Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence

Dynamical systems with multiple, hierarchically long-delayed feedback are introduced and studied extending our previous work [Yanchuk and Giacomelli, Phys. Rev. Lett. 112, 174103 (2014)]. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable time scales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delay size and the features of the transition between small and large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.

Figure 1

Time series of $\left|z\right(t\left)\right|$ from (1), shown at different levels of zoom. Time is measured in units of ${\tau }_2$ in (a) and (b) and in units of ${\tau }_1$ in (c) and (d). The parameter values are $a=-0.985, b=0.4, c=0.6$ (corresponding to $P=0.015$), $d=-0.75+i, {\tau }_1=100$, and ${\tau }_2=10000$. Initial conditions are chosen randomly.

Figure 2

Same as in Fig. 1 for $d=-0.1+i$.

Figure 3

Spatiotemporal representation (see text) of the dynamics of system (1) in Fig. 1: spiral defects. (a) Snapshot of the spatial profile in the pseudospace coordinates $(x,y)$, plotted for ${\theta }_0=0.4$. (b) Constant level lines for the phase of $z$; the circles denote the positions of defects. The phase is considered in the range $[-\pi ,\pi ]$, and the levels for negative and positive phases are shown as the dashed and solid lines, respectively. The spatial coordinates $x$ and $y$ are different time scales given by Eq. (37).

Figure 4

Spatiotemporal representation (see text) of the system dynamics in Fig. 2: defects turbulence. Plots are as in Fig. 3. The jumps of the phase from $-\pi$ to $\pi$ are emphasized by thicker lines.

Figure 5

Autocorrelation function $R$ of $z$ for the solution of Eq. (2) with $a=-0.625, b=0.4, c=0.225, d=-0.1+i, {\tau }_1=\tau =20$, and ${\tau }_2={\tau }^2=400$. $R$ is plotted using three different rescalings for the shift (see inset).

Figure 6

Geometric representation of solutions to Eqs. (18) and (19). The solutions are given as the intersection points of two functions: $\varphi =-{\omega }_0/\varepsilon \text{mod}2\pi$ and $\varphi ={\varphi }_k\left({\omega }_0\right)$, where the latter is given implicitly by Eq. (19). Since the distance between the neighboring solutions is $\sim \varepsilon$, the distance from any point of the domain to a solution is also $\sim \varepsilon$.

Figure 7

Snapshots for the solutions of the normal form equation (38). (a) Spiral defects, parameter values: $p=250,a_1=1.11, a_2=-1.22, a_3=1.39, a_4=1.11, a_5=0.56$, and $d=-0.75+i$. (b) Defect turbulence, parameter values are the same except for $d=-0.1+i$. Initial values are random and close to zero.

Figure 8

Drift in the propagation of defect structures, from the integration of (1). Parameters are those used in Fig. 1. Using the STR with the uncorrected pseudospace variables, the defects are moving in pseudotime (a). The components of the (vectorial) drift can be evidenced by plotting two successive ${\tau }_2$-long (b) and ${\tau }_1$-long slices (c). The separations between the two vertical dashed lines are $\overline{y}$ (b) and $\overline{x}$ (c) drift components, respectively.

Figure 9

Temporal series from system (1) for different values of the delays. The parameters used are those of Fig. 2. Left column: time series of $\left|z\right|$ with amplitude rescaled by ${\tau }_1$, for increasing values of ${\tau }_1$ (from top to bottom: ${\tau }_1=3$, 5, 8, and 15). Time is expressed in units of ${\tau }_2$. Right: maximum and minimum of the amplitude $\left|z\right|$ (a) and of the rescaled amplitude $\left|z\right|/\varepsilon =\left|z\right|{\tau }_1$ (b), evaluated after a transient.

Figure 10

Autocorrelation function $R$ for $\left|z\left(t\right)\right|$. Dependence of the autocorrelation on the delay $\tau$.