Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems

In this paper, Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems is studied, a new algorithm of the formal series for the flow on center manifold is discussed, from this, a recursion formula for computation of the singular point quantities is obtained for the corresponding bifurcation equation, which is linear and then avoids complex integrating operations, therefore the calculation can be readily done with using computer symbol operation system such as Mathematica, and more the algebraic equivalence of the singular point quantities and corresponding focal values is proved, thus Hopf bifurcation can be considered easily. Finally an example is studied, by computing the singular point quantities and constructing a bifurcation function, the existence of 5 limit cycles bifurcated from the origin for the flow on center manifold is proved.