Dynamical systems and nonlinear differential equations

VO 250010-0 (5ECTS) and PS 250010-1 (2ECTS)

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Proseminar by: Prof. Henk Bruin

Announcements

Due to Covid measures, you will have to register at this course on u-space.

The last class was on June 15 2021. The last proseminar will be on June 18 2021.

As agreed during the lecture, the exam will be oral. You can make an appointment by sending me an email.

Rector's Day is March 12
Easter Break is from March 29 to April 9
Pentacost is on May 24 & 25 (Vorlesungsfrei).

Schedule

In the below schedule, the location HS13 might actually be replace by online lectures, subject to the current Covid19-Measures.

Day	Time	Room	from	to
Tuesday	15:00-16:30	HS13	2.3.2021	29.06.2021
Friday	9:45-10:30	HS13	5.3.2021	25.06.2021
Friday (Proseminar)	10:30-11:15	HS13	5.3.2021	25.06.2021

Contents of the course

Basic notions for continuous and discrete dynamical systems; flows, attractors, and chaos; stability of stationary points by linearization and by Lyapunov functions, Poincaré-Bendixson theory, bifurcations.

Weekly Progress

Notes of lecture 6

Day	Material
Week 1
2/3/2021	Introduction, 1-D ODE Stationary points and their stability simple polulation growth models	Notes of lecture 1 Lecture on a population dynamics model by Nagar.
5/3/2021	Quadratic family Cobweb diagram Feigenbaum diagram	Notes of lecture 2 (with typos corrected). The link of recording is sent to you in the email of March 5. Applets used in the lecture: cobweb diagram, the bifurcation diagram and another one , all for the logistic family. Lecture by Strogatz A video on the Feigenbaum diagram. (Feigenbaum constant are not part of the course material.)
Week 2
9/3/2021	Bifurcations in 1D (transcritical, saddlenode/fold, pitchfork)
12/3/2021	No class (Rektorstag)
Week 3
16/3/2021	Sensitive dependence on initial conditions Mathematical chaos (in the sense of Devaney) Lyapunov exponents	Notes of lecture 3 Lecture on sensitive dependence on initial conditions by Nagar. Lecture on Devaney chaos by Nagar.
19/3/2021	Proseminar	Exercises 1-5
Week 4
23/3/2021	Linear systems Structural stability Hartman-Grobman Theorem	Lecture by Strogatz Exercise + solution on Structural stability
27/3/2021	Lotka-Volterra Predator-Prey model Lyapunov functions	Lecture by Strogatz
Week 5	This is after the Easter break
13/4/2021	Limit cykels Van der Pol equation Hopf bifurcation	Lecture by Strogatz
16/4/2021	Proseminar	Exercises 6-10
Week 6
20/4/2021	Symbolic dynamics and Markov partitions Sharkovskii's Theorem	Notes of the lecture on symbolic dynamics Nagar's Lecture 23 on introduction to symbolic dynamics. Nagar's Lecture 17 on transitivity and Devaney chaos and Sharkovskiy's Theorem. Some typed notes on Symbolic dynamics
23/4/2021	Smale's Horseshoe	Notes of the lecture on Smale's horse-shoe Nagar's Lecture 12 on dynamics of the horseshoe attractor.
Week 7
27/4/2021	Poincaré-Bendixson Theorem	Notes of the Poincaré-Bendixson Theorem Lecture by Strogatz
30/4/2021	Proseminar	Exercises 11-13
Week 8
4/5/2021	Circle homeomorphisms rotation numbers	Notes of the lecture on circle homeomorphisms
7/5/2021	canceled
Week 9
11/5/2021	The cusp bifurcation hysteresis	Notes of the cusp bifurcation Recorded lecture on the cusp bfurcation will be posted here. Lecture by Strogatz
14/5/2021	canceled
Week 10
18/5/2021
21/5/2021	Proseminar: Exercises 13-16.
Week 11
25/5/2021	No class Vorlesungsfrei
28/5/2021	Oscillators	Notes of the lecture on oscillators Lecture by Wyslouch
Week 12
1/6/2021	Hamiltonian dynamics Euler-Lagrange equations	Notes of the lecture on Hamiltonian dynamics Online lecture by Strogatz
4/6/2021	Proseminar	Exercises 15-18.
Week 13
8/6/2021	Noether's Theorem n-body problem.	Notes of the lecture on Noether's Theorem Further text on Hamiltonian dynamics and the n-body problem
11/6/2021	Kepler's Problem 2-body problem
Week 14
15/6/2021	Lorenz attractors	Lectures by Strogatz on the Lorenz system and another one. Notes of the lecture on the Lorenz system
18/6/2021	Proseminar	Exercises 19-22

