Short delay limit of the delayed Duffing oscillator

Area of Science:

• Nonlinear Dynamics
• Delay Differential Equations
• Bifurcation Theory

Background:

• The delayed Duffing equation exhibits a Hopf bifurcation that becomes singular under specific conditions (ε→0, τ=O(ε)→0).
• This singularity poses challenges in analyzing systems like laser stability, which share similar mathematical structures.
• Existing methods struggle to accurately capture the bifurcation behavior in this singular limit.

Purpose of the Study:

• To develop an asymptotic theory for resolving the singular Hopf bifurcation in the delayed Duffing equation.
• To derive a simplified system of ordinary differential equations (ODEs) that accurately represents the bifurcation.
• To validate the theory by comparing it with existing asymptotic solutions for arbitrary delays.

Main Methods:

• Taylor expansion of the delay term x(t-τ) in powers of τ.
• Derivation of a minimal system of ODEs from the original delay differential equation.
• Analysis of the asymptotic behavior and matching with solutions for fixed delays.

Main Results:

• A novel asymptotic theory successfully resolves the singularity of the Hopf bifurcation.
• Expansion of the delay term up to the third order was found to be essential, contrary to initial expectations.
• The derived ODE system accurately captures the Hopf bifurcation branch, validated by overlap with other asymptotic solutions.

Conclusions:

• The developed asymptotic theory provides an effective method for analyzing singular Hopf bifurcations in delayed systems.
• The necessity of third-order expansion highlights a subtle aspect of delayed dynamics.
• This work offers a simplified model for understanding complex phenomena in laser stability and other fields.