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Introduction to Bifurcation Theory:

Differential Equations, Dynamical Systems and Applications

fall 2013


PhD Tadeas Priklopil

(Speaks English and Finnish; tutoring/discussions/advice may be given in both languages)


10 sp.


Advanced studies

About the course

Mathematics is playing an ever more important role in physical and biological sciences. Of special interest is the modeling of events that change over time, for example the movement of particles, the spread of a virus or economic growth. Mathematical framework of dynamical systems provides a way to describe and analyze such changes.

A perturbation of model parameters, such as an increase in replication ability of a virus, will cause a change in the dynamics. Often such a perturbation results in qualitatively similar behavior. But real fun starts when certain singularity conditions are satisfied and a perturbation can alter the dynamics radically. This is where bifurcation theory is needed.

This course focuses on the theory of bifurcations but also on its applications. As the analysis of a model by pure analytical methods is often a formidable task, we will complement the theory with numerical methods.    

The course will cover the necessary principles of ordinary differential equations and linear algebra, equilibria and their stability, invariant manifolds and their use, and local and global bifurcations. The course is suitable for master students as well as advanced bachelor students. 

rough teaching plan: in the first period we give an introduction to dynamical systems, differential equations and linear algebra and then in the second period we study bifurcations. 


Weeks 37-42 and 44-50, Monday 14-16 in room CK111 and Wednesday 14-16 in room B321. Two hours of exercise classes per week.

Exams and getting credits

To pass the course and to get credits choose either option 1 or 2:

  1. Do the following three things (a)  exercises (max. 6 points)
                                               (b) project: this consists of analyzing a model, and using a computer software to complement the results. (max. 6 points)
                                               (c) exam on 18th of December: this will be focused on conceptual understanding of the topics of the course (max. 24 points)
    30 points is the maximum (yes, you can get all-together 36, but the points above 30 are just "extra"): grade 5 for above 27p, grade 4 between 24-27p. etc. You need half of the points (15p) to pass the course.
    EXAM TIME/PLACE: 18th of December, 14.15, room B321

  2. Do a normal exam. The exam date is one of the public examination dates. Let the lecturer know in advance!

Lecturer is advising to choose the option 1 (if you can)! 

Grades: Those who chose option 2 and those who chose option 1 (and already returned the projects!) may ask for their grade by writing an e-mail to the lecturer! Grades will be officially registered during the week 6th to 10th of January.


Lecture notes. Lecture notes are composed from several sources. Main references are Hirsch, Smale and Devaney (2004) and Wiggins (2000).


Did you forget to register?  What to do?

1.Thursday 10-12 B321 Tadeas Priklopil 


Below you will find a link to a feedback page, where you can express (anonymously) your thoughts about the course. ALL COMMENTS ARE WELCOME!



Under the following link you find the model and instructions: The system of Lorenz

Correction (6.12.): constants sigma and beta are assumed fixed and positive!

Lecture notes:

Lecture notes (updated 30.11.): Introduction to Bifurcation Theory (corrected typos, 16.12.) 

Some handwritten notes so that you can study! These are really just my personal notes, apologies for any misprints and swearwords!

Hopf Bifurcation

Normal forms & Codimensions

Final Lecture: SIR-model


Lecture topics - period I
week 37
9.9. Introductory lecture - introductory examples
11.9. SIR-model with vaccinations - Computer session
week 38
16.9. Planar linear systems: intro - part I
18.9. Planar linear systems: intro - part II
Exercises 1
Solutions 1  Maple Solutions 1
week 39
23.9. Planar linear systems: phase portraits, complex eigenvalues
25.9. Planar linear systems: phase portraits, repeated eigenvalues
Exercises 2
Solutions 2  Maple Solutions 2 
week 40
30.9. Planar linear systems: change of coordinates
2.10. Planar linear systems: summary and classification
Exercises 3
Solutions 3
week 41
7.10. Dynamical Conjugacy
Exercises 4
Solutions 4
week 42
14.10. Higher dimensional linear algebra
16.10. Higher dimensional linear systems
no exercises
week 43




lecture topics - period II
week 44
28.10. Nonlinear systems: introduction + properties
30.10. Taylor expansion, Stability concepts, Variational Equation
Exercises 5
Solutions 5 Maple solutions Ex. 21
week 45
4.11.  Linearization
6.11.  no lecture
Exercises 6 (will be held week later)
week 46
11.11. Invariant sets, subspaces and manifolds
13.11. Invariant manifolds: example, Structural stability
Exercises 7
Solutions 6 and 7
week 47
18.11. Structural stability, Bifurcations
20.11. Bifurcations + Center Manifolds: reducing the dimensionality
Exercises 8
Solutions 8 Maple solutions Ex.31-34
week 48
25.11. Bifurcations + Center Manifolds: reducing the dimensionality
27.11. Bifurcations: Saddle-node, pitchfork, transcritical
Exercises 9 (corrected)
Solutions 9
week 49
2.12.  Bifurcations: Hopf 
4.12.  Bifurcations: normal forms, codimension 
Exercises 10 (corrected)
Solutions 10 (ex. 38,39 updated!)
week 50
9.12.  Last lecture: Summary of the course


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