Course Info:

• Course: Applied Dynamical Systems 1MA151
• Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se
• First Day Handout

• Textbooks:

• Primary text: Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in nonlinearity), Steven H. Strogatz, Publisher: Westview Press 2001-01-19, 512 Pages, ISBN: 0738204536

Things on the current agenda:

Example of a program that finds an attracting cycle in the quadratic family, in SageMath, here

TAKEHOME, all problems here

LORENZ SYSTEM

An applet, and some applete in the wiki page

The Lorinz attractor folding onto itself

CONSERVATIVE MAPS, HORSESHOES AND CHAOS

Various applets here

My notes about hyperbolic sets and the horseshoe

Iterations and the bifurcation diagram for the quadratic map

Another iteration engine for the quadratic map

Interactive cobweb plot for the quadratic map by Andre Burbanks

CHAOS

Double pendulum applet

Equations of motion for a double pendulumt

Paper by J. Banks et al, "On Devaney's Difinition of Chaos". A proof of full Sharkovsky theorem, a paper in which Sharkovsky's theorem was rediscovered.

FRACTALS

An IFS applet.

IFS,explanation.

Fractal image compression.

A Barnsley fern, explanation.

A Barnsley fern, applet.

Some fractal applets here and many more here

Box counting dimension of coastline, applet

Covering coastline of Norway, applet

Examples of fractals in nature here and here

Devil's staircase for the Cantor set here , here and here

Course Objectives/Outcomes:

1) Understand several fundamental concepts of the dynamical systems, specifically:

• dynamics of the quadratic family,
• topological dynamics; recurrence, mixing and transitivity,
• stable and unstable manifolds, homoclinic and heteroclinic intersections, horseshoes,
• fractals,
• one and two dimensional flows, phase space, limit cycles and Poincar\'e-Bendixson Theorem,
• bifurcations in flows and maps.
4) Understand and be able to explain/present some applications of the theory.

3) Solve representative problems in the above-mentioned topics.

4) Carry out numerical studies of dynamical systems.

A take home final, 100%.

Schedule

1. 2015-10-26
• Introduction, course information.
• What is "dynamics"?
• Continous and descrete dynamical systems.
• The quadrtic family as the fundamental example of a chaotic dynamical system.

2. 2015-10-28
• The quadratic family continued. Periodic and stochastic behaviour.
• Functions vs maps.
• Periodic orbits and their stability.

3. 2015-11-3
• Renormalization

4. 2015-11-4
• Intermittency and period 3.
• What is chaos?

Reading: Ch 10 and p 323

5. 2015-11-6
• Fractals.
• Cardinality and Cantor set.

6. Stable/unstable manifold theorem
7. 2015-11-9
• Fractals.
• Iterated Function System.
• Fractal dimensions.

8. 2015-11-10
• Hausdorff dimension.
• Rotation and devil's staircase.
• An exmaple of a "dynamical Cantor set": the Feigenbaum attractor.

9. 2015-11-12
• An exmaple of a "dynamical Cantor set": the Feigenbaum attractor.
• Bifurcation theory

Reading: "An exposition about the dynamics of the quadratic map" in the "current agenda". Ch 3 in Strogatz.

10. 2015-11-13
• Bifurcation theory.

11. 2015-11-13
• Introduction to ODE's.
• Stable/unstable manifold theorem.

Reading: Ch 5. Ch 6.0-6.3. Ch 3,4 in Hirsch, Smale, Devaney,

12. 2015-11-24
• Stable/unstable manifold theorem
• Stability of equilibria. Lyapunov function.
• Ex: Pendulum

Reading: Ch 7.2. Ch 9.2 in Hirsch, Smale, Devaney. Ch 6.7

13. 2015-11-25
• Lyapunov function for damped pendulum and Lorenz system.
• Limit sets and cycles.
• Poincare-Bendixson theorem
• Oscilating reactions, Selkov system and Hopf bifurcations