When all ε-order focal values are zero, we compute (varepsilon^{2} -order focal varepsilon^{2} -order).
In this part, we give the definition of the focal values for a three-dimensional system.
According to the method in the article [11, 12], to compute the Liyapunov constants (or focal values) of the origin of system (2.8), we obtain the Lyapunov constants (or focal values) of the origin of (2.8) (namely the focal values of the equilibrium ( 1, 0 ) of model (1.2)) as follows.
In Section 2, we give some preliminaries about Liapunov constants, the focal values and singular point quantities on center manifold for a three-dimensional system (1).
In [3, 21 24], by considering the normal forms of (1.1), the authors tried to study the computation problem of focal values.
So when all ε-order focal values are zero, there exist four limit cycles which could be bifurcated from the origin of system (1.3).
By including all the probabilities of the hypothesis called focal values P k as a function of m, we consider a power set 2Φ and satisfy the conditions as follows: 1.
By similar discussion, we could conclude that for any sufficiently small (|varepsilon|neq0) there exist at most four limit cycles which could be bifurcated from the origin of system (1.4) when all ε-order focal values are zero.
