From dynamical systems with time-varying delay to circle maps and Koopman operators

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Operator Theory

Background:

• Time-delay systems exhibit complex dynamics influenced by delays.
• Time-varying delays introduce unique challenges compared to constant delays.
• Understanding these dynamics is crucial for various scientific and engineering fields.

Purpose of the Study:

• To investigate the impact of time-varying delays on the dynamics of delay systems.
• To identify and characterize different universality classes arising from time-varying delays.
• To develop a theoretical framework connecting delay system dynamics with Koopman operator theory.

Main Methods:

• Decomposition of the delay system's solution operator using an operator-theoretic framework.
• Introduction of the Koopman operator to describe time-varying delay access.
• Analysis of the dynamics of the associated one-dimensional iterated map (access map).

Main Results:

• Discovery of two distinct universality classes for time-varying delays.
• Demonstration of fundamental differences in dynamical quantities, such as the Lyapunov spectrum.
• Establishment of a connection between delay differential equations and the theory of circle maps.

Conclusions:

• The dynamics of the access map determine the universality classes of infinite-dimensional delay systems.
• The operator-theoretic framework provides new insights into the behavior of time-delay systems.
• This work extends previous findings by detailing mathematical aspects and Lyapunov vector behavior.