Anticipated synchronization in systems with distributed delay

Abstract

Anticipated synchronisation occurs when a driven dynamical system synchronises with the future state of the driver system to which it is unidirectionally coupled. Previous theoretical and experimental studies have focused on setups with a single delay time in the coupling term, for which exact anticipation can arise as a solution. Here we extend this framework to configurations with distributed delay times. Our main result is that, for a given delay distribution, approximate anticipated synchronisation can emerge over a range of coupling strengths. We analyse this phenomenon analytically for systems of linear oscillators, where we identify simple cases exhibiting exact synchronisation--up to a constant amplitude factor. Numerical simulations of nonlinear chaotic systems reveal stable forms of approximate anticipated synchronisation.

Figures (15)

• Figure 1: Illustration of the driver $\mathbf{x}(t)$ and the driven system $\mathbf{y}(t)$. The driver operates autonomously. The driven system is influenced unidirectionally by the driver through a term $K\mathbf{x}$. There is also a delay term $K\mathbf{y}(t-\tau)$ feeding into the dynamics of $\mathbf{y}$ at time $t$. As explained in the text, we refer to $\underline{\underline{\mathbf{K}}} \left[ \mathbf{x}\mleft( t \mright) - \mathbf{y}\mleft( t - \tau \mright) \right]$ as the 'delayed coupling term'.
• Figure 2: Real part and imaginary parts [panels (a) and (b) respectively] of the leading eigenvalues of the driven system as a function of the average delay $\overline\tau$ for a setup of two harmonic oscillators, with delay distribution $g(\tau) = \frac{1}{2}\left[ \delta\mleft( \tau - \tau_1 \mright) + \delta\mleft( \tau - \tau_2 \mright)\right]$. We set $\Omega = 1$, $b = 0.1$ and $K=0.8$. We also fixed the difference between the two delays $\tau_2 - \tau_1 = 0.2$. The leading eigenvalue is shown by a solid line, and subleading eigenvalue as dotted line. The eigenvalue of the driver system $-b \pm i\Omega_0$ is plotted in blue, while in orange the other leading eigenvalue of the driven system $s_{\max} \pm \Omega_1$ is shown.
• Figure 3: Sample trajectories from numerical integration of the system in Eqs. (\ref{['eq:driver_osc']}, \ref{['eq:driven_osc']}) with the delay distribution of Eq. \ref{['eq:g2taus']}, and initial conditions $y_i(t\le 0)=0$, $x_1(0)=1,x_2(0)=0$. Parameters were chosen to exemplify the whole phase space. I) The driven system shows anticipated synchronisation of the driver: $\tau_1 = 0.7$, $\tau_2 = 0.9$, II) the driven system is not synchronised with the driver but it decays to zero: $\tau_1 = 1$, $\tau_2 = 1.2$, and III) the driven system diverges: $\tau_1 = 1.3$, $\tau_2 = 1.5$. The other parameters are the same for all the panels: $\Omega = 1$, $b = 0.1$, $K_1 = 0.8$ and $K_2 = 0$. The solid orange lines correspond with the driven system's variable ($y_1$), the dashed blue lines with the driver system's variable ($x_1$), and the dotted green lines with the coupling mismatch $\Delta_1$, see Eq. \ref{['eq:Delta_def']}. An inset is added to each of the panels to show the correlation function between the $x_1$ and the $y_1$ variables.
• Figure 4: Stability diagram for the setup of two damped harmonic oscillators Eqs. (\ref{['eq:driver_osc']}, \ref{['eq:driven_osc']}) with the delay distribution of Eq. \ref{['eq:g2taus']}. Below the green dashed line the theory predicts the presence of AS (region I). Between the two dashed lines the analytical calculation shows decaying behaviour of the driver and the driven system, but no AS (region II). Above the blue dashed line finally, we expect the driven system to diverge (region III). The background colour shading in regions I and II indicates the value of $\left[ \max_{\Delta t}\mleft\{\rho_{x_1, y_1}(\Delta t)\mright\} + \max_{\Delta t}\mleft\{\rho_{x_2, y_2}(\Delta t)\mright\} \right]/2$, obtained from numerical integration of Eqs. (\ref{['eq:driver_osc']}, \ref{['eq:driven_osc']}). Combinations of $\overline\tau$ and $K$ are shown in red when the height of the maxima of the driven system exceeds that of the driver by a factor of at least $100$ at the end of the simulation $T_{\mathrm{sim}} = 6000$. We use this as an indicator of divergence in the driven system. Model parameters are $\Omega = 1$ and $b = 0.1$. We fix $\tau_2-\tau_1 = 0.2$.
• Figure 5: Anticipation time ($\tau^\ast$) from the theory and the numerical integration of the system of the system in Eqs. (\ref{['eq:driver_osc']}, \ref{['eq:driven_osc']}) with the delay distribution of Eq. \ref{['eq:g2taus']}. The results are shown as a function of $\Delta\tau = \tau_2 - \tau_1$. The parameters are: $\overline{\tau} = 0.9$, $\Omega = 1$, $b = 0.1$, $K_1 = 0.8$ and $K_2 = 0$. The lines are the results from the theory, Eq. \ref{['eq:taubar_2delays']}, and the points are obtained from numerical integration of the original system. Note that $\Delta\tau=1.8$ is the maximum possible value given $\overline\tau=0.9$.