Dynamical Systems

2025/2026

Учебный год

ENG

Обучение ведется на английском языке

3

Кредиты

Кто читает:

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

Статус:

Курс обязательный

Когда читается:

3-й курс, 1, 2 модуль

Преподаватели

Гуревич Елена Яковлевна

Сараев Илья Александрович

Course Syllabus

Abstract

The program introduces fundamental concepts and methods of dynamical systems theory, which are essential for solving various problems in modern mathematics and its applications. Topics covered include phase portraits, stability analysis, bifurcation theory, chaotic dynamics, and ergodic theory. Emphasis is placed on the application of these techniques to real-world scenarios such as physics, engineering, biology, and economics. Through this course, students will gain a deep understanding of how to model, analyze, and predict the behavior of complex systems over time.

Learning Objectives

• Undestanding a concept and methods of classical and modern theory of dynamical systems

Expected Learning Outcomes

• A student understand definitions of dynamical system with continous and discreet time, orbit, fixed and periodic point, topological eqiuivalence, phase trajectory and phase portrait. A strdent can plot the phase portait of the model flows on the real line and on the plane, using the geometrical meaning of the derivative.
• A student can solve with separable variables and homogeneous equations and prove the existence and uniqueness solution of the Cauchy problem for such equations, knows properties of integral curves of homogeneous equations
• Strudent can determine Exact Differential Equations, solve them and know solution properties. Strudent can construct and research differential equations of Volterra-Lotka System and Nonlinear Oscillator Equation.
• A strudent can solve model problems in geometry and phisics using differential equations
• A strudent knows and can prove basic theorems on the structure of the solution of homogenuous and non-homogenuous linear equations and systems, and can solve such equations.
• Student knows statement and can prove the following theorems. Theorem on the existence and uniqueness of the Cauchy problem. Theorem on the continuous dependence of the solution on the initial conditions and the right-hand side. Theorem on the differentiability of the solution. Theorem on the continuation of the solution until it reaches the boundary of a compact set. Theorems of the straightening.
• A strudent knows statements, can prove and appy the following theorem. Definition of Lyapunov and asymptotic stability. Criterion for stability of equilibrium state of a system of linear homogeneous equations. Lyapunov function. Lyapunov and Chetaev theorems on stability. Theorem on stability by first approximation.
• A strudent knows and can prove basic properties of solutions and phase trajectories of autonomous systems.
• A student can research model discrete dynamical systems.

Course Contents

• DS-1. First definitions and examples of dynamic systems
• DS-4 Applications
• DS-2. Elementary methods of first order ODE solution
• DS-3 Conservative Systems and Exact Differential Equations
• DS-5 Linear Theory
• DS-6. Basic ODE Theorems
• DS-7. Stability
• DS-8. Autonomous ODE systems. Properties of solutions and phase trajectories
• DS-9. Dynamics of discrete dynamical systems. Deterministic chaos.

Assessment Elements

• контрольная работа
• экзамен

Interim Assessment

• 2025/2026 2nd module
  0.5 * контрольная работа + 0.5 * экзамен

Bibliography

Recommended Core Bibliography

• Differential equations, Blanchard, P., 2011
• Differential equations, dynamical systems, and linear algebra, Hirsch, M. W., 1974
• Introduction to the modern theory of dynamical systems, Katok, A., 1999
• Введение в теорию дифференциальных уравнений, учебник, 4-е изд., 239 с., Филиппов, А. Ф., 2015

Recommended Additional Bibliography

• A course in ordinary differential equations, Swift, R. J., 2007
• Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed., XIII, 513 p., Strogatz, S. H., 2019
• Геометрические методы в теории обыкновенных дифференциальных уравнений, Арнольд, В. И., 2000

Authors

• Gurevich Elena Iakovlevna