@misc{WulffHohmannDeuflhard, author = {Wulff, Claudia and Hohmann, Andreas and Deuflhard, Peter}, title = {Numerical Continuation of Periodic Orbits with Symmetry.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1421}, number = {SC-94-12}, abstract = {We consider periodic orbits of autonomous parameter dependent ODE's. Starting from a shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincar\'e-section we develop a pathfollowing algorithm for periodic solutions based on a tangential continuation method with implicit reparametrization. For ODE's equivariant w.r.t. a finite group we show that spatial as well as spatio-temporal symmetries of periodic orbits can be exploited within the (multiple) shooting context. We describe how turning points, period doubling bifurcations and Hopf points along the branch of periodic solutions can be handled. Furthermore equivariant Hopf points and generic secondary bifurcations of periodic orbits with \$ Z_m\$-symmetry are treated. We tested the code with standard examples, e.g., the period doubling cascade in the Lorenz equations. To show the efficiency of the described methods we also used the program for an application from electronics, a ring oscillator with \$n \$ inverters. In this example the exploitation of symmetry reduces the amount of work for the continuation of periodic orbits from \${\cal O}(n^2)\$ to \${\cal O}(n)\$}, language = {en} } @misc{HohmannWulff, author = {Hohmann, Andreas and Wulff, Claudia}, title = {Modular Design of Extrapolation Codes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4909}, number = {TR-92-05}, abstract = {In this paper we describe a modular implementation of the well-established extrapolation codes EULEX, EULSIM and DIFEX for initial value problems of ordinary differential equations. The basic module embodies an abstract extrapolation method with order and stepsize control. Based on this module the particular integration codes only have to provide the underlying discretization schemes.}, language = {en} }