Non-existence conditions of local bifurcations for rational systems with structured uncertainties

Sufficient conditions are presented for the avoidance of bifurcations for uncertain rational systems. A necessary condition for a steady-state bifurcation to occur is that the system´s Jacobian has a zero eigenvalue, and a necessary condition for a Hopf bifurcation to occur is that the system´s Jacobian has a pair of pure imaginary eigenvalues. Based on the structured singular value μ and skewed structured singular value ν, this paper derives guaranteed non-existence conditions for a zero eigenvalue and pure imaginary eigenvalues of the system´s Jacobian. Conditions for systems with mixed parametric and dynamic uncertainties to maintain stability/instability are also derived. An example illustrates the determination of regions guaranteed to have no bifurcations.