# Coursebooks

## Dynamical system theory for engineers

Thiran Patrick

English

#### 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).
• 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.

#### 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%
2. 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.

#### Prerequisite for

Classes using methods from dynamical systems.

### Reference week

MoTuWeThFr
8-9  GCC330
9-10
10-11  CE1106
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German