Limit cycle uniqueness for a class of planar dynamical systems. (English) Zbl 1188.34040
The paper contains the proof of a theorem concerning the uniqueness of a limit cycle for the system
\[ \dot{x}=\beta(x)(\varphi(y)-F(x,y)),\quad \dot{y}=-\alpha(y)g(x), \]
where \(\alpha(y)\) and \(\beta(x)\) are assumed to be positive. The theorem is applied to such systems in the case \(F(x,y)=F(x)v(y),\) where \(v(y)\) vanishes at some point.
\[ \dot{x}=\beta(x)(\varphi(y)-F(x,y)),\quad \dot{y}=-\alpha(y)g(x), \]
where \(\alpha(y)\) and \(\beta(x)\) are assumed to be positive. The theorem is applied to such systems in the case \(F(x,y)=F(x)v(y),\) where \(v(y)\) vanishes at some point.
Reviewer: Alexander Grin (Grodno)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
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