Adjoint method provides phase response functions for delay-induced oscillations.

Area of Science:

• Dynamical Systems and Mathematical Biology
• Theoretical Physics

Background:

• Time delays are common in natural and engineered systems, often leading to limit-cycle oscillations.
• Existing phase-reduction theories are insufficient for systems with time delays and infinite-dimensional phase spaces.

Purpose of the Study:

• To develop a systematic phase-reduction theory for limit-cycle oscillations induced by time delay.
• To provide a practical framework for calculating the phase response function (Z(θ)) for such systems.

Main Methods:

• Developed a theoretical framework applicable to delay differential equations with infinite-dimensional phase spaces.
• Utilized the adjoint equation and a bilinear form to obtain Z(θ) as a zero eigenfunction.
• Validated the framework using two distinct biological oscillator models.

Main Results:

• Successfully calculated the phase response function (Z(θ)) for delay-induced limit cycles.
• Demonstrated the framework's validity and applicability to biological systems.
• The derived phase equation accurately predicts multimodal locking behavior.

Conclusions:

• The proposed framework offers a practical approach to analyzing phase dynamics in systems with time delays.
• This work advances the understanding of oscillations in complex systems, with implications for biology and physics.
• The prediction of multimodal locking behavior highlights the unique dynamics introduced by time delays.