Universal Dichotomy for Dynamical Systems with Variable Delay

We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.

Figure 1

Visualization of the dichotomy in the parameter space of any delay system [Eq. (1)] with sawtooth-shaped delay [Eq. (6a)]: In the black regions a transformation to a constant delay is possible, whereas in the white areas (Arnold tongues) the access map Lyapunov exponent $\mu$ is negative, and, therefore, no transformation to a constant delay exists. Because the white regions are dense, almost no variation of the parameters $A$ or ${\tau }_0$ is possible without crossing an Arnold tongue.

Figure 2

(a) For DDEs with a conservative delay (crosses), the logarithmic scaling of the Lyapunov exponents ${\lambda }_l$ for large $l$ is well known [25], but a qualitatively different scaling appears for dissipative delay (dots). From top to bottom, the spectra correspond to Eqs. (7a), (7b), and (7c). (b) By subtracting the logarithmic behavior ${\lambda }_{\widehat{I},l}$, for DDEs with a dissipative delay a linear scaling with slope $\alpha$ is identified, which is independent of the specific system or attractor.

Figure 3

Decomposition of the solution operator for DDEs with time-varying delay. After the Koopman operator ${\widehat{C}}_i$ maps the solution segment ${\mathbf{x}}_{i-1}$ to the segment ${\overline{\mathbf{x}}}_i$ in the next interval, which is deformed according to the access map $R$, the integral operator ${\widehat{I}}_i$ generates the solution ${\mathbf{x}}_i$ from ${\overline{\mathbf{x}}}_i$.

Figure 4

Asymptotic slope $\widehat{\alpha }$ calculated from approximations ${\mathbf{D}}^{(N)}$ of the operator $\widehat{D}$ expanded in the eigenbasis of the DDE vs measured slope $\alpha$ of the Lyapunov spectrum of DDEs with various dissipative sinusoidal delays. The slopes of the Lyapunov spectra are identified as illustrated in Fig. 2. One finds $\widehat{\alpha }\simeq \alpha$, implying that the Koopman operator determines the asymptotics of the Lyapunov spectrum.