A modal decomposition of the Hopf normal form coefficient

The local stability of a nonlinear dynamical system at an equilibrium point with a pair of purely imaginary eigenvalues can be assessed through the co mputation of a cubic Hopf normal form coefficient, assuming the remaining e igenvalues have negative real parts, In this paper, a modal decomposition o f the Hopf coefficient is proved. The decomposition provides a new methodol ogy for analyzing the Hopf cubic normal form coefficient in a formal way, T he framework is illustrated by nonlinear stability analysis of two control designs where it is shown that the Hopf coefficient can be stabilized throu gh modal nonlinear feedbacks.