The project component of your grade can be about any of the following:
- Read an article and summarize its content. You can ask me for one or propose one of your choice.
- Make some numerical exploration of a dynamical system. For instance, the 3D discrete map at page 39 of the textbook or the planar restricted 3-body system problem in Sect. 2.1 of the textbook.
- Study the proof of some non-trivial theorem.
The project consists into working on one of the tasks above, writing a short essay on them and presenting them in class (each presentation can be up to 30 minutes long).
The format can be either TeX (e.g. on
overleaf or
cocalc), HTML or
PreTeXt.
Below is a list of possible topics:
Period-3 implies chaos
Read the seminal article
Period-3 implies chaos by Li and Yorke or study a proof of
Sharkovskii's theorem, e.g. from
The Sharkovsky Theorem: A Natural Direct Proof by Burns and Hasselblatt.
Invariant sets of the logistic map
Read
The graph of the logistic map is a tower to learn about which type of invariant set can the logistic map have and what is the dynamical
relation between them. Possibly write some code to visualize such sets.
Bimodal maps
Study numerically the qualitative dynamics of some family of bimodal maps, i.e. maps of the unit interval into itself having exactly two extremal points (hence a local max and a local min).
An example is the family $f_{a,b}(x) = 1 - ax(1-x^2)(x-b)$.
Lorenz system
Read
Infinite towers in the graph of a dynamical system to learn about the Poincar´ map of the Lorenz system and its bifurcation diagram.
Another interesting article would be
What's New on Lorenz Strange Attractors? by Viana.
KAM theorem
Read Kolmogorov's article
Preservation of conditionally periodic movements with small change in the Hamilton function.
This article is considered the beginning of KAM's theory.
Arnold tongues
Learn about
Arnold's tongues, possibly do some numerical analysis of the system yourself.
Some example of elementary sources:
CIRCLE MAPS AND THE ARNOLD FAMILY,
Circle maps and Arnold Tongues.
Strange attractors
Read first
What is a strange attractor? by Ruelle.
Then read Ott's article
Strange attractors and chaotic motions of dynamical systems.
Another interesting article would be
What's New on Lorenz Strange Attractors? by Viana.
A Lorenz system on the 3-sphere
Read
Generalized Lorenz equations on a three-sphere by Saiki, Sander and Yorke
about some system of ODEs on the 3-sphere introduced by Lorenz in 60s.
Dynamics of real and complex Newton maps
Read
CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON $\Bbb R^2$ about the dynamics of real and complex Newton maps.
Possibly write code to numerically investigate some concrete case.
Arnold's cat map
Read about
Arnold's cat map.
Chaos in the solar system
Read
The arches of chaos in the Solar System, by Todorovic, Wu and Rosengren, or
the more analytical
ORBITAL RESONANCES AND CHAOS IN THE SOLAR SYSTEM by Malhotra.
