## Pertubation Theory (22/23)

 Faculteit Science and Engineering Jaar 2022/23 Vakcode WMMA032-05 Vaknaam Pertubation Theory (22/23) Niveau(s) master Voertaal Engels Periode semester I a ECTS 5 Rooster rooster.rug.nl

Uitgebreide vaknaam Pertubation Theory (tweejaarlijks 22/23)
Leerdoelen At the end of the course, the student is able to:
1. Distinguish when a perturbation is regular or singular
2. Compute the asymptotic expansion of a solution and verify its validity as a solution of a perturbation problem
3. Determine if a given model is structurally stable
4. Identify and apply appropriate perturbation methods/techniques/results (such as averaging, WKB, Fenichel, KAM, etc.) to specific examples
5. Compute normal forms of specific model examples
6. Identify the types of bifurcations present on specific model examples
7. Carry-out a local analysis of a bifurcation of a point
8. Formulate the main results of perturbation theory and persistence results in several contexts such as Hamiltonian systems, gradient systems, multi-time scale systems, etc.
Omschrijving Brief description of the course
Please provide a short and to the point description of your course (max. 2000 characters).
Briefly describe the content of the course unit. You can use the same text that is used in Ocasys (copy the text under the heading ‘Overview’). If the ocasys text is outdated or insufficient in covering the course’s description, you can update it by using the field below.
Perturbation theory, having its roots in the celestial mechanics, is concerned with the study of dynamical
systems by viewing these as perturbations of systems that are “simpler” or “ideal”, in the sense that the latter are
solvable and/or well-understood. The main objective of perturbation theory is to approximate a dynamical system
by a more manageable one, regarding the former as a “small” perturbation of the latter. With such an idea, one
aims at deducing dynamical properties of the “perturbed system” from the “unperturbed” one.

In this course, concepts like “ideal system”, or “perturbation”, among others, are made rigorous. Our goal is
to provide an introduction to several useful perturbation methods and to the fundamental results of regular and of singular perturbation theory. Along the course we shall cover several examples appearing in physical and life sciences
Uren per week
Onderwijsvorm Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
Toetsvorm Practisch werk (PR), Schriftelijk tentamen (WE)
(The two take-home exams account for up to 8/10 points (max. 4 points each) of the final grade. The remaining of the 2/10 points are obtained by actively participating in the tutorial sessions. Although attendance is not mandatory to pass the course, to be eligible for the 2/10 points, a student must proactively attend at least 4/8 tutorial sessions.)
Vaksoort master
Coördinator dr. H. Jardon Kojakhmetov
Docent(en) dr. H. Jardon Kojakhmetov
Verplichte literatuur
Titel Auteur ISBN Prijs
Mandatory: Lecture Notes on Perturbation Theory
Recommended: Introduction to Perturbation Methods
Mark H. Holmes (2013) 9781489996138
Recommended: Mathematical Physics: Classical Mechanics
Andreas Knauf (2017) 9783662557723
Entreevoorwaarden The course assumes knowledge from: Mathematics Bachelor and the course Dynamical Systems
Opmerkingen
Opgenomen in
Opleiding Jaar Periode Type
MSc Mathematics  (Specialisatie: Mathematical Physics) - semester I a keuzegroep
MSc Mathematics  (Specialisatie: Analysis and Dynamical Systems) - semester I a keuzegroep
MSc Mathematics and Physics (double degree)  (Mathematics and Complex Dynamical Systems (50 ects)) - semester I a keuzegroep
MSc Mathematics: Mathematics and Complex Dynamical Systems - semester I a keuzegroep
MSc Mathematics: Science, Business and Policy  (Science, Business and Policy: Mathematics and Complex Dynamical Systems) - semester I a keuzegroep