Pseudolaminar chaos from on-off intermittency

In finite-dimensional, chaotic, Lorenz-like wave-particle dynamical systems one can find diffusive trajectories, which share their appearance with that of laminar chaotic diffusion [Phys. Rev. Lett. 128, 074101 (2022)] known from delay systems with lag-time modulation. Applying, however, to such systems a test for laminar chaos, as proposed in [Phys. Rev. E 101, 032213 (2020)], these signals fail such a test, thus leading to the notion of pseudolaminar chaos. The latter can be interpreted as integrated periodically driven on-off intermittency. We demonstrate that, on a signal level, true laminar and pseudolaminar chaos are hardly distinguishable in systems with and without dynamical noise. However, very pronounced differences become apparent when correlations of signals and increments are considered. We compare and contrast these properties of pseudolaminar chaos with true laminar chaos.

Figure 1

Time series of pseudolaminar chaos and laminar chaos. In (a) and (c) pseudolaminar chaos generated from Eq. (1) is shown. True laminar chaos generated from Eq. (2) is shown in (e). The corresponding derivatives of the time series are plotted below in Figs. (b), (d), and (f). For (a) and (b) we used the parameters $\sigma =5, b=1, A_r=40$, and $T=400$. In (c) and (d) we have $\sigma =5, b=1, A_r=15$, and $T=10$. For the laminar chaos in (e) and (f), we used $\mathrm{\Theta }=50, \mu =0.9, {\tau }_0=1, A_{\tau }=0.76$, and $T=1$. For a sufficient relaxation of the transient dynamics, plotting of the time series begins after 100 periods of the time-varying parameter $r\left(t\right)$ or of the delay $\tau \left(t\right)$. The dotted lines in (b) and (d) correspond to $r\left(t\right)<1$, where a stable fixed point exists at $(X,Y,Z)=(0,0,0)$ of the system in Eqs. (1).

Figure 2

Return maps of the levels $x_n$ of the (pseudo) laminar phases and of the increments ${\delta }_n=x_{n+1}-x_n$. The parameters were chosen as in Fig. 1, where (a) and (b) correspond to Fig. 1, (c) and (d) correspond to Fig. 1, and (e) and (f) correspond to Fig. 1 as indicated by the colors. The levels are estimated by setting $x_n=x\left(nT\right)$, where $T$ is the period of $r\left(t\right)$ or of the time-varying delay $\tau \left(t\right)$ for pseudolaminar and laminar chaos, respectively. For the laminar chaos considered in (e) and (f), the time instants $t_n^{*}=nT$ are the members of the drifting attractive orbits of the access map, Eq. (3) (see text).

Figure 3

Histogram of the increments ${\delta }_n=x_{n+1}-x_n$ shown in Fig. 2 and Gaussian approximation (solid line) with zero mean, where the standard deviation was computed from the data.

Figure 4

Effect of dynamical noise on time series of pseudolaminar chaos and laminar chaos. In (a) and (c) pseudolaminar chaos and in (e) true laminar chaos is shown, where additive Gaussian white noise with a small standard deviation $\epsilon =0.01$ was added to the right-hand side of Eq. (1b) for pseudolaminar chaos and to Eq. (2) for laminar chaos. The corresponding derivatives of the time series are plotted in (b), (d), and (f), where, for laminar chaos, the approximate derivative ${\mathrm{\Delta }}_h\left(x\right)\left(t\right)=h^{-1}[x(t+h)-x\left(t\right)]$ with $h\approx 0.0023$ is shown. The parameters were chosen as in Fig. 1.

Figure 5

Return maps of the levels $x_n$ of the (pseudo) laminar phases and of the increments ${\delta }_n=x_{n+1}-x_n$ in systems with noise. While the features of laminar chaos shown in (e) and (f) persist, the line-like structures obtained for pseudolaminar chaos, which are shown in Figs. 2 and 2, are destroyed by the noise as illustrated in (c) and (d). The parameters were chosen as in Fig. 1, where (a) and (b) correspond to Fig. 1, (c) and (d) correspond to Fig. 1, and (e) and (f) correspond to Fig. 1, as indicated by the colors. Additive Gaussian white noise with a small standard deviation $\epsilon =0.01$ was added to the right-hand side of Eq. (1b) for pseudolaminar chaos and to Eq. (2) for laminar chaos. The levels are estimated by setting $x_n=x\left(nT\right)$, where $T$ is the period of $r\left(t\right)$ or of the time-varying delay $\tau \left(t\right)$ for pseudolaminar and laminar chaos, respectively. For the laminar chaos considered in (e) and (f), the time instants $t_n^{*}=nT$ are the members of the drifting attractive orbits of the access map, Eq. (3), where the fluctuations due to the noise are minimal (see Sec. 3).

Figure 6

Time series of pseudolaminar chaos in the switched system. (a) Pseudolaminar chaos generated from Eq. (1), where the parameters were chosen as in Fig. 1. The sinusoidal variation of parameter $r=r\left(t\right)$ was replaced by a piecewise constant function, which equals $A_r$ if the equilibrium $(X,Y,Z)=(0,0,0)$ of the system in Fig. 1 is unstable (no dashed black line) and $-A_r$ if it is stable (black dashed line visible). (b) The corresponding derivative of the time series. For a sufficient relaxation of the transient dynamics, we begin plotting of the time series after 100 periods of the time-varying parameter $r\left(t\right)$.

Figure 7

Return maps of the levels $x_n$ of the pseudolaminar phases and of the increments ${\delta }_n=x_{n+1}-x_n$. The parameters were chosen as in Fig. 6.