Stability of R 3-dynamical systems with symmetry.

Abstract

The study of the stability of a periodic solution p of a vector field using either the linear variational equations (associated to the vector field at p ), or the Poincaré map on a cross section, is known to present some difficulties. This work provides some techniques to ascertain the stability of the closed curve C={p 0 (t): t∈R} in the case of an R 3 analytic vector field X → possessing symmetries. It is assumed that one or more symmetry vectors S → are known (the Lie derivative of S → along the streamlines of X → , L X → (S → ) , is zero modulus X → ). One of the cases for which the stability of the closed curve can be determined is that of a divergence-free field X → having a known symmetry S → satisfying L X → (S → )=λ(x)X → and divS → =λ(x) . This is an interesting case because many devices used in the confinement of plasma possess symmetries of this type (X → is the magnetic induction vector B → ) with λ(x)=0 . This type of symmetry implies torus-like magnetic surfaces. It is noted that it constitutes an interesting (and difficult) problem to find examples of vector fields with symmetries for which λ≠0 . All the proofs are simple, and the technique is very nice.