Dynamical system theory for engineers

COM-502 / 6 credits

Teacher: Thiran Patrick

Language: English

Remark: Cours biennal

Summary

Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the qualitative analysis of nonlinear systems, both in discrete-time and continuous-time.

Content

  • Introduction: Dynamics of linear and non linear systems. Definitions; Unicity of a solution; Limit Sets, Attractors.
  • Linear Systems: Solutions; Stability of autonomous systems, Geometrical analysis, connection with frequency domain analysis.
  • Nonlinear Systems: Solutions; Examples.
  • Large-scale notions of stability: Lyapunov functions. Hamiltonian systems, gradient systems.
  • Small-scale notions of stability: Linearization; stability and basin of attraction of an equilibrium point, stability of periodic solutions, Floquet Multipliers. Graphical methods for the analysis of low-dimensional systems.
  • Structural stability, Bifurcation theory.
  • Introduction to chaotic systems: Lyapunov exponents, computation using ergodic theory, period doubling route to chaos, topological conjugacy.
  • Time permitting: chosen advanced topic (e.g, a topic in stochastic approximation, mean-field approach, optimization, network dynamics, ...).
  • The class is methodology-driven. It may present some limited examples of applications, but it is not application-driven.

Keywords

Dynamical Systems, Attractors, Equilibrium point, Limit Cycles, Stability, Lyapunov Functions, Bifurcations, Lyapunov exponents, Measure Theory.

Learning Prerequisites

Required courses

  • Linear algebra (MATH 111 or equivalent).
  • Analysis I, II, III (MATH 101, 106, 203 or equivalent).
  • Circuits & Systems II (EE 205 or equivalent) or a Systems & Signals class (MICRO 310/311 or equivalent).
  • A first-year Probabilty class, such as MATH-232, MATH-231, MATH-234(b), MATH-234(c), or equivalent.

Recommended courses

  • Analysis IV (MATH 207 or equivalent)

Important concepts to start the course

  • Linear Algebra (vector spaces, matrix operations, including matrix inversion and eigendecomposition).
  • Calculus (linear ordinary differential equations; Fourier, Laplace and z-Transforms).
  • Notions of topology.
  • Notions of probability.

Learning Outcomes

By the end of the course, the student must be able to:

  • Analyze a linear or nonlinear dynamical system.
  • Anticipate the asymptotic behavior of a dynamical system.
  • Assess / Evaluate the stability of a dynamical system.
  • Identify the type of solutions of a dynamical sytem.
  • Analyze a linear or nonlinear dynamical system
  • Anticipate the asymptotic behavior of a dynamical system
  • Assess / Evaluate the stability of a dynamical system
  • Identify the type of solutions of a dynamical system

Teaching methods

  • Lectures (blackboard, no video), 3h per week
  • Exercise session, 1h per week

Expected student activities

Exercises in class and at home (paper and pencil, and Python)

Assessment methods

  1. Mid-term 20%
  2. Final exam 80%

Supervision

Office hours No
Assistant.e.s No
Forum Yes

Resources

Bibliography

Course notes; textbooks given as reference on the moodle page of the course.

Notes/Handbook

Course notes, exercises and solutions provided on the moodle page of the course.

Moodle Link

Prerequisite for

Classes using methods from linear and nonlinear dynamical systems.

In the programs

  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Dynamical system theory for engineers
  • Courses: 3 Hour(s) per week x 14 weeks
  • Exercises: 1 Hour(s) per week x 14 weeks
  • Type: optional

Reference week

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