then there exist two positive number δ* and γ*, such that for 0 < |δ| < δ*, 0 < ∥γ - γ0∥ < γ*, in a neighborhood of the origin, system (2.24) has at most m limit cycles which enclose the origin (an elementary node) O 0, 0).
then there exist two positive numbers δ ∗ and γ ∗, such that for 0 < | δ | < δ ∗, 0 < ∥ γ − γ 0 ∥ < γ ∗, in a neighborhood of the origin, system (2.24) has at most m limit cycles which enclose the origin (an elementary node) O ( 0, 0,).
When 0 < δ ≪ 1, in a small neighborhood of the origin, system (1.5) has exact three critical points: O 0, 0), (x1, y1) and (x2, y2), where O 0, 0) is an elementary critical point, another two critical points are complex with x 1, 2 = ± i δ + o , y 1, 2 = ∓ i δ 2 + o ( δ 2 ), (1.6).
Firstly, by using the descriptor observer approach, the origin system is transformed into a descriptor system.
For this aim, first the truncation of the Taylor expansions for unknown functions was substituted in the origin system.
For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system.
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