Highest weak focus order for trigonometric Liénard equations. (English) Zbl 1450.34022
In this paper, the authors generalize some results about the highest weak focus order of the system
\[\dot x=y, \quad \dot y=g(x)+yf(x),\]
where \(f\) and \(g\) are polynomials with given degrees.
Concretely, the authors consider the systems \[\dot \theta=y, \quad \dot y=G'(\theta)+yF'(\theta),\] where \(F\) and \(G\) are trigonometric polynomials. They use the transformation \(\tan \theta=x\) to change the above systems to polynomial systems and try to use the results of polynomial systems, at last they can estimate the smallest upper bounds of weak focus order, sometimes, the upper bounds are sharp.
Concretely, the authors consider the systems \[\dot \theta=y, \quad \dot y=G'(\theta)+yF'(\theta),\] where \(F\) and \(G\) are trigonometric polynomials. They use the transformation \(\tan \theta=x\) to change the above systems to polynomial systems and try to use the results of polynomial systems, at last they can estimate the smallest upper bounds of weak focus order, sometimes, the upper bounds are sharp.
Reviewer: Changjian Liu (Zhuhai)
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 |
| 34C25 | Periodic solutions to ordinary differential equations |
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