18 research outputs found
Three ways of treating a linear delay differential equation
This work concerns the occurrence of Hopf bifurcations in delay differential equations (DDE). Such bifurcations
are associated with the occurrence of pure imaginary characteristic roots in a linearized DDE. In this work
we seek the exact analytical conditions for pure imaginary roots, and we compare them with the approximate
conditions obtained by using the two variable expansion perturbation method. This method characteristically
gives rise to a “slow flow” which contains delayed variables. In analyzing such approximate slow flows, we
compare the exact treatment of the slow flow with a further approximation based on replacing the delayed
variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. By
comparing these three approaches we are able to assess the accuracy of making the various approximations. We
apply this comparison to a linear harmonic oscillator with delayed self-feedback