Research on the properties of some planar polynomial differential equations. (English) Zbl 1251.34048

Consider the differential equation \[ x' = X(t,x), \] where \(X\) is a differentiable function. Let \(\phi(t;t_0,x_0)\) denote the general solution. The reflecting function is defined by \(F(t,x)=\phi(-t;t,x)\). If the equation is \(2\omega\)-periodic in \(t\) then the Poincaré mapping is \(T(x)=F(-\omega,x)\). Hence, a solution \(\phi(t;-\omega,x_0)\) is \(2\omega\)-periodic if and only if \(x_0\) is a fixed point of \(T\). The author uses the method of reflecting functions to study the behavior of solutions and gives sufficient conditions for the equations to have linear or fractional reflecting functions. The results are used to derive sufficient conditions for a center of polynomial two-dimensional systems.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Alwash, M.A.M.; Lloyd, N.G., Non-autonomous equations related to polynomial two-dimensional systems, Proc. roy. soc. Edinburgh sec. A, 105, 129-152, (1987) · Zbl 0618.34026
[2] Lijun, Yang; Yuan, Tang, Some new results on Abel equations, J. math. anal. appl., 261, 100-112, (2001) · Zbl 0995.34031
[3] Decline, J.; Lloyd, N.G.; Pearson, J.M., Cubic systems and Abel equation, J. diff. eq., 147, 435-454, (1998) · Zbl 0911.34020
[4] Mironenko, V.I., Analysis of reflective function and multivariate differential system, (2004), University Press Gomel · Zbl 1079.34518
[5] Arnold, V.I., Ordinary differential equation, (1971), Science Press, Moscow, 198-240
[6] Mironenko, V.I., The reflecting function of a family of functions, Differ. eq., 36, 12, 1636-1641, (2000) · Zbl 1001.34027
[7] Alisevich, L.A., On linear system with triangular reflective function, Differ. eq., 25, 3, 1446-1449, (1989)
[8] Musafirov, E.V., Differential systems, the mapping over period for which is represented by a product of three exponential matrixes, J. math. anal. appl., 329, 647-654, (2007) · Zbl 1136.34003
[9] Mironenko, V.V., Time symmetry preserving perturbations of differential systems, Differ. eq., 40, 20, 1395-1403, (2004) · Zbl 1087.34521
[10] Verecovich, P.P., Nonautonomous second order quadric system equivalent to linear system, Differ. eq., 34, 12, 2257-2259, (1998)
[11] Maiorovskaya, S.V., Quadratic systems with a linear reflecting function, Differ. eq., 45, 2, 271-273, (2009) · Zbl 1180.34046
[12] Zhengxin, Z., On the reflective function of polynomial differential system, J. math. anal. appl., 278, 1, 18-26, (2003) · Zbl 1034.34008
[13] Zhengxin, Z., The structure of reflective function of polynomial differential systems, Nonlinear anal., 71, 391-398, (2009) · Zbl 1179.34039
[14] Zhengin, Z., On the qualitative behavior of periodic solutions of differential systems, J. comput. appl. math., 232, 600-611, (2009)
[15] Lloyd, N.G.; Pearson, J.M.; Romanovsky, V.G., Computing integrability conditions for a cubic differential system, Comput. math. appl., 32, 99-107, (1996) · Zbl 0871.34003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.