In the present investigation, we revisit the problem of bifurcation of limit cycles. The problem of limit cycle was studied intensively. For Liénard, we can read  - , and for non Liénard we can read  - .
We give criterion for the non Liénard system to have or not to have limit cycles with some parameters. We also demonstrate that the system exhibits a Hopf-bifurcation. Now we consider the following Liénard equation
The above equation may be written in two dimensional autonomous dynamical system
Therefore, the above equations can be written in the Liénard plane as
Theorem 1.1  Suppose that for system (1.1), there exist such that , for , for , for , and , then (1.1) has at most one limit cycle in D, which is simple and stable, if exists.
Theorem 1.2  If, in system (1.1), (or ), and equality holds only for at most a finite number of points, then (1.1) has no closed orbits in closed region .
In Section 2, the main system equations results have been presented, the section has been divided in two cases.
The case I considered the conditions that the system has a limit cycle when is an anti saddle.
Finally, the case of saddle point with limit cycle is presented in theorems and lemmas in Section 4 along with the concluding remakes.
2. The Basic System Equations and Results
The main part of this paper is devoted to explain the existence and uniqueness of limit cycles of the following differential equations system
the singular points of the system are and .
The Jacobian matrix
has the determinant for the origin O is anti saddle and for the origin O is saddle for more details (see  ).
The system (2.2) needs to change to the Liénard system (1.1).
Let so after simplify and substitute so that we have
After change z to y we can get system (1.1) as follows
The system (2.4) is considered in two cases.
Case I: The Origin is an anti Saddle
The case under consideration is , in this case and as above the system (2.4) has unique equilibrium point which is an anti saddle.
For system (2.4) has no limit cycle.
Let , then and . Thus and by using theorem (1.2) there is no limit cycle so we just look for .
In the case of O becomes saddle also for or ( ) O is node in two cases no limit cycles surround O. Thus, in the sequel, we only need to consider .
Consider the polynomial Liénard system of degree n
Lemma 2.2 
For system (2.5) with , the first three focal values at are
where is positive constant.
By scaling and [where new , and ], then system (2.4) becomes
Therefore the three focal values of and by using Lemma 2.2 namely are
If then O, is strong focus which is unstable for and stable if , and for , then O is weak focus of order one which is stable.
By using Hopf-bifurcation (by changing of stability), for no limit cycle because no change of stability if , then O is weak focus of order one which is stable. Thus as a decreasing from −1 O becomes unstable and one stable limit cycle appears from Hopf-bifurcation.
For the system (2.4) has a unique stable limit cycle.
Now we apply theorem (1.1) consider since So has only one root which is . For
the roots are . The roots of are and has minimum at .
Since we have , then we deduce that for
since then the term and the value of in the interval so for . Finally since so we have .
Case II: The Origin is a saddle
In this case, we discuss system (2.4) when and as above the system has three equilibrium points and where trance the to the Origin by the relation
Let , , (2.4) is converted into
By using Hopf-bifurcation, for no limit cycle because no change of stability if , then O becomes weak focus of order one which is stable.
Thus for fixe b is bifurcate value so as increasing, O becomes unstable and one stable limit cycle appear from Hopf-bifurcation.
equivalent to .
Assume that since , then we have contradiction. Thus for also we get .
Lemma 3.2 
If there exists a constant such that for , System (2.8) has at most one limit cycle.
We have by putting , and after simplify we have
Let so we have
Since , we can delete from upper equation and for suitable a as small enough we have
A non-Liénard system is studied and analyzed by adapting Hopf-bifurcation theory. It has been proved that the system has unique limit cycle under some change of parameters under two cases. Bendixons theorem is used to prove non-existence of limit cycles.
I would like to express my thanks to Prof. V.P. Sing, Depr. of Mathematics Albaha University for his voluble suggestion for improving the paper.
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