Abstract: By a newly developed method for finding the normal form, this paper deduces the normal form for the mutiple Hopf bifurcations of a high dimensional nonlinear dynamical system in noninternal resonance cases when its Jacobian matrix has many pairs of imaginary eigenvalues, and gives the expressions of the coefficients in the normal form by those inthe original system.

Key words: normal form, noninternal resonance, mutiple Hopf bifurcations