# Coursebooks

## Dynamics and bifurcation

#### Lecturer(s) :

Vela Arevalo Luz Vianey

English

#### Summary

Introduction to local and global behavior of nonlinear dynamical systems arising from maps and ordinary differential equations. Theoretical and computational aspects studied.

#### Content

One dimensional flows

Elementary bifurcations

One dimensional maps

Systems of ordinary differential equations: planar systems and phase portraits, non linear systems, Lyapunov stability, mechanical systems, La-Salle invariance principle, index in two dimensional vector fields, periodic orbits and limit cycles, Poincaré-Andronov-Hopf bifurcation, structural stability.

#### Keywords

Systèmes dynamiques à temps discrets et à temps continu, discrete and continuous dynamical systems, elementary bifurcations: saddle-node, transcritical, hysteresis, pitchfork; Lyapunov stability, Poincaré maps, mechanical systems.

#### Learning Prerequisites

##### Required courses

Analyse I, Analyse II, Algebre linéaire

##### Recommended courses

Equations différentielles ordinaires

#### Learning Outcomes

By the end of the course, the student must be able to:
• Analyze dynamical systems in continue and discrete time
• Describe local and global behavior of differential equations and maps
• Work out / Determine stability properties as a function of one or more parameters

#### Transversal skills

• Communicate effectively, being understood, including across different languages and cultures.
• Communicate effectively with professionals from other disciplines.
• Use a work methodology appropriate to the task.

#### Teaching methods

Lectures 2 hours a week and exercise sessions 2 hours a week.

#### Expected student activities

Attendance to lectures, attendance to exercise sessions, solution of homework problems that may involve theoretical or numerical solutions, give a final exam.

#### Assessment methods

Final exam.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés

#### Supervision

 Office hours No Assistants Yes Forum Yes

#### Resources

##### Bibliography

Introduction to Dynamical Systems: continuous and discrete, by Clark Robinson, 2012.

Dynamics and bifurcations, by J. Hale and H. Kocak, 1991.

##### Notes/Handbook

Lecture notes will be given, to be completed by the students.

### Reference week

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

Lecture
Exercise, TP
Project, other

### legend

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