Dynamical system theory for engineers
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.
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).
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 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), 2h per week
- Exercise session, 1h per week
Expected student activities
Exercises in class and at home (paper and pencil, and Matlab)
Assessment methods
- Mid-term 20% (conditionally on the Covid situation allowing for it to be taken at EPFL).
- Final exam 80%
Supervision
Office hours | Yes |
Assistants | Yes |
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 dynamical systems.
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Dynamical system theory for engineers
- Cours: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 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: 3 Heure(s) hebdo x 14 semaines
- Exercices: 1 Heure(s) hebdo x 14 semaines
- Type: optionnel
Semaine de référence
Lu | Ma | Me | Je | Ve | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
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19-20 | |||||
20-21 | |||||
21-22 |
Légendes:
Cours
Exercice, TP
Projet, labo, autre