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Course Syllabus

Магистратура 2025/2026

Анализ нелинейных динамических систем

Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Курс обязательный (Математика)
Кто читает: Кафедра фундаментальной математики
Где читается: Факультет информатики, математики и компьютерных наук (Нижний Новгород)
Когда читается: 1-й курс, 1, 2 модуль
Охват аудитории: для всех кампусов НИУ ВШЭ
Преподаватели: Корякин Владислав Андреевич, Станкевич Наталия Владимировна
Язык: английский
Кредиты: 6
Контактные часы: 60

Course Syllabus

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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
  • Multi-dimensional chaos and quasi-periodicity.
  • 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 Test 1
  • 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 Test 2
Interim Assessment

Interim Assessment

  • 2025/2026 2nd module
    0.05 * Analysis of equilibrium states in multi-dimensional systems + 0.05 * Numerical simulation of dynamics of 2D and 3D flows + 0.05 * Analisys of fixed points stability and bifurcations of 2D maps + 0.05 * Calculation of the spectrum of Lyapunov exponents + 0.05 * Numerical simulation of dynamics of 1D and 2D maps + 0.25 * Test 2 + 0.25 * Test 1 + 0.05 * Asymptotic methods for finding solution, determing of synchronization area in parameter space + 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 * Numerical bifurcational analysis (XPPAUT) + 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

  • KORYAKIN VLADISLAV ANDREEVICH
  • Stankevich Nataliia Vladimirovna