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September 16, 2009

Speaker: Dr. Bart Oldeman, Department of Computer Science and Engineering, Concordia University 

Title: Numerical continuation of orbit segments with application to connecting orbits in the restricted three body problem and one-dimensional maps in a biological model

Abstract: The method of successive continuation of orbit segments (also known as the homotopy method) is a very powerful method for determining how a trajectory varies as initial conditions change. Instead of shooting from a set of initial conditions we pose a boundary value problem (BVP). First we compute one orbit segment that satisfies one initial condition, which can actually be done by continuation in its 'period' T, from 0 until the orbit satisfies a certain end condition, for instance, for its length, time, or its intersection with a plane. Next the initial condition of the orbit can be varied, keeping the end condition from the first step. Keeping the initial and end conditions well-posed the orbit is then continued as a whole, which means that all points on the orbit are taken into account, and not just the initial conditions. For example, sometimes the initial conditions might not vary numerically but the segment changes a lot at its end or elsewhere, and this method has no problems with that. This method can be implemented in standard continuation software such as AUTO, and has been used to compute, for example, the Lorenz manifold. I show here its application to the computation of homoclinic and heteroclinic orbits between relative periodic orbits in the circular planar restricted three body problem, and of one-dimensional Poincaré maps in a reduced ionic model of a cell from the rabbit sinoatrial node. 

Disciplines: Computing, Mathematics, Physics