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An algorithm for providing the normal forms of spatial quasi-homogeneous polynomial differential systems. (English) Zbl 1428.34049

Summary: Quasi-homogeneous systems, and in particular those 3-dimensional, are currently a thriving line of research. But a method for obtaining all fields of this class is not yet available. The weight vectors of a quasi-homogeneous system are grouped into families. We found the maximal spatial quasi-homogeneous systems with the property of having only one family with minimum weight vector. This minimum vector is unique to the system, thus acting as identification code. We develop an algorithm that provides all normal forms of maximal 3-dimensional quasi-homogeneous systems for a given degree. All other 3-dimensional quasi-homogeneous systems can be trivially deduced from these maximal systems. We also list all the systems of this type of degree 2 using the algorithm. With this algorithm we make available to the researchers all 3-dimensional quasi-homogeneous systems.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
12H05 Differential algebra
12H20 Abstract differential equations
12-08 Computational methods for problems pertaining to field theory
68W30 Symbolic computation and algebraic computation

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References:

[1] Álvarez, M.; Gasull, A.; Prohens, R., Global behaviour of the period function of the sum of two quasi-homogeneous vector fields, J. Math. Anal. Appl., 449, 2, 1553-1569 (2017) · Zbl 1369.34046
[2] Chiba, H., Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field, J. Differ. Equ., 259, 12, 7681-7716 (2015) · Zbl 1329.35012
[3] García, B.; Llibre, J.; Pérez del Río, J. S., Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differ. Equ., 255, 10, 3185-3204 (2013) · Zbl 1336.34003
[4] García, B.; Lombardero, A.; Pérez del Río, J. S., Classification and counting of planar quasi-homogeneous differential systems through their weight vectors, Qual. Theory Dyn. Syst., 17, 3 (2018) · Zbl 1400.37021
[5] Geng, F.; Lian, H., Bifurcation of limit cycles from a quasi-homogeneous degenerate center, Int. J. Bifurc. Chaos, 25, 01 (2015) · Zbl 1309.34066
[6] Han, M.; Romanovski, V. G., Isochronicity and normal forms of polynomial systems of odes, J. Symb. Comput., 47, 10, 1163-1174 (2012) · Zbl 1248.34046
[7] Huang, J.; Zhao, Y., The limit set of trajectory in quasi-homogeneous system in \(R^3\), Appl. Anal., 91, 7, 1279-1297 (2012) · Zbl 1286.37020
[8] Kozlov, V., Rational integrals of quasi-homogeneous dynamical systems, J. Appl. Math. Mech., 79, 3, 209-216 (2015) · Zbl 1432.37081
[9] Li, S.; Wu, K., On the limit cycles bifurcating from piecewise quasi-homogeneous differential center, Int. J. Bifurc. Chaos, 26, 07 (2016) · Zbl 1343.34043
[10] Liang, H.; Huang, J.; Zhao, Y., Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dyn., 78, 3, 1659-1681 (2014) · Zbl 1345.34068
[11] Liang, H.; Torregrosa, J., Centers of projective vector fields of spatial quasi-homogeneous systems with weight \((m, m, n)\) and degree 2 on the sphere, Electron. J. Qual. Theory Differ. Equ., 2016, 103, Article 1 pp. (2016)
[12] Llibre, J.; Zhang, X., Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15, 4 (2002) · Zbl 1024.34001
[13] Tang, Y.; Wang, L.; Zhang, X., Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35, 5, 2177-2191 (2015) · Zbl 1327.34054
[14] Xiong, Y.; Han, M.; Wang, Y., Center problems and limit cycle bifurcations in a class of quasi-homogeneous systems, Int. J. Bifurc. Chaos, 25, 10 (2015) · Zbl 1326.34069
[15] Yoshida, H., Necessary condition for the existence of algebraic first integrals, Celest. Mech. Dyn. Astron., 31, 4, 363-399 (1983) · Zbl 0556.70015
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