| Abstrakt: |
Consider the polynomial differential system of degree m of the form x ˙ = − y (1 + μ (a 2 x − a 1 y)) + x (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , y ˙ = x (1 + μ (a 2 x − a 1 y)) + y (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , where μ and ν are real numbers such that (μ 2 + ν 2) (μ + ν (m − 2)) (a 1 2 + a 2 2) ≠ 0 , m > 2 and Ωm−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ. [ABSTRACT FROM AUTHOR] |
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