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Two-dimensional homogeneous cubic systems: classification and normal forms. III. (English. Russian original) Zbl 1373.34054

Vestn. St. Petersbg. Univ., Math. 50, No. 2, 97-110 (2017); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 4(62), No. 2, 179-192 (2017).
Summary: This article is the third in a series of works devoted to two-dimensional homogeneous cubic systems. It considers the case where the homogeneous polynomial vector on the right-hand side of the system has a quadratic common factor with real zeros. The set of such systems is divided into classes of linear equivalence, in each of which a simplest system being a third-order normal form is distinguished on the basis of appropriately introduced structural and normalization principles. In fact, this normal form is determined by the coefficient matrix of the right-hand side, which is called a canonical form (CF). Each CF is characterized by an arrangement of nonzero elements, their specific normalization, and a canonical set of admissible values of the unnormalized elements, which ensures that the given CF belongs to a certain equivalence class. In addition, for each CF, (a) conditions on the coefficients of the initial system are obtained, (b) nonsingular linear substitutions reducing the right-hand side of a system satisfying these conditions to a given CF are specified, and (c) the values of the unnormalized elements of the CF thus obtained are given.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C41 Equivalence and asymptotic equivalence of ordinary differential equations
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[1] V. V. Basov, “Two-dimensional homogeneous cubic systems: Classification and normal forms. I,” Vestn. St. Petersburg Univ.: Math. 49, 99-110 (2016). · Zbl 1383.15009 · doi:10.3103/S1063454116020023
[2] V. V. Basov, “Two-dimensional homogeneous cubic systems: Classification and normal forms II,” Vestn. St. Petersburg Univ.: Math. 49, 204-218 (2016). · Zbl 1388.34029 · doi:10.3103/S1063454116030031
[3] V. V. Basov and A. S. Chermnykh, “Canonical forms of two-dimensional homogeneous cubic systems with a common square factor,” Differ. Uravn. Protsessy Upr., No. 3, 66-190 (2016). http://www.math.spbu.ru/diffjournal/EN/numbers/2016.3/article.1.7.html. Accessed March 03, 2017. · Zbl 1360.34085
[4] Markeev A.P., “Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation,” Nelineinaya Din. 10, 447-464 (2014). · Zbl 1372.70053 · doi:10.20537/nd1404005
[5] A. P. Markeev, “On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian,” Regular Chaotic Dyn. 20, 309-316 (2015). · Zbl 1378.70022 · doi:10.1134/S1560354715030077
[6] K. S. Sibirskii, Introduction to the Algebraic Theory of Invariants of Differential Equations (Shtiintsa, Kishinev, 1982; Manchester Univ. Press, Manchester, 1988).
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