The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations

Abstract

We are interested in nonlinear delay differential equations which have a Hopf bifurcation. We assume zero is a steady state for the problem, and so a Hopf bifurcation point lies on the boundary of the region of asymptotic stability for the zero solution. We investigate whether discrete versions of the nonlinear delay differential equation also exhibit Hopf bifurcations. We use the boundary locus method as a tool both for the delay differential equation and for numerical analogues. We use the information obtained about the stability domain to assist in identifying Hopf bifurcations. We demonstrate the following: • For certain linear multistep methods, the boundary of the region of stability for the zero solution of the original equation is approximated by the boundary of the region of stability for the zero solution of the numerical analogue equation to the order of the method. • The boundary locus method enables us to determine precise parameter values at which any Hopf bifurcations arise in the discrete equations. We prove that Hopf bifurcation points for the true equation are approximated to the order of the method by corresponding points in the discrete scheme. • Further calculations are necessary to determine the precise nature of bifurcation points identified in this way.