The course will be given in English

Contents of the course (B = Exercises , S = Schmeiser's notes, T = Teschl's book), St = Strogatz lecture on youtube.

Bifurcations (S§5. T§6.5. T§11.1.)
Chaos (T§10.3)
Cusp bifurcation (S§5.4, St)
Existence, uniqueness, continuity of solutions of ODEs (S§1. T§2.2. T§2.4.)
Euler-Lagrange equations (T§8.3. S§10 and B)
Feigenbaum (bifurcation diagram) (S§6. T§11.1, St)
Hamiltonian systems (S§10. T§8.3. St and B)
Harmonic and other oscillators (S§8.3.)
Homoclinic orbits and tangles (S Def 10. T§13.2)
Hopf bifurcation (S§8.4. T p219-220)
Inhomogeneous linear ODEs (S§2.1. T§3.2.)
(In)stability of equilibria, sink, sources, saddles, centers (T§3.2.)
(In)stability of fixed points (T§6.5, T§10.2.)
Iteration of maps, cobweb diagrams (T§10.2.)
Kepler problem = two-body problem (T§8.5. S§10 and B )
Lagrangian = Lagrange function (T§8.3. S§10 and B )
Legendre transform (T§8.3. S§10 and B )
Limit cycles (S§8), see also Van der Pol equation and Poincaré-Bendixson.
Linear ODEs (S§2. T§3.2. T§3.3 St)
Linearization, the Hartman-Grobman theorem (S Thm 8. T§9.3.)
Logistic differential equation (T§10.1.)
Logistic (= quadratic) map (T§10.1. St)
Lorenz system (S§9. T§8.2., St and St)
Lotka-Volterra system (preditor-prey) T§7.1.
Lyapunov functions (S§7. T§6.6. St))
Orbits, omega-limit sets (T§6.3 T§8.1. S Def 4.)
Pendulum (T§6.7. S§10 (Example 8) and B .)
Phase portraits (T§3.2.)
Poincaré maps (T§6.4. T§12.2)
Poincaré-Bendixson Theorem (S§8.5. T§7.3, St)
Sensitive dependence on initial conditions (T§10.3)
Sharkovskiy's Theorem (T§11.2)
Smale's horseshoe (T§13.1.)
Stable/unstable manifolds (S Thm 7. T§9.2. T§12.3.)
Structural stability
Symbolic dynamics (T§11.5.)
Strange attractors (T§11.6.)
Van der Pol equations (S§8, T§7.2, St)

References/Background Reading

Kathy Alligood, Tim Sauer and James Yorke, Chaos, an Introduction to Dynamical Systems, Springer 1996 ISBN 0-387-94677-2
Michael Brin and Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002 ISBN 0-521-80841-3
Robert Devaney, An Introduction to Cahotic Dynamical Systems, Benjamin and Cummings Publishing 1986, ISBN 0-8053-1601-9
Clark Robinson, Dynamical Systems (Stability, Symbolic Dynamics, and Chaos), CRC Press 1995 ISBN 0-8493-8493-1.
Steven Strogatz, Nonlinear dynamics and chaos, with applications to physics, biology and engineering, CRC Press, 2015, ISBN-13: 978-0813349107 or ISBN-10: 0813349109
G. Teschl, Ordinary Differential Equations and Dynamical Systems ODE Classnotes (or in fact book).

