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# Analysis of nonlinear dynamical systems

2022/2023
Учебный год
ENG
Обучение ведется на английском языке
6
Кредиты

### 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

• Analysis of nonlinear mappings
• Analysis of nonlinear flow dynamical systems
• Base analysis of nonlinear systems
• Numerical simulation of nonlinear mappings
• Numerical simulations of 2D and 3D flow dynamical systems
• Complete synchronization in autonomous ensembles of oscillators
• Numerical simulations of multi-dimensional dynamical systems
• Numerical simulation of models with hyperbolic dynamics
• Assimpotic methods for detecting complete synchrinization
• Test - Nonlinear dynamical systems

#### Interim Assessment

• 2022/2023 2nd module
0.05 * Test - Nonlinear dynamical systems + 0.15 * Base analysis of nonlinear systems + 0.125 * Numerical simulation of nonlinear mappings + 0.05 * Complete synchronization in autonomous ensembles of oscillators + 0.05 * Analysis of nonlinear flow dynamical systems + 0.15 * Numerical simulations of multi-dimensional dynamical systems + 0.125 * Numerical simulations of 2D and 3D flow dynamical systems + 0.1 * Numerical simulation of models with hyperbolic dynamics + 0.05 * Analysis of nonlinear mappings + 0.15 * Assimpotic methods for detecting complete synchrinization

#### 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 NATALIYA VLADIMIROVNA