Seminars & Colloquia Calendar

Abstract:  I will discuss an automated method for locating/excluding connecting orbits for differential equations.  The method is based on formal series calculations
and results in an atlas for a large portion of the (local) stable/unstable manifold.   In this talk I will focus on manifolds associated with saddle-focus equilibrium solutions.   The atlases are ``mined'' for connections but can also be used to rule out their existence.  The major drawback of the method its high cost in both CPU time and memory resources.   For dynamical systems with many parameters it may be prohibitively expensive to run such calculations at all parameter values of interest. 

I will discuss some recent work with Shane Kepley, where we use numerical continuation methods to find connecting orbits over a wide range of parameters values, starting from an ensemble of connections computed using the more deliberate methods discussed above.  As an example I will focus on a model problem coming from celestial mechanics. 

I will also discuss a little the history of this problem and some interesting results which follow from the existence of homoclinic connections in Hamiltonian systems.