Analysis of nonlinear dynamical systems

2024/2025

Учебный год

ENG

Обучение ведется на английском языке

6

Кредиты

Кто читает:

Кафедра фундаментальной математики (Факультет информатики, математики и компьютерных наук (Нижний Новгород))

Статус:

Курс обязательный

Когда читается:

1-й курс, 1, 2 модуль

Преподаватели

Бобровский Андрей Александрович

Станкевич Наталия Владимировна

Course Syllabus

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

• 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

• 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

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

Assessment Elements

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

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

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