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Theory of center-focus for a class of higher-degree critical points and infinite points. (English) Zbl 1012.34027

Here, the real planar autonomous differential systems with higher-degree critical points (including elementary critical points and infinite points) \[ \frac{dx}{dt}=\sum_{k=2n+1}^{\infty}X_{k}(x,y)=X(x,y),\quad \frac{dy}{dt}=\sum_{k=2n+1}^{\infty}Y_{k}(x,y)=Y(x,y), \] and \[ \frac{dx}{dt}=\sum_{k=0}^{2n+1}X_{k}(x,y),\quad \frac{dy}{dt}=\sum_{k=0}^{2n+1}Y_{k}(x,y), \] are studied, where \(X_{k}(x,y), Y_{k}(x,y)\) are homogeneous polynomials of degree \(k\) and the functions \(X(x,y), Y(x,y)\) are analytic at the origin. The results extend statements concerning the detection of center or focus, successor function, central integration, integration factor, focal values, values of singular points and bifurcation of limit cycles.

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
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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References:

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