The concept of "graceful" transition through a Hopf bifurcation for a system of nonlinear
ordinary differential equations (ODEs) is introduced. The key idea is to control the
system such that its state space trajectory is close to the branch of equilibrium
solutions or to the branch of periodic solutions associated with a Hopf bifurcation.
This kind of evolution is called "graceful" and can be generated by formulating and
solving optimization control problems.