Computing bifurcation points via characteristics gain loci

The authors show how to check the crossing on the imaginary axis by the eigenvalues of the linearized system of differential equations depending on a real parameter μ via feedback system theory. E. Hopf's theorem (1942) refers to a system of ordinary differential equations depending on the real parameter μ in which, when a single pair of complex conjugate eigenvalues of the linearized equations cross the imaginary axis under the parameter vibration, near this critical condition periodic orbits appear. The authors present simple formulas for both static (one eigenvalue zero) and dynamic or Hopf (a single pure imaginary pair) bifurcations, and show some singular conditions (degeneracies) by continuing the bifurcation curves in the steady-state manifold. The bifurcation curves and singular sets of an interesting chemical reactor which possesses multiplicity in the equilibrium solutions and in the Hopf bifurcation points are described