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Classification and counting of planar quasi-homogeneous differential systems through their weight vectors. (English) Zbl 1400.37021

Summary: The quasi-homogeneous systems have important properties and they have been studied from various points of view. In this work, we provide the classification of quasi-homogeneous systems on the basis of the weight vector concept, especially in terms of the minimum weight vector, which is proved to be unique for any quasi-homogeneous system. Later we obtain the exact number of different forms of non-homogeneous quasi-homogeneous systems of arbitrary degree, proving a nice relation between this number and Euler’s totient function. Finally, we provide software implementations for some of the above results, and also for the algorithm, recently published by B. García et al. [J. Differ. Equations 255, No. 10, 3185–3204 (2013; Zbl 1336.34003)], that generates all the quasi-homogeneous systems.

MSC:

37C10 Dynamics induced by flows and semiflows
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
33E05 Elliptic functions and integrals
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations

Citations:

Zbl 1336.34003

Software:

counterQH.m; Matlab
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Full Text: DOI

References:

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