COM-502 / 6 crédits

Enseignant:

Langue: Anglais

Remark: Cours biennal, pas donné en 2023-24

## 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. Introduction to structural stability, Bifurcation theory. Introduction to chaotic systems (Lyapunov exponents); time permitting: a review of tools of measure theory to compute Lyapunov exponents.
• 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.

## 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).

## Recommended courses

• A first-year Probabilty class, such as MATH-232, MATH-231, MATH-234(b), MATH-234(c), or equivalent.
• Analysis IV (MATH 207 or equivalent)

## Important concepts to start the course

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

## Teaching methods

• Lectures (blackboard), 2h per week
• Exercise session, 1h per week

## Expected student activities

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

## Assessment methods

1. Mid-term 20% (conditionally on the Covid situation allowing for it to be taken at EPFL).
2. Final exam 80%

## Supervision

 Office hours Yes Assistants Yes Forum Yes

## 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.

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Dynamical system theory for engineers
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 1 Heure(s) hebdo x 14 semaines
• Type: optionnel

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