The transition between strong and weak chaos in delay systems: Stochastic modeling approach

We investigate the scaling behavior of the maximal Lyapunov exponent in chaotic systems with time delay. In the large-delay limit, it is known that one can distinguish between strong and weak chaos depending on the delay scaling, analogously to strong and weak instabilities for steady states and periodic orbits. Here we show that the Lyapunov exponent of chaotic systems shows significant differences in its scaling behavior compared to constant or periodic dynamics due to fluctuations in the linearized equations of motion. We reproduce the chaotic scaling properties with a linear delay system with multiplicative noise. We further derive analytic limit cases for the stochastic model illustrating the mechanisms of the emerging scaling laws.

Figure 1

Maximum Lyapunov exponent [upper curve, blue (gray)] and sub-Lyapunov exponent [lower curve (inset), black] for the logistic map with delayed feedback for $\tau =100$ as a function of the feedback strength $k$. The inset shows the sub-LE over a wider range, including three critical points.

Figure 2

Difference between maximum LE and sub-LE in strong chaos at $k=0.27$ as a function of $\tau$. Lower dashed line (green): Steady-state approximation. Solid line (blue): True $\lambda -{\lambda }_0$ from chaotic map. Line with markers (red): Stochastic model.

Figure 3

Delay-normalized maximum Lyapunov exponent in the weak-chaos regime as a function of the sub-LE for $\tau ={10}^3$ (lower curve of each pair) and $\tau ={10}^6$ (upper curve of each pair). Coupling strength $k$ has been varied from the critical point $k=0.328$ to $k=0.5$ to achieve the variation in ${\lambda }_0$. Pair of dashed lines (green): Steady-state approximation. Pair of solid lines (blue): True $\lambda \tau$ from chaotic map. Pair of lines with markers (red): Stochastic model.

Figure 4

Delay-normalized maximum Lyapunov exponent at critical point $k=0.328$ as a function of $\tau$. Lower dashed line (green): steady-state approximation. Solid line (blue): true $\lambda \tau$ from chaotic map. Line with markers (red): stochastic model.

Figure 5

Example trajectories of the stochastic system (16) for parameters $D=1$, $k=1$, and $\tau =10$ (solid fluctuating curves) together with the long-term average trend (dashed lines). Equation (16) was integrated with a stochastic Heun method and step size $dt=0.001$. Left: Strong-chaos case with ${\lambda }_0=1$ and $\lambda \approx 1.009$. Right: Weak-chaos case with ${\lambda }_0=-1$ and $\lambda \approx 0.087$.

Figure 6

Noise dependence of $\lambda -{\lambda }_0$ in strong chaos for $\kappa =1$ and different pairs of $({\lambda }_0,\tau )$ (black curves with markers). Red (upper) curves: Analytic limit case after Eq. (25). Blue (lower) curves: Extension of the scaling law at critical point by coordinate transformation after Eq. (C1). Vertical dotted lines indicate $D={\lambda }_0$. Left panel: ${\lambda }_0=1$, $\tau =4.28$. The limit of $D\ll {\lambda }_0$ is well approximated by Eq. (25). Right panel: ${\lambda }_0=0.1$, $\tau =78.5$. The limit of $D\gg {\lambda }_0$ is well approximated by Eq. (C1).

Figure 7

Panel (a): Evolution of the probability distribution for the distance $w\left(t\right)-w(t-\theta )$ for a displacement $\theta \in [0,\tau ]$. Parameters are ${\lambda }_0=-1$, $\kappa =1$, $D=1$, and $\tau =50$. Blue (gray) ensemble of curves: Density $\rho (w-w_{\theta })$. The red curve (thick, rightmost) highlights the distribution at $\theta =\tau$. Black (outer) and light green (gray, inner) curves at the bottom indicate spread of a Wiener process and the standard deviation of $\rho$, respectively. Panel (b): For same parameters as panel (a), except $D=0.1$, the trajectory of $w\left(t\right)$ (black curve) in the vibrating potential given by $w(t-\tau )$ is shown for $t\in [0,\tau ]$. The white highlighted region is a tube with radius $r=0.7$ around the potential minimum at $w_{\tau }+ln(-\kappa /{\lambda }_0)$. Iterating this delay-tunneled process leads to a meandering effect over successive delay windows.

Figure 8

Panel (a): Analytic limit curve (red) after Eqs. (28) and (29) and numerical values (black dots) for $\kappa =1$ and $\tau ={10}^3$. Values of ${\lambda }_0\in [-1,-0.1]$ and $D\in [0,2]$ have been chosen independently from a uniform probability distribution on the intervals. Normalization of axes see text. Panel (b): $\lambda \tau$ versus ${\lambda }_0$ at $D=1$, $\kappa =1$, and $\tau =54.6$ (black curve with markers). Comparison of finite delay approximations after Eqs. (C2) (red, left asymptotic curve) and (C1) (blue, right asymptotic curve).

Figure 9

Delay-normalized exponent at the critical point ${\lambda }_0=0$. Panel (a): Delay dependence for $\kappa =1$ and $D=1$. Numerical curve (black with markers) converges for $\tau \to \infty$ to the analytical limit curve (red) after Eq. (30). Panel (b): Dependence on coupling strength for $D=1$ and $\tau =100$. Numerical curve (black with markers) and analytical limit curve (red) are indistinguishable.

Figure 10

Gradual change between different scaling behaviors in the skew Bernoulli map. Panel (a): Parameter plane of $(k,r)$ with lines of constant ${\lambda }_0$ as labeled. Squares, circles, and thick solid lines mark the parameter values examined in (b)–(d), respectively. Lowest curve in each of (b)–(d) corresponds to $r=0$ (with no fluctuations in the linearized equations), the middle curve to $r=0.3$, and the upper curve to $r=0.5$. Panel (b): Strong chaos ${\lambda }_0>0$ (close to transition point); fluctuations decrease the decay constant $\alpha$ of $\lambda -{\lambda }_0\propto exp(-\alpha \tau )$. Panel (c): Critical point ${\lambda }_0=0$; fluctuations change scaling from $\lambda \tau \propto W\left(\tau \right)$ to $\lambda \tau \propto \sqrt{\tau }$. Panel (d): Weak chaos ${\lambda }_0<0$ at $\tau =1.5\times {10}^4$. Fluctuations change scaling from $\widehat{\mu }\propto ln|k/{\lambda }_0|$ towards $\widehat{\mu }\propto {\lambda }_0^{-1}$ for ${\lambda }_0\to 0-$.