Singular Hopf bifurcation in a differential equation with large state-dependent delay

Area of Science:

• Control Theory
• Nonlinear Dynamics
• Differential Equations

Background:

• State-dependent delay (SDD) differential equations are crucial in control theory and nonlinear dynamics.
• Understanding sustained oscillations and bifurcations in these systems is challenging due to large delays.

Purpose of the Study:

• To investigate the onset of sustained oscillations in classical state-dependent delay differential equations.
• To analyze the transition from sinusoidal to sawtooth oscillations past the instability threshold.
• To develop explicit analytical solutions for SDD equations with large delays.

Main Methods:

• Application of singular perturbation techniques.
• Asymptotic analysis to capture the evolution of oscillation profiles.
• Derivation of connections to established oscillators like Rayleigh and van der Pol.

Main Results:

• Identified a singular Hopf bifurcation leading to rapid sawtooth oscillations.
• Explicitly described the gradual transition from sinusoidal to sawtooth profiles.
• Derived van der Pol's equation for small-amplitude oscillations in nonlinear delay-dependent cases.

Conclusions:

• Singular perturbation techniques are effective for analyzing SDD equations with large delays.
• This work provides rare explicit analytical constructions for such complex systems.
• The findings contribute to the theoretical understanding of delay differential equations and their applications.