Assessment

Assessment of the course will be by oral exam, as discussed in class. The default language is English, but German is possible too.

Course material (Hand-outs)

Class notes in pdf (written by Christian Schmeiser)
Exercises for the Proseminar.
An Applet for cobweb diagrams for the logistic family.
An Applet for the bifurcation diagram for the logistic family. Also this one .
One applet and another about period doubling bifurcations in the logistic family.
Hand-out about structural stability

Some bifurcation diagrams:


Pitchfork bifurcation	Period doubling bifurcation	Transcritical bifurcation

Spruce budworm (Caterpillar and Moth)	Two saddle node (= fold) bifurcations meeting in a cusp (x,k)-plane	Bifurcation surface for cusp (projecting to (r,k)-plane

as well as a lecture by Strogatz on the same topic.

half

Hand-out about symbolic itineraries
Some phase portraits for the Van der Pol oscillator:

and also some youtube videos here, here, and a lecture by Strogatz (Cornell University).
Some videos about oscillators such as a driven pendulum, another one , and an inverted pendulum. Spectular outcome of resonance is demonstrated by Tacoma Bridge and in colour.
Some youtube videos on Huygens resonance here, here, here, and here, and if you still didn't have enough, also here.
Some youtube videos on Smale and his horseshoe: here and here.


Smale's horseshoe	A homoclinic tangle	Another homoclinic tangle

Some youtube videos on Lorenz and the Lorenz attractor here, here, here, here and the Wikipedia page
Written out class notes of June 18 in pdf.
Some youtube videos on three-body choreographies, here, here, and here and here. Or this one.
Some youtube videos on the Kepler problem (motion by the Earth and Sun), on gravitation, by Feynman (sort of), and general.
Some online source on the pendulum
Some online source on Hamiltonian/Lagragian/Newtonian mechanics
Online Lectures on Chaotic Dynamical Systems by Anima Nagar (Indian Institute of Technology, Delhi)

Preview
Lecture 1 general introduction and motivation of dynamical systems
Lecture 2 on elementary concepts.
Lecture 3 on periodic orbits, their hyperbolicity, stable/unstable sets, Schwarzian derivative.
Lecture 4 more on orbits.
Lecture 5 on periods of periodic points.
Lecture 6 on scrambled sets.
Lecture 7 on sensitive dependence on initial conditions.
Lecture 8 on a population dynamics model.
Lecture 9 on bifurcations, tent-maps and topological conjugacy.
Lecture 10 on nonlinear systems, solenoids, Smale's horseshoe.
Lecture 11 on horseshoe attractors.
Lecture 12 on dynamics of the horseshoe attractor.
Lecture 13 on recurrence.
Lecture 14 on recurrence continued.
Lecture 15 on transitivity.
Lecture 16 on Devaney chaos.
Lecture 17 on transitivity and Devaney chaos, Sharkovskiy's Theorem.
Lecture 18 on stronger forms of transitivity, mixing and weak mixing.
Lecture 19 on the chaotic properties of mixing systems.
Lecture 20 on weakly mixing and chaos.
Lecture 21 on strong transitivity and locally eventually onto.
Lecture 22 on stronger forms of transitivity continued.
Lecture 23 on introduction to symbolic dynamics.
Lecture 24 on symbolic dynamics, subshifts, continued fractions,
Lecture 25 on subshift of finite type.
Lecture 26 on subshift of finite type continued.
Lecture 27 on measuring chaos, topological entropy.
Lecture 28 on Adler's et al. definition of topological entropy.
Lecture 29 on Bowen's definition of topological entropy.
Lecture 30 on equivalence of the two definitions of topological entropy.
Lecture 31 on linear systems in dimension 2.
Lecture 32 on asymptotic properties of orbits of linear systems in dimension 2.
Lecture 33 hyperbolic toral automoprhisms.
Lecture 34 on chaos in toral automorphisms.
Lecture 35 on attractors of the Hénon map.

Updated February 26 2021