Claudia Wulff

Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations

Summary.We prove that a standard second order finite difference uniform space discretization of the semilinear wave equation with periodic boundary conditions, analytic nonlinearity, and analytic initial data conserves momentum up to an error which is exponentially small in the stepsize. Our estimates are valid for as long as the trajectories of the full semilinear wave equation remain real analytic. The method of proof is that of backward error analysis, whereby we construct a modified equation which is itself Lagrangian and translation invariant, and therefore also conserves momentum. This modified equation interpolates the semidiscrete system for all time, and we prove that it remains exponentially close to the trigonometric interpolation of the semidiscrete system. These properties directly imply approximate momentum conservation for the semidiscrete system. We also consider discretizations that are not variational as well as discretizations on non-uniform grids. Through numerical example as well as arguments from geometric mechanics and perturbation theory we show that such methods generically do not approximately preserve momentum.

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 Poincare 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 properties of the Hamiltonian system is described, and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.

Bifurcation from relative periodic solutions

Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries. In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.

Center-manifold reduction for spiral waves*

Abstract Spiral waves are rotating waves of reaction-diffusion equations on the plane. In this Note, a center-manifold reduction for the dynamics of spiral waves is presented. Bifurcations of rigidly-rotating spiral waves are then described by ordinary differential equations, which are equivariant under the (special) Euclidean group SE(2). Several difficulties arise in this analysis because SE(2) is not compact and does not induce a strongly continuous group action on the underlying function space.

Translational Symmetry-Breaking for Spiral Waves

Summary. Spiral waves are observed in numerous physical situations, ranging from Belousov-Zhabotinsky (BZ) chemical reactions, to cardiac tissue, to slime-mold aggregates. Mathematical models with Euclidean symmetry have recently been developed to describe the dynamic behavior (for example, meandering) of spiral waves in excitable media. However, no physical experiment is ever infinite in spatial extent, so the Euclidean symmetry is only approximate. Experiments on spiral waves show that inhomogeneities can anchor spirals and that boundary effects (for example, boundary drifting) become very important when the size of the spiral core is comparable to the size of the reacting medium. Spiral anchoring and boundary drifting cannot be explained by the Euclidean model alone.In this paper, we investigate the effects on spiral wave dynamics of breaking the translation symmetry while keeping the rotation symmetry. This is accomplished by introducing a small perturbation in the five-dimensional center bundle equations (describing Hopf bifurcation from one-armed spiral waves) which is SO(2)-equivariant but not equivariant under translations. We then study the effects of this perturbation on rigid spiral rotation, on quasi-periodic meandering and on drifting.

Numerical Continuation of Symmetric 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 symmetric dynamical systems. However, there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatio-temporal 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 spatio-temporal 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 Poincare section and a tangential continuation method with implicit reparametrization.

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 first step toward 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 path following 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.

Persistence of Hamiltonian relative periodic orbits

Abstract We prove a persistence result for Hamiltonian relative periodic orbits with generic drift–momentum pairs in the case of non-compact non-free group actions. Our starting point is a relative periodic orbit which is non-degenerate modulo isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety, the space of drift–momentum pairs, which is determined solely by the symmetry group of the problem. We apply our results to relative periodic solutions of deformable bodies in fluids.

Numerical Bifurcation of Hamiltonian Relative Periodic Orbits

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and the motion of rigid bodies. RPOs are solutions which are periodic orbits of the symmetry-reduced system. In this paper we analyze certain symmetry-breaking bifurcations of RPOs in Hamiltonian systems with compact symmetry group and show how they can be detected and computed numerically. These are turning points of RPOs and relative period-doubling and relative period-halving bifurcations along branches of RPOs. In a comoving frame the latter correspond to symmetry-breaking/symmetry-increasing pitchfork bifurcations or to period-doubling/period-halving bifurcations. We apply our methods to the family of rotating choreographies which bifurcate from the famous figure eight solution of the three-body problem as angular momentum is varied. We find that the family of choreographies rotating around the

Pinning and locking of discrete waves

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