Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities

Abstract In this paper we investigate the linearizability problem for the two-dimensional time-reversible complex system x = x + P ( x , y ) , y = − y + Q ( x , y ) , where P and Q are homogeneous quartic polynomials. The necessary and sufficient conditions for the linearizability of this system are found. As a corollary, the necessary and sufficient conditions for the origin to be an isochronous center of the planar time-reversible real system u = − v + F ( u , v ) , v = u + G ( u , v ) , where F and G are homogeneous quartic polynomials, are obtained.

[1]  Christiane Rousseau,et al.  Linearization of Isochronous Centers , 1995 .

[2]  Alexandra Fronville,et al.  Solution of the 1 : −2 resonant center problem in the quadratic case , 1998 .

[3]  Valery G. Romanovski,et al.  The Sibirsky component of the center variety of polynomial differential systems , 2003, J. Symb. Comput..

[4]  Jaume Giné,et al.  Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials , 2000 .

[5]  Jiang Qian Ying,et al.  Conditions for Strong Stabilizabilities of n-Dimensional Systems , 1998, Multidimens. Syst. Signal Process..

[6]  C. Rousseau,et al.  Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbb{C}^2$ , 2001 .

[7]  Dongming Wang,et al.  Differential equations with symbolic computation , 2005 .

[8]  Marco Sabatini,et al.  A survey of isochronous centers , 1999 .

[9]  Valery G. Romanovski,et al.  Linearizability conditions for a cubic system , 2007, Appl. Math. Comput..

[10]  V. Romanovski.  The Linearizable Centers of Time-Reversible Polynomial Systems , 2003 .

[11]  Jaume Giné,et al.  Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial , 1999 .

[12]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[13]  Christiane Rousseau,et al.  Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in $$\mathbb{C}^2 $$ , 2003 .

[14]  Valery G. Romanovski,et al.  The centre and isochronicity problems for some cubic systems , 2001 .

[15]  Valery G. Romanovski,et al.  Time-Reversibility in Two-Dimensional Polynomial Systems , 2005 .

[16]  B. Buchberger.  Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems , 1970 .

[17]  一松 信,et al.  R.C. Gunning and H.Rossi: Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965, 317頁, 15×23cm, $12.50. , 1965 .

[18]  Anatolii A. Logunov,et al.  Analytic functions of several complex variables , 1965 .