Dynamics of commuting systems on two-dimensional manifolds. (English) Zbl 0941.34018
Consider the two planar autonomous systems \(S_V\) and \(S_W\) generated by the vector fields \(V\) and \(W\), respectively. Assume that (i) the fields have isolated (and the same) critical points; (ii) they are transverse at noncritical points; (iii) their flows commute. Some results by N. A. Lukashevich [Differ. Equations 1, 220-226 (1966; Zbl 0178.43301); translation from Differ. Uravn. 1, 295-302 (1965)] are reproved and some new results are obtained. For example, the author shows that, for the system \(S_V\), (i) every nonempty limit set is a critical point; (ii) if \(\operatorname {div} V=0\) in a neighborhood of a critical point, then this point is a center; (iii) if a critical point is asymptotically stable, then its basin of attraction is unbounded. It is shown that if a two-dimensional compact connected orientable manifold admits two vector fields \(V\) and \(W\) as above, then this manifold is either a sphere or a torus.
Reviewer: S.Yu.Pilyugin (St.Peterburg)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C10 | Dynamics induced by flows and semiflows |
| 34C40 | Ordinary differential equations and systems on manifolds |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
Keywords:
planar autonomous systems; vector fields; noncritical points; flows; critical point; center; attraction; sphere; torusCitations:
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