| Normal form theory is one of the most important methods for reducing ordinarydifferential equations (ODE). It plays an important role in the study of the dynamicalbehavior of the original system in the vicinity of the critical equilibrium. This methodcarves out a new way for solving the nonlinear dynamical system.A brief introduction to the characteristic of the basic methods for calculatingnormal form of ODE is given in this paper. Expansion and application of this theoryare also discussed. The Liapunov-Schmidt reduction,the Central Manifold theoremand normal form theory are compared with each other. The former program forcalculating the Central Manifold of nonlinear systems was improved in this paper.This program becomes more universal,simpler and more efficient than the formerone.The simplest normal forms of the non-resonant double Hopf bifurcation (DHB)with parameter are obtained without truncating the original differential equation, i.e.the perturbation parameter of the system is treated as one-dimensional state variable.The computation of the simplest normal form is not only based on near-identitynonlinear transformations, but also based on parameter rescaling. The expressions ofthe simplest normal forms from order two to order five for non-resonant DHB for thesystem with parameter are gotten with the help of computer algebra programMathematica. From which, the simplest normal forms of this system up to any highorder is deduced.The simplest normal form of resonant DHB was studied based on lie operator.The coefficients of the simplest normal forms of resonant DHB and the non-lineartransformations in terms of the original system coefficients are given explicitly. Ageneral program was achieved with the help of the computer algebra languageMathematica. This program can not only compute the simplest normal form ofresonant DHB but also can compute the non-resonant form. This program couldcompute the simplest normal forms of DHB up to seventh order.
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