Stability of R 3-dynamical systems with symmetry.



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Gonzalez Gascón, F. and Romero Ruiz del Portal, Francisco (1999) Stability of R 3-dynamical systems with symmetry. Il Nuovo Cimento della Società Italiana di Fisica. B, 114 (3). pp. 273-280. ISSN 0369-3554

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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.

Item Type:Article
Subjects:Sciences > Mathematics > Topology
ID Code:21780
Deposited On:11 Jun 2013 12:46
Last Modified:12 Dec 2018 15:13

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