Relative periodic orbits in point vortex systems

Relative Periodic Orbits of Symmetric Lagrangian Systems

Hamiltonian Systems Near Relative Periodic Orbits

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry propertie...

Reversible Relative Periodic Orbits

We study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector eld on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along g...

Numerical Bifurcation of Hamiltonian Relative Periodic Orbits

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur for example in celestial mechanices, molecular dynamics and rigid body motion. RPOs are solutions which are periodic orbits of the symmetry-reduced system. In this paper we analyze certain symmetry-breaking bifurcations of Hamiltonian relative periodic orbits and show how they can be detected and computed ...

Numerical Continuation of Hamiltonian Relative Periodic Orbits

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 ...