# Dynamical system theory for engineers

COM-502 / **6 credits**

**Teacher: **

**Language:** English

**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.

## 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 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

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Dynamical system theory for engineers**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**1 Hour(s) per week x 14 weeks

## Reference week

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