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

In this paper, we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore, we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In this paper, we extend our previous work [A. Otto, D. Müller, and G. Radons, Phys. Rev. Lett. 118, 044104 (2017)] by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.

Figure 1

Parameter space of an arbitrary system with time-varying delay $\tau \left(t\right)={\tau }_0+\frac{A}{2\pi }sin\left(2\pi t\right)$. The white regions correspond to a conservative delay, where the system is equivalent to a system with constant delay. On the other hand, the black regions correspond to a dissipative delay, where the system is not equivalent to a system with constant delay.

Figure 2

Comparison of the Lyapunov spectra between systems with conservative delay (filled) and dissipative delay (empty), computed by the semidiscretization method [2, 26]. (a) The Lyapunov spectra are shown for three different systems, where the systems from top to bottom are given by Eq. (10b) (circles), Eq. (10c) (triangles), and Eq. (10a) (squares). (b) The convergence properties of the method are illustrated for Eq. (10b), where $m$ denotes the number of equidistant points approximating the state of the delay system. As a guide to the eye, the asymptotic scaling behavior for dissipative delay, which we derive in Sec. 5, is represented by the dashed gray line. The jumps in the spectra related to conservative delay and the deviation of the spectra from the dashed line for dissipative delay, respectively, indicate the bounds on the number of well approximated exponents. For all computations, we have chosen the delay $\tau \left(t\right)={\tau }_0+0.9/\left(2\pi \right)sin\left(2\pi t\right)$ with the mean delay ${\tau }_0=1.51$ and 1.54 for dissipative and conservative delay, respectively.

Figure 3

Illustration of the decomposition of the solution operator into the sequential action of the access operator ${\widehat{C}}_k$ and the constant delay evolution operator ${\widehat{I}}_k$ for (a) constant delay and (b) time-varying delay. For constant delay ${\tau }_0$ the access operator ${\widehat{C}}_k$ reduces to a time shift by ${\tau }_0$ and for time-varying delay the access operator changes the length of the state interval and deforms the state on the time domain.

Figure 4

(a) Separation of the state intervals into the basins of attraction $U_{k,j}=[u_{k,j-1},u_{k,j}]$, where $u_{k,p-1}=t_k$, of the attractive fixed points of the $q\mathrm{th}$ iteration $R^q\left(t\right)$ of an exemplary access map $R\left(t\right)$ with rotation number $\frac{p}{q}=\frac{3}{2}$. (b) Evolution of a function $x_0\left(t\right)$ inside the $U_{k,j}$ by the successive application of the access operators ${\widehat{C}}_k$. The basic types of symbols of ${\widehat{C}}_{k,j}$, i.e., segments of $R\left(t\right)$, are illustrated by the alternating dashed and solid lines.

Figure 5

(a) Sawtooth-shaped delay and (b) corresponding reduced access map (solid line) and bisectrix (dashed line).

Figure 6

(a) Triangle delay characterized by two slopes $A$ and $B$ with (b) corresponding reduced access map (solid line) and bisectrix (dashed line).

Figure 7

(a) Asymptotic slope of the Lyapunov spectrum vs delay parameters $A$ and $B$ and (b) comparison of the asymptotic slope from the finite section approximation of the Koopman operator using an orthonormal basis (empty circles) and the Lyapunov basis (dots) vs direct measurement at the Lyapunov spectrum (line), for all $A$ and $B$ from (a). The inset shows the same analysis, whereby the Lyapunov basis was used, for sine-shaped delay $\tau \left(t\right)=1+\frac{A_s}{2\pi }sin\left(2\pi t\right)$ under variation of $A_s$.

Figure 8

(a) Lyapunov vectors (real part), i.e., eigenvectors of the monodromy operator, and (b) inverse participation ratio (IPR) of the periodic part $p_n\left(t\right)$ of the Floquet decomposition for exemplary conservative (dots) and dissipative (triangles) delays of Eq. (63) with $\tau \left(t\right)={\tau }_0+(0.9/2/\pi )sin\left(2\pi t\right)$. For conservative delay, the mean delay was set to ${\tau }_0=1.51$ and for dissipative delay ${\tau }_0=1.54$. The IPR is plotted versus the index of the corresponding Lyapunov exponent ordered from the largest to the smallest.

Figure 9

(a) Comparison of the real part (dots) and imaginary part (triangles) of the Fourier amplitudes of a Lyapunov vector (index $n=29$) with the corresponding envelope of the Fourier transform $X\left(f\right)\propto X_1\left(f\right)$ (lines), with $X_1\left(f\right)$ given by Eq. (81). (b) Lyapunov spectrum from semidiscretization (empty circles) compared with modeled spectrum (dots) computed by fitting Eq. (81) to the Fourier amplitudes of the Lyapunov vectors. The system (63) with $R\left(t\right)=t-\frac{0.1}{2\pi }sin\left(2\pi t\right)-1$ was used.

Figure 10

Comparison of the convergence properties of the Lyapunov spectra shown in Fig. 2 in Sec. 2 between systems with conservative delay (filled) and dissipative delay (empty), computed by the semidiscretization method [2, 26] for (a) the turning model, given by Eq. (10c) (triangles) and (b) the Hutchinson equation, given by Eq. (10a) (squares). The parameter $m$ denotes the number of equidistant points approximating the state of the delay system. As a guide to the eye, the asymptotic scaling behavior for dissipative delay, which is derived in Sec. 5, is represented by the dashed gray line.