The Problem of the Centre for Cubic Systems with Two Parallel Invariant Straight Lines and One Invariant Conic

Abstract

Abstract. For cubic differential systems with two parallel invariant straight lines and at least one invariant conic it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first three Liapunov quantities Lj, j = 1, 2, 3 vanish. Mathematics Subject Classification (2000). Primary 34C05; Secondary 58F14.

Darboux integrability of a cubic differential system with two parallel invariant straight lines

“…Note that the integrability conditions (i), (ii), (vi), (x) were obtained in [7] and the integrability conditions (iii), (iv), (v) were determined in [6].…”

“…The application of the method of Darboux to prove centers in all cases of quadratic differential systems was firstly proved in [16] and for cubic differential systems (1.2) with two invariant straight lines and one invariant conic was shown in [6]. Using Definition 2.1 for determining the invariant algebraic curves, the method of Darboux integrability and the identities (4.11), (4.14) we prove the following Theorem: Theorem 4.2.…”

“…Then by a rotation of axes we can make them parallel to the axis of ordinates (Oy) and the linear part of (1.2) preserves the form. According to [6] the cubic system (1.2) has two invariant straight lines l 1 and l 2 parallel to the axis Oy if and only if the following coefficient conditions are satisfied (2.4)…”

“…For cubic system (2.5) the problem of the existence of an invariant straight line was studied in [7], of an invariant conic in [6] and of an invariant cubic in [9].…”

“…Using the method of Darboux integrability and the rational reversibility, the problem of the center was solved for cubic system (1.2) with: four invariant straight lines [5]; three invariant straight lines [7], [17]; two invariant straight lines and one irreducible invariant conic [6]; two invariant straight lines and one irreducible invariant cubic [10]. The center conditions for a cubic system (1.2) with two distinct invariant straight lines by using the method of Darboux integrability and rational reversibility were found in [8].…”