Exact Response Theory for Delay Equations

Area of Science:

• Theoretical Physics
• Chemical Physics
• Dynamical Systems Theory

Background:

• Exact response theory (Transient Time Correlation Function formalism) analyzes system dynamics perturbations.
• Linear response theory is limited to small perturbations.
• Generic dynamical systems require advanced response formalisms.

Purpose of the Study:

• Adapt exact response theory for time-lagged systems.
• Illustrate the adapted theory with delay equations.
• Compare linear and exact response approaches.

Main Methods:

• Extension of exact response theory to incorporate time-lagged dynamics.
• Mapping delay-dependent dynamics into a higher-dimensional, autonomous augmented phase space.
• Utilizing techniques for time-dependent perturbations in deterministic and stochastic systems.

Main Results:

• The adapted theory provides a framework for computing exact responses in time-lagged systems.
• Demonstrated applicability using simple delay equations with memory terms.
• Established the autonomy of the dynamics in the augmented phase space.

Conclusions:

• Exact response theory is successfully extended to handle time-lagged systems.
• The augmented phase space approach simplifies computation of exact responses.
• Further investigation into linear versus exact approaches is warranted.