A Simple Criterion for the Non-Existence of Limit Cycles of a Lienard System

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Abstract:

In this paper, as an application in our result, the non-existence of limit cycles for the Liénard  system   with
  ,   
is discussed by the simple criterion.
J.Graef[Gr :1971] has studied the uniformly boundedness of the solution orbits under the condition
(C1)  , 
Moreover, he also proved the existence of limit cycles under the conditions (C1) and
(C2) ∃k>0 s.t. F(x)≥A (x ≥k")," F(x)" ≤"-"A (" x ≤-k") for some fixed constant A."
Recently, M.Cioni and G.Villari[2015] gave the same result as in [Gr] under the conditions (C1) and
(C3) ∃K_1>〖∃K〗_2 s.t. F(x)≥K_1 (x>"β > 0)," F(x)" ≤" 〖 K〗_2 "(" x "< α < 0)."
Note that (C2) is included to (C3). Our aim here is to discuss on the case of which (C1) is satisfied, but (C3) is not satisfied. We shall give the simple criterion for the non-existence of limit cycles for a Liénard system with these conditions.

Keywords:

Liénard  system, Limit cycle, Global asymptotic stability