1MA151 Homepage
A collection of links to various Dynamical Systems resources
Maple worksheets
Timetable


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 20010119, 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
QUADRATIC MAP
An exposition about the dynamics of the quadratic map
Iterations and the bifurcation diagram for the quadratic map
Another iteration engine for the quadratic map
Intermitency in the quadratic map
Period doubling cascade
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
Serpinski Gasket.
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\'eBendixson 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 abovementioned topics.
4) Carry out numerical studies of dynamical systems.
Grading:
A take home final, 100%.
Schedule
 20151026
 Introduction, course information.
 What is "dynamics"?
 Continous and descrete dynamical systems.
 The quadrtic family as the fundamental example of a chaotic dynamical system.
Reading: Ch 10 in Strogatz
 20151028
 The quadratic family continued. Periodic and stochastic behaviour.
 Functions vs maps.
 Periodic orbits and their stability.
Reading: Ch 10
 2015113
 The bifurcation cascade.
 Renormalization
Reading: Ch 10
 2015114
 Intermittency and period 3.
 What is chaos?
Reading: Ch 10 and p 323
 2015116
 Fractals.
 Cardinality and Cantor set.
Reading: Ch 11
 Stable/unstable manifold theorem
 2015119
 Fractals.
 Iterated Function System.
 Fractal dimensions.
Reading: Ch 11
 20151110
 Hausdorff dimension.
 Rotation and devil's staircase.
 An exmaple of a "dynamical Cantor set": the Feigenbaum attractor.
Reading: Ch 11
 20151112
 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.
 20151113
Reading: Ch 3
 20151113
 Introduction to ODE's.
 Stable/unstable manifold theorem.
Reading: Ch 5. Ch 6.06.3. Ch 3,4 in Hirsch, Smale, Devaney,
 20151124
 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
 20151125
 Lyapunov function for damped pendulum and Lorenz system.
 Limit sets and cycles.
 PoincareBendixson theorem
 Oscilating reactions, Selkov system and Hopf bifurcations
Reading: Ch 7.
 20151126
 PoincareBendixson theorem
 Oscilating reactions, Selkov system and Hopf bifurcations
 2D bifurcations.
Reading: Ch 7. Ch 8.2, 8.3. Read about bifurcations here
