Lyapunov Spectrum of Linear Delay Differential Equations with Time-Varying Delay

To understand the dynamics of systems with fluctuations in the retarded argument, the Lyapunov spectra of a scalar, linear delay differential equation (DDE) are studied. Depending on the structure of the deviated argument some Lyapunov exponents can be equal to minus infinity, which indicate the difference between the dimension of the state space and the asymptotic solution space. In order to validate the results of this dimensional collapse in systems with time-varying delay, the continuous DDE is analyzed by the method of steps. The iterated map of the stepwise retarded access by the deviated argument up to values of the initial function can characterize the dimensional behavior of DDE with time-varying delay.