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Linearizability conditions of time-reversible cubic systems. (English) Zbl 1192.34037

The authors obtain necessary and sufficient conditions for the linearizability of the planar time-reversible cubic complex system
\[ \dot x= x+ P(x, y),\quad \dot y= -y+ Q(x, y). \]
From these conditions they derive necessary and sufficient conditions for the origin to be an isochronous center of the time-reversible cubic real system
\[ \begin{aligned} \dot x & = -y+ \alpha_1 xy+ \alpha_2 x^2 y+\alpha_3 y^3,\\ \dot y & = x+ \beta_1 x^2+ \beta_2 y^2+ \beta_3 x^3+ \beta_4 xy^2.\end{aligned}\tag{1} \]
Theorem: The real system (1) has an isochronous center at the origin if and only if its coefficients lie in the variety of one of the following ideals
(1) \(\langle\alpha_1- 2\beta_1- \beta_2, \alpha_3, 2\beta_1(3\beta_1+ \beta_2)(4\beta_1+ \beta_2)- \alpha_2(15\beta_1+ 4\beta_2), 3\alpha_2- \beta_4, 2\alpha_2- \beta_1(4\beta_1+ \beta_2)+ 3\beta_3\rangle\),
(2) \(\langle\beta_1, \alpha_1- \beta_2, \alpha_3, \alpha_2- \beta_4, \beta_3\rangle\),
(3) \(\langle\beta_1+ \beta_2, \alpha_1+ 2\beta_1, \alpha_3, \alpha_2- \beta_4, \beta_3\rangle\),
(4) \(\langle\alpha_1- 2\beta_1- \beta_2, \alpha_3, \beta_1(4\beta_1+ \beta_2)- \alpha_2, 2\alpha_2- \beta_4, \beta_3\rangle\),
(5) \(\langle\beta_1, 9\alpha_3+ (\alpha_1- 4\beta_2)(\alpha_1- \beta_2), \alpha_2, \beta_3, \beta_4\rangle\),
(6) \(\langle\beta_1+ \beta_2, \alpha_1+ 2\beta_1, \alpha_2+ 3\alpha_3, \alpha_2- \beta_4, \alpha_3- \beta_3\rangle\),
(7) \(\langle\alpha_1+ \beta_1- \beta_2, \alpha_2- \beta_1(\beta_1+ \beta_2), \alpha_2+ \alpha_3, 2\alpha_3+ \beta_4, \beta_3\rangle\),
(8) \(\langle\beta_1+ \beta_2, 2\alpha_1+ \beta_1, \alpha_3, \alpha_2, \beta_4, \beta^2_1- 4\beta_3\rangle\),
(9) \(\langle\beta_1+ 2\beta_2, 2\alpha_1+ 3\beta_1, \alpha_3, 2\alpha_2+ \beta^2_1, \beta_4, \beta^2_1- 2\beta_3\rangle\),
(10) \(\langle 3\beta_1+ 4\beta_2, \alpha_1+ \beta_1, \alpha_3, \alpha_2, \beta_4, \beta^2_1- 3\beta_3\rangle\),
(11) \(\langle \beta_1, \beta_2, \alpha_3, \alpha^2_1- 9\beta_4, \alpha_2+ 2\beta_4, \beta_3\rangle\),
(12) \(\langle \beta_1, \beta_2, \alpha_1, 2\alpha_2+ 9\alpha_3, 3\alpha_3+ 2\beta_4, \beta_3\rangle\).

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms

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References:

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