RT Journal Article
T1 Stability of R 3-dynamical systems with symmetry.
A1 Gonzalez Gascón, F.
A1 Romero Ruiz del Portal, Francisco
AB 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.
PB Società Italiana di Fisica
SN 0369-3554
YR 1999
FD 1999
LK https://hdl.handle.net/20.500.14352/58558
UL https://hdl.handle.net/20.500.14352/58558
DS Docta Complutense
RD 7 may 2024