Center problem for cubic differential systems with the line at infinity of multiplicity four

Abstract

In this paper the center problem for cubic differential systems with the line at infinity of multiplicity four is solved.

Real cubic differential systems with a linear center and multiple line at infinity

“…According to [11], the line at infinity in complex variables 𝐴, 𝐵, ..., 𝑅, 𝑆 has the multiplicity at least four for cubic system {(1), (2)} if and only if the coefficients of {(1),(2)} verify one of the following two sets of conditions:…”

“…In [11], using the variables 𝐴, 𝐵, ..., 𝑅, 𝑆, it was shown that the line at infinity in (1) has multiplicity two if and only if one of the following three sets of conditions holds:…”

“…-in Case (7): We change the coefficients 𝑙 → −3𝑙, 𝑛 → −𝑛/3 (respectively, {𝑙 → −3𝑙, 𝑚 → −𝑚/3} and {𝑚 → 𝑚/9, 𝑝 → −𝑝/3, 𝑞 → −𝑞/3, 𝑟 → −𝑟/3, 𝑠 → −3𝑠}) in (1). This implies that the set of conditions (11) (respectively, ( 12) and ( 13)) is equivalent with the set of conditions (8) (respectively, ( 9) and ( 10)). Conditions (8) are contained in (14), while {(9), 𝑚 = 0}, {(9), 𝑚 ≠ 0}, { (10), 𝑘 = 0}, { (10), 𝑘 ≠ 0} are contained in ( 14), ( 16), ( 14), (18), respectively.…”

“…(see [11]). Using (4), we will solve in the real coefficient 𝑎, 𝑏, 𝑐, 𝑑, 𝑓 , 𝑔, 𝑘, 𝑙, 𝑚, 𝑛, 𝑝, 𝑞, 𝑟, 𝑠 of (1) the equalities (19)-(24).…”

Center conditions for a cubic system with two homogeneous in variant straight lines and exponential factors

“…The problem of the center was solved for some families of cubic differential systems having invariant algebraic curves (invariant straight lines, invariant conics, invariant cubics) in [5], [7], [9], [10], [12], [13], [16], [19], [20], [21]. Center conditions were determined for some cubic systems having integrating factors in [8], [11], [14], for some reversible cubic systems in [2] and for a few families of the complex cubic system in [15].…”