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 computation of a cubic Hopf normal form coefficient, assuming the remaining eigenvalues have negative real parts. In this paper, a modal decomposition of the Hopf coefficient is proved. The decomposition provides a new methodology for analyzing the Hopf cubic normal form coefficient in a formal way. The framework is illustrated by nonlinear stability analysis of two control designs where it is shown that the Hopf coefficient can be stabilized through modal nonlinear feedbacks.