Symmetries of center singularities of plane vector fields

• Main Author: Maksymenko, S.I.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2010
• Series: Нелінійні коливання
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/174925

Let D² ⊂ R² be a closed unit 2-disk centered at the origin O ∈ R², and F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a «center» singularity. Let θ : D² \ {O} → (0, +∞ ) be the function associating with each z ≠ O its period with respect to F. In general, such a function can not be even continuously defined at O. Let also D⁺(F) — be the group of diffeomorphisms of D², which preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D² if and only if the 1-jet of F at O is a «rotation», that is, j¹F(O) = −y(∂/∂x) + x(∂/∂y). Then D⁺(F) is homotopy equivalent to a circle.