
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

  • . Kuznetsov, Sergey. Strange Nonchaotic Attractors : Dynamics Between Order and Chaos in Qua-siperiodically Forced Systems [Электронный ресурс] / Sergey Kuznetsov, Arkady Pikovsky, and Ul-rike Feudel. – World Scientific Publishing Co Pte Ltd, 2014, . – ISBN: 9789812566331 (Print).
  • • R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/Cum-. (2015). Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.20873EF4