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Regular version of the site
Master 2023/2024

Analysis of nonlinear dynamical systems

Type: Compulsory course (Mathematics)
Area of studies: Mathematics
When: 1 year, 1, 2 module
Mode of studies: offline
Open to: students of all HSE University campuses
Master’s programme: Mathematics
Language: English
ECTS credits: 6
Contact hours: 56

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

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
  • Complex behavior in dynamical systems
  • Synchronization
  • Hyperbolic chaos
Assessment Elements

Assessment Elements

  • non-blocking Home Task 1.1. “Analysis of fixed points stability and bifurcations of 1D maps”
  • non-blocking Home Task 1.2. “Analisys of fixed points stability and bifurcations of 2D maps”
  • non-blocking Laboratory Work 1.1. “Numerical simulation of nonlinear maps”
  • non-blocking Home Task 1.3. “Analysis of 2D nonlinear flow dynamical systems”
  • non-blocking Home Task 1.4. “Analysis of 3D nonlinear flow dynamical systems”
  • non-blocking Laboratory Work 1.2 “Numerical simulation of nonlinear flow dynamical systems”
  • non-blocking Test “Base analysis of nonlinear systems”
  • non-blocking Home Task 2.1. “Analysis of equilibrium states in multi-dimensional systems”
  • non-blocking Numerical bifurcation analysis of 4D dynamical systems. Skeleton of chaotic attractor
  • non-blocking Laboratory Work 2.2. “Numerical simulations of different types of chaos”
  • non-blocking Home Task 2.2. “Complete synchronization in autonomous ensembles of oscillators”
  • non-blocking Home Task 2.3. “Transition from continuous to discrete dynamical systems”
  • non-blocking Nonlinear dynamical systems
Interim Assessment

Interim Assessment

  • 2023/2024 2nd module
    0.03 * Home Task 1.1. “Analysis of fixed points stability and bifurcations of 1D maps” + 0.02 * Home Task 1.2. “Analisys of fixed points stability and bifurcations of 2D maps” + 0.05 * Home Task 1.3. “Analysis of 2D nonlinear flow dynamical systems” + 0.05 * Home Task 1.4. “Analysis of 3D nonlinear flow dynamical systems” + 0.05 * Home Task 2.1. “Analysis of equilibrium states in multi-dimensional systems” + 0.05 * Home Task 2.2. “Complete synchronization in autonomous ensembles of oscillators” + 0.05 * Home Task 2.3. “Transition from continuous to discrete dynamical systems” + 0.05 * Laboratory Work 1.1. “Numerical simulation of nonlinear maps” + 0.05 * Laboratory Work 1.2 “Numerical simulation of nonlinear flow dynamical systems” + 0.05 * Laboratory Work 2.2. “Numerical simulations of different types of chaos” + 0.25 * Nonlinear dynamical systems + 0.05 * Numerical bifurcation analysis of 4D dynamical systems. Skeleton of chaotic attractor + 0.25 * Test “Base analysis of nonlinear systems”
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