Center Conditions for a Cubic Differential System With an Invariant Conic

Abstract

We find conditions for a singular point O(0, 0) of a center or a focus type to be a center, in a cubic differential system with one irreducible invariant conic. The presence of a center at O(0, 0) is proved by constructing integrating factors.

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].…”

“…Theorem 4.1. The cubic system (1) with two invariant straight lines 𝑥 ∓ 𝑖𝑦 = 0 and an exponential factor of the form (14) has a center at the origin 𝑂 (0, 0) if and only if the first Lyapunov quantity vanishes.…”