Анализ нелинейных динамических систем

2024/2025

Статус: Маго-лего

Кто читает: Кафедра фундаментальной математики

Когда читается: 1, 2 модуль

Охват аудитории: для всех кампусов НИУ ВШЭ

Преподаватели: Бобровский Андрей Александрович, Станкевич Наталия Владимировна

Язык: русский

Кредиты: 6

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Аннотация

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

Цель освоения дисциплины

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.

Планируемые результаты обучения

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.

Содержание учебной дисциплины

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

Элементы контроля

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

Промежуточная аттестация

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

Список литературы

Авторы

Станкевич Наталия Владимировна

Программа дисциплины