Laminar Chaos

We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems, the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.

Figure 1

Exemplary trajectories of DDE (1) with $f(z)=4z(1-z)$ and sinusoidal delay $\tau (t)={\tau }_0+A\sin (2\pi t)$ for (a) a conservative delay (${\tau }_0=1.54$) leading to turbulent chaos known from systems with constant delay [23], and for (b) a dissipative delay (${\tau }_0=1.50$) leading to laminar chaos characterized by burstlike transitions at the attractive periodic points of the map $R^{-1}(t)$ mod 1 (vertical lines) and constant laminar phases between them [$T=200$, $A=0.9/(2\pi )$].

Figure 2

Construction of the solution of Eq. (1) for large $T$ via the exact formula Eq. (3) (a),(c) and the limit map Eq. (4) (b),(d). For a conservative delay (a),(b), the operator $\mathcal{C}$ has no significant effect, and each iteration of $\mathcal{F}$ creates stronger chaotic oscillations resulting in turbulent chaos. For dissipative delays (c),(d), the periodic repulsive and attracting properties of the map $R^{-1}$ [21] lead to laminar phases and irregular bursts, because the chaotic oscillations are compressed at the attracting points of $R^{-1}$. For finite $T$ (a),(c), the additional operator ${\mathcal{T}}_n$ smooths the solutions compared to the case $T\to \infty$ in (b),(d).

Figure 3

A laminar chaotic solution $z(t)$ and the inverse access map Eq. (6a) of the corresponding dissipative delay with rotation number $\rho =(p/q)$ are shown (here $\rho =\frac{3}{2}$). The arrows indicate the mapping of the boundaries of the laminar phases to the next state interval, which correspond to the attractive periodic points of $R^{-1}$ mod 1 (the attractive fixed points of $R^{-q}(t)-p$, where the subtraction of $p$ removes the drift). Thus, the width and the location of the plateaus is specified by the variable delay $\tau (t)$, whereas its intensity levels are given by the nonlinearity $f$ (see text).

Figure 4

a) Lyapunov chart of the access map $R$ for $\tau (t)={\tau }_0+A\sin (2\pi t)/(2\pi )$, where the contours correspond to a constant Lyapunov exponent $\lambda [R]({\tau }_0,A)$. For Eq. (1) with $f(z)=2z/(1+z^{10})$ (Mackey-Glass equation) the criterion (8) for laminar chaos is fulfilled above the dashed (red) line, which in this case corresponds to the contour level $-\lambda [f]\approx -0.51$. (b) Kaplan-Yorke dimension $D_{KY}$ of this system ($T=2000$) for the sinusoidal delay with $A=0.9$ [horizontal dashed line in (a)]. For conservative delays very high attractor dimensions are possible. In contrast, for mean delays ${\tau }_0$, where Eq. (8) holds, $D_{KY}$ is very small indicating laminar chaos.