Jan A. Sanders

Integrable systems in n-dimensional Riemannian geometry

Summary. In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.

Integrable systems in three-dimensional Riemannian geometry

Summary. In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.

Integrable systems and their recursion operators

In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchies of infinitely many commuting symmetries of the operator. Moreover, the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots.

Normal form theory and spectral sequences

Abstract The concept of unique normal form is formulated in terms of a spectral sequence. As an illustration of this technique some results of Baider and Churchill concerning the normal form of the anharmonic oscillator are reproduced. The aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory.

On recursion operators

Abstract We observe that application of a recursion operator of the Burgers equation does not produce the expected symmetries. This is explained by the incorrect assumption that D x −1 D x =1. We then proceed to give a method to compute the symmetries using the recursion operator as a first-order approximation.

Are higher order resonances really interesting

It is shown that higher order resonances of equilibria of conservative Hamiltonian systems do not give rise to an exchange of energies between the different degrees of freedom for most of the initial values on a certain long time-scale. The topology of these resonances is analysed, using Bott-Morse theory.

Limit cycles in the Josephson equations

Using techniques from bifurcation theory, we find the bifurcation diagram and corresponding phase portraits on

On the Classification of Automorphic Lie Algebras

TS^1

On the Passage through Resonance

of the Josephson equation:

Coupled cell networks: Semigroups, Lie algebras and normal forms

\dot \phi = y,\dot y = - \sin \phi + \varepsilon (a - (1 + \gamma \cos \phi )y)