Dynamics and bifurcation
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
- 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.
Ressources en bibliothèque
- Introduction to Dynamical Systems: continuous and discrete / Robinson
- (electronic version)
- Dynamics and bifurcations / Hale & Kocak
Notes/Handbook
Lecture notes will be given, to be completed by the students.
Websites
Videos
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Ecrit (session d'été)
- Matière examinée: Dynamics and bifurcation
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
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 | |||||
| 17-18 | |||||
| 18-19 | |||||
| 19-20 | |||||
| 20-21 | |||||
| 21-22 |
Légendes:
Cours
Exercice, TP
Projet, autre