The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for symmetric dynamical systems. But there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatiotemporal symmetries of periodic orbits can be exploited numerically. We describe methods for the computation of symmetry breaking bifurcations of periodic orbits for free group actions and show how bifurcations increasing the spatiotemporal symmetry of periodic orbits (including period halving bifurcations and equivariant Hopf bifurcations) can be detected and computed numerically. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a a tangential continuation method with implicit reparametrization.

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity.But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step towards a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous Figure Eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating Eight family and compute the rotating choreographies bifurcating from it.

Abstract. This paper describes the mechanical and electrical design, as well as the control strategy, of the FU-Fighters robots, a F180 league team that won the second place at RoboCup'99. It explains how we solved the computer vision and radio communication problems that arose in the course of the project. The paper mainly discusses the hierarchical control architecture used to generate the behavior of individual agents and the team. Our reactive approach is based on the Dual Dynamics framework developed by H. Jäger, in which activation dynamics determines when a behavior is allowed to influence the actuators, and a target dynamics establishes how this is done. We extended the original framework by adding a third module, the perceptual dynamics. Here, the readings of fast changing sensors are aggregated temporarily to form complex, slow changing percepts. We describe the bottom-up design of behaviors and illustrate our approach using examples from the RoboCup domain.

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