Riccati equation and special third-order systems without movable critical points. (English. Russian original) Zbl 0911.34028

Lukashevich, N. A.; Martynov, I. P.

Differ. Equations 33, No. 5, 710-712 (1997); translation from Differ. Uravn. 33, No. 5, 707-708 (1997).

The paper deals with the third-order system of nonlinear ODEs \(\alpha'= a_2 (\alpha \gamma-\beta)+\varphi (\alpha)\), \(\beta'=(a_1 \alpha+a_2\gamma +b_1+b_2)\beta+c_1 \gamma+c_2\alpha\), \(\gamma'=a_1(\alpha\gamma-\beta)+\psi(\gamma)\), whose coefficients \(a_i, b_i,c_i\), \(i=1,2\), are analytic functions of \(t\in D\), \(a_1a_2\not\equiv 0\), and \(\varphi(\alpha)\), \(\psi(\gamma)\) are second-degree polynomials in \(\alpha\) and \(\gamma\), respectively. The authors prove that this system has no movable critical points and use its solutions \(\alpha,\beta\), and \(\gamma\) for finding a relationship among solutions to two corresponding Riccati equations.

Reviewer: V.Chernyatin (Szczecin)

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems, general theory
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

Keywords:

analytic coefficients; nonlinear ODEs; second-degree polynomials