Applied theory of dynamical systems

2024/2025

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:

4 year, 3 module

Instructor

Шемахин Александр Юрьевич

Course Syllabus

Abstract

The course is designed as an introduction to the applied aspects of the modern theory of dynamical systems. A rigorous presentation of the theory of global bifurcations in multidimensional systems and its application to problems of biology will be given. The fundamentals of the dynamics of partial differential equations and the theory of dynamical systems under the action of random perturbations will be presented.

Learning Objectives

• Introduction to the theory of dynamical systems and introduction to some applications

Expected Learning Outcomes

• Know the basic definitions of dynamics systems
• know the main results and examples
• be able to solve problems on the topic

Course Contents

• Elementary dynamics
• Periodic forcing and quasiperiodic dynamics
• Two-dimensional dynamics
• Synchronization theory
• Chaos
• Saddles and homoclinic structures
• Measure and dimensions
• Logistic map

Assessment Elements

• Домашнее задание
• Экзамен

Interim Assessment

• 2024/2025 3rd module
  0.4 * Домашнее задание + 0.6 * Экзамен

Bibliography

Recommended Core Bibliography

• Differentiable Dynamical Systems : An Introduction to Structural Stability and Hyperbolicity, XI, 192 p., Wen, L., 2016
• Dynamical Systems : Stability, Symbolic Dynamics, and Chaos, 2nd ed., 504 p., Robinson, C., 1999
• Hasselblatt, B., Takens, F., & Broer, H. W. (2010). Handbook of Dynamical Systems. Amsterdam: North Holland. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=344991
• Introduction to the Modern Theory of Dynamical Systems, With a supplement by Anatole Katok and Leonardo Mendoza, XVIII, 802 p., Katok, A., Hasselblatt, B., 1996
• Strogatz, S. H. (2000). Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering (Vol. 1st pbk. print). Cambridge, MA: Westview Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421098

Recommended Additional Bibliography

• • 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
• Dynamical Systems on 2- and 3-Manifolds, XXVI, 295 p., Grines, V. Z., Medvedev, T. V., Pochinka, O. V., 2016
• M. I. Freidlin, & A. D. Wentzell. (2012). Random Perturbations of Dynamical Systems (Vol. 1984). Springer.

Authors

• Трехлеб Ольга Юрьевна
• Nozdrinova Elena Viacheslavovna