Center Problem for Several Differential Equations via Cherkas' Method☆

Cherkas' method characterizes centers for analytic Lienard differential equations. We extend his method to degenerate Lienard differential equations and we apply this extension to solve the center problem for several families of polynomial differential equations. In particular, we give all centers for some differential equations given by a vector field which is the sum of two quasi-homogeneous ones.

[1]  Joan Torregrosa,et al.  Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity , 1999 .

[2]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[3]  J. Giné,et al.  Integrability of a linear center perturbed by a fifth degree homogeneous polynomial , 1997 .

[4]  Jaume Giné,et al.  Integrability of Centers Perturbed by Quasi-Homogeneous Polynomials , 1997 .

[5]  Armengol Gasull,et al.  Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields , 1997, Canadian Journal of Mathematics.

[6]  N. G. Lloyd,et al.  Five limit cycles for a simple cubic system , 1997 .

[7]  Noel G. Lloyd,et al.  Algorithmic Derivation of Centre Conditions , 1996, SIAM Rev..

[8]  Noel G. Lloyd,et al.  Small-amplitude limit cycles in polynomial Liénard systems , 1996 .

[9]  Jaume Giné,et al.  Integrability of a linear center perturbed by a fourth degree homogeneous polynomial , 1996 .

[10]  Francesc Mañosas,et al.  Cyclicity of a family of vector fields , 1995 .

[11]  J. Z. Zhang,et al.  On the Center of the Liénard Equation , 1993 .

[12]  Stephen Lynch,et al.  Small-amplitude limit cycles of Liénard systems , 1990 .

[13]  N. G. Lloyd,et al.  The number of small-amplitude limit cycles of Liénard equations , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Roberto Conti,et al.  Recent Advances in Differential Equations , 1981 .

[15]  U. Staude.  UNIQUENESS OF PERIODIC SOLUTIONS OF THE LIENARD EQUATION , 1981 .

[16]  C. Pugh,et al.  On Liénard's equation , 1977 .

[17]  A. Andronov,et al.  Qualitative Theory of Second-order Dynamic Systems , 1973 .

[18]  A. Barut.  Studies in mathematical physics , 1973 .

[19]  A. Andronov.  Theory of bifurcations of dynamic systems on a plane = Teoriya bifurkatsii dinamicheskikh sistem na ploskosti , 1971 .

[20]  R. Conti,et al.  Nonlinear Differential Equations , 1964 .

[21]  N. N. Bautin,et al.  On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , 1954 .

[22]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .