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

Статус:: Курс обязательный

Когда читается:: 1-й курс, 1, 2 модуль

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.

Course Contents

Introduction
Discrete dynamical systems
Flow dynamical systems
Numerical methods for analyzing dynamical systems
Multi-dimensional chaos and quasi-periodicity.
Synchronization

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
Analysis of steady state stability in dynamical systems
Analysis of equilibrium states in multi-dimensional systems
Numerical bifurcational analysis (XPPAUT)
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

2026/2027 2nd module
0.05 * Analisys of fixed points stability and bifurcations of 2D maps + 0.05 * Numerical simulation of dynamics of 2D and 3D flows + 0.25 * Analysis of nonlinear dynamical systems + 0.05 * Numerical bifurcational analysis (XPPAUT) + 0.05 * Transition from continous to discrete dynamical system via asymptotic method + 0.05 * Analysis of equilibrium states in multi-dimensional systems + 0.05 * Numerical simulation of dynamics of 1D and 2D maps + 0.25 * Analysis of steady state stability in dynamical systems + 0.05 * Analisys of fixed points stability and bifurcations of 1D maps + 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 * Calculation of the spectrum of Lyapunov exponents

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
Bagaev Andrei Vladimirovich

Магистерская программа «Математика»

Адрес:

Analysis of nonlinear dynamical systems

Course Syllabus