Lecture "Introduction to Numerical Continuation"

The data deposit contains an introductory  video lecture by Uwe Thiele, WWU Münster, Institut für Theoretische Physik on "Introduction to Numerical Continuation" together with the slides in pdf format and this info file.

The lecture introduces in a step-by-step manner the basic concept of pseudo arclength path-continuation in a form suitable for advanced Bachelor students, Master students and beginning PhD students of the natural sciences and other interested people. It is thought as a prelude to the "Münsteranian Torturials" a series of hands-on tutorials on continuation for selected soft matter related problems available on Zenodo and also hosted at https://www.uni-muenster.de/CeNoS/Lehre/Tutorials/continuation.html

The lecture is self-contained, lists some starting points in the literature, further readings and available software. The lecture has been given in a slowly evolving form over a number of years. Its basic structure follows several available lecture notes, e.g., by Doedel and Keller. The material is sufficiently detailed to enable everyone to create their own basic numerical continuation code.

After motivating why one would like to numerically solve bifurcation problems, Newton’s root finding method is recapitulated for one dimensional systems before coming to its multi-dimensional version. Then simple parameter continuation is explained, detailing tangent predictor and Newton corrector steps. It is illustrated for a predator-prey model where it continues some fixed points but fails at a saddle-node bifurcation. Subsequently, to alleviate the problem pseudo-arclength continuation is introduced, again explaining  tangent predictor and Newton corrector steps in some detail. The predator-prey model is revisited and solved for fixed points, this time successfully. Finally an outlook is given to further capabilities of continuation and relevant software packages are mentioned.