Short delay limit of the delayed Duffing oscillator

The delayed Duffing equation, x^{″}+ɛx^{'}+x+x^{3}+cx(t-τ)=0, admits a Hopf bifurcation which becomes singular in the limit ɛ→0 and τ=O(ɛ)→0. To resolve this singularity, we develop an asymptotic theory where x(t-τ) is Taylor expanded in powers of τ. We derive a minimal system of ordinary differential equations that captures the Hopf bifurcation branch of the original delay differential equation. An unexpected result of our analysis is the necessity of expanding x(t-τ) up to third order rather than first order. Our work is motivated by laser stability problems exhibiting the same bifurcation problem as the delayed Duffing oscillator [Kovalev et al., Phys. Rev. E 103, 042206 (2021)2470-004510.1103/PhysRevE.103.042206]. Here we substantiate our theory based on the short delay limit by showing the overlap (matching) between our solution and two different asymptotic solutions derived for arbitrary fixed delays.