Computations in Dynamics

Course Number: 
Hours - Lecture: 
Hours - Lab: 
Hours - Recitation: 
Hours - Total Credit: 
Typical Scheduling: 
Not typically scheduled

This is a 2 week intense course, with lectures in the morning and computational lab in the afternoon. 

Students will be asked to have a laptop with Matlab on it.

The course will meet in the mornings for lecture, break for  lunch, and reconvene for the lab in the afternoons.


Rudiments of Matlab. 

Undergraduate ODE course like Math 2552.

Course Text: 

No prescribed textbook. 

Topic Outline: 

The emphasis will be on problem-driven exploration of computational methods for dynamics. For instance, based on Lorenz system, students will explore basic dynamical system notions, numerical simulation of ODEs, and more advanced topics such as the computation and continuation of various invariant sets, like equilibria.

The computational part, in particular, will be done slowly and carefully, as it is the key and uniqueness of the course, and the instructors will assist the GTAs to make sure this part is done well.

On the first week the topics are:

(1) Lorenz system and review of elementary dynamical system notions 

(2) Using MATLAB to simulate the system: ODE integrators,
(3) Linearization, equations of variations,

(4) Simulating a simple evolutionary PDE: finite difference and spectral method.

On the second week the advanced topics are:

(5) Computing periodic orbits and their stability,
(6) Bifurcations and their computation,

(7) Computing hyperbolic structures (saddle point and stable/unstable manifolds, hyperbolic periodic orbit),

(8) Homoclinic/heteroclinic connections and their computations,

(9) Hamiltonian systems and geometric numerical integration.
(10) Excursion into piecewise-smooth systems.

Topics (1)-(7) are mandatory, the rest as time permits.