Steady state bifurcations and exact multiplicity conditions via Carleman linearization

Abstract

The problem of steady-state bifurcations of vector fields under parameter perturbation is resolved by a linear algebraic method. Exact multiplicity conditions for any steady state are obtained in terms of the system parameters. No reduction of the steady-state system to one equation is required. Instead the one-dimensional case is included as a subspace in this generalized framework. The key point that this paper highlights is that the order of the steady multiplicity at bifurcation can be determined by examining the dimension of the kernel of the successive Carleman linear operators for all cases of practical interest. In particular, the dimension of the kernel of any Carleman linear operator of order l, equals l if l is less than the multiplicity, μ. However, the μth order Carleman operator retains a (μ - 1)-dimensional kernel. © 1987.