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Course Syllabus

Master 2024/2025

Analysis of nonlinear dynamical systems

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
Open to: students of all HSE University campuses
Instructors: Andrei Bobrovskii, Natalya Stankevich
Master’s programme: Mathematics
Language: English
ECTS credits: 6

Course Syllabus

Full SyllabusAsk Question

Abstract

The course will study numerical and analytical methods for the study of various nonlinear phenomena in dynamical systems. The phenomenon of dynamic chaos, including multidimensional, synchronization, multistability and others will be considered. Numerical modeling of the behavior of dynamic systems is planned as part of the course
Learning Objectives

Learning Objectives

  • The purpose of the course is to gain knowledges of the analysis of nonlinear dynamic systems, both analytical and numerical methods.
  • Get acquainted with various non-linear dynamical systems and study their complex behavior.
  • Study nonlinear phenomena: multistability and synchronization.
Expected Learning Outcomes

Expected Learning Outcomes

  • A student knows the history of the discipline and subfields
  • A student studies analytical methods for the analysis of nonlinear mappings. Learn the main bifurcations of non-linear mappings. Study application package for numerical bifurcation analysis of nonlinear mappings - XPP AUTO. Prepare programs for the analysis of nonlinear mappings.
  • A student studies analytical methods for the analysis of nonlinear flow dynamical systems. Learn types of equilibrium points, main bifurcation. Study application package for numerical bifurcation analysis.
  • A student learns multi-frequency and chaotic behavior. Make numerical simulations of models with chaotic and multi-frequency quasiperiodic oscillations.
  • A student studies phenomena synchronization. Learn asymptotic methods for analyzing synchronization in ensembles of coupled oscillators.
  • A student learns models with hyperbolic chaos. Study models, and character time series and phase portraits.
Course Contents

Course Contents

  • Introduction
  • Discrete dynamical systems
  • Flow dynamical systems
  • Numerical methods for analyzing dynamical systems
  • Complex behavior in dynamical systems
  • Synchronization
  • Hyperbolic chaos
Assessment Elements

Assessment Elements

  • non-blocking Analisys of fixed points stability and bifurcations of 1D maps
  • non-blocking Analisys of fixed points stability and bifurcations of 2D maps
  • non-blocking Numerical simulation of dynamics of 1D and 2D maps
  • non-blocking Analisys of equilibrium points stability and bifurcations of 2D and 3D flows
  • non-blocking Numerical simulation of dynamics of 2D and 3D flows
  • non-blocking Bifurcation analysis of fixed points in maps and equilibrium points in flows
  • non-blocking Numerical bifurcational analysis (XPPAUT)
  • non-blocking Analysis of equilibrium states in multi-dimensional systems
  • non-blocking Calculation of the spectrum of Lyapunov exponents
  • non-blocking Asymptotic methods for finding solution, determing of synchronization area in parameter space
  • non-blocking Transition from continous to discrete dynamical system via asymptotic method
  • non-blocking Analysis of nonlinear dynamical systems
Interim Assessment

Interim Assessment

  • 2024/2025 2nd module
    0.05 * Analisys of equilibrium points stability and bifurcations of 2D and 3D flows + 0.05 * Analisys of fixed points stability and bifurcations of 1D maps + 0.05 * Analisys of fixed points stability and bifurcations of 2D maps + 0.05 * Analysis of equilibrium states in multi-dimensional systems + 0.25 * Analysis of nonlinear dynamical systems + 0.05 * Asymptotic methods for finding solution, determing of synchronization area in parameter space + 0.25 * Bifurcation analysis of fixed points in maps and equilibrium points in flows + 0.05 * Calculation of the spectrum of Lyapunov exponents + 0.05 * Numerical bifurcational analysis (XPPAUT) + 0.05 * Numerical simulation of dynamics of 1D and 2D maps + 0.05 * Numerical simulation of dynamics of 2D and 3D flows + 0.05 * Transition from continous to discrete dynamical system via asymptotic method
Bibliography

Bibliography

Recommended Core Bibliography

  • Differential dynamical systems, Meiss, J. D., 2007
  • Discrete dynamical systems, Galor, O., 2010
  • Dynamical systems and chaos, Broer, H., 2011

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

Authors

  • STANKEVICH Nataliia VLADIMIROVNA