Chaos synchronization by resonance of multiple delay times

Chaos synchronization may arise in networks of nonlinear units with delayed couplings. We study complete and sublattice synchronization generated by resonance of two large time delays with a specific ratio. As it is known for single-delay networks, the number of synchronized sublattices is determined by the greatest common divisor (GCD) of the network loop lengths. We demonstrate analytically the GCD condition in networks of iterated Bernoulli maps with multiple delay times and complement our analytic results by numerical phase diagrams, providing parameter regions showing complete and sublattice synchronization by resonance for Tent and Bernoulli maps. We compare networks with the same GCD with single and multiple delays, and we investigate the sensitivity of the correlation to a detuning between the delays in a network of coupled Stuart-Landau oscillators. Moreover, the GCD condition also allows detection of time-delay resonances, leading to high correlations in nonsynchronizable networks. Specifically, GCD-induced resonances are observed both in a chaotic asymmetric network and in doubly connected rings of delay-coupled noisy linear oscillators.

Figure 1

(a) Sketch of a three-unit directed ring with double time delay. (b) Complete synchronization regions for directed rings of Bernoulli maps with ${\tau }_2=2{\tau }_1$ and $a=1.05$. Synchronization regions are nested “tongues” of smaller area for an increasing number of ring units: two (blue), three (red), four (green), and five (yellow). The color code and nested structure is common to all subsequent subfigures. The maps are obtained by solving Eq. (4) numerically for every perpendicular mode $n>0$ and then intersecting the stable regions. (c) Complete synchronization maps for the same networks after a detuning $({\tau }_1,{\tau }_2)\to ({\tau }_1-1,{\tau }_2-1)$. (d) Equivalent complete synchronization regions for rings of Tent maps with equal local Lyapunov exponents: $b=0.008458,{\lambda }_L=log1.05$. Squares indicate a completely synchronized trajectory after a small perturbation of magnitude ${10}^{-3}$ at the SM and $40000$ map iterations.

Figure 2

Master stability function $\lambda (\gamma ,\gamma )$ for a double-delayed network as obtained by solving Eq. (4), with $2{\tau }_1={\tau }_2$ (left) and $3{\tau }_1={\tau }_2$ (right). Parameters are $\kappa =\epsilon =0.5$ and $a=1.1$. The solid line marks the contour $\lambda =0$. Thus, the eigenmodes whose corresponding eigenvalue lies inside the contour are stable and those staying outside are unstable. The eigenvalues ${\gamma }_n=exp(2\pi in/N)$ are indicated for $N=3$ (triangles), $N=4$ (squares), and $N=5$ (inverted triangles). The eigenvalue ${\gamma }_0=1$ along the synchronization manifold is indicated with a circle.

Figure 3

(a) Four-unit directed ring with two delay times $3{\tau }_1={\tau }_2=100$. Units 1 and 3, and 2 and 4, respectively, belong to synchronized sublattices. (b) Sublattice synchronization region for Bernoulli maps with $a=1.05$ obtained by solving Eq. (4) numerically for modes 2 and 3, ${\gamma }_{2,3}=\pm i$, and intersecting their stability regions. Squares mark stable sublattice synchronization for equivalent Tent maps, obtained by simulation as in Fig. 1.

Figure 4

(a) Three-unit ring with auxiliary units analogous to the network depicted in Fig. 1. The double delay $2{\tau }_1$ is substituted by two links of delay length ${\tau }_1$ mediated by auxiliary units. (b) The synchronization region, in light red, was obtained by solving Eq. (9) $\left(a=1.05\right)$. It is smaller than the corresponding from the double-delay case, in darker red, identical to the one shown in Fig. 1.

Figure 5

Correlation peaks corresponding to time-delay resonances for (a) two mutually coupled and (b) a doubly coupled directed ring of three chaotic Stuart-Landau oscillators with ${\tau }_1=50$.

Figure 6

(a) Nonsynchronizable asymmetric network of Bernoulli maps. (b) Finite correlations among units: $C(1,2)$ (green triangles), $C(2,3)$ (blue squares), and $C(1,3)$ (red circles) for $\epsilon =\kappa =0.85,\tau =50$ after a transitory of $40000$ time steps, averaged over 100 trials and over a time window of $10000$ time steps.

Figure 7

(a) Variances of the in-phase eigenmode $\langle |v_0\left(t\right){|}^2\rangle$ (black curve) and the first out-of-phase eigenmode $\langle |v_1\left(t\right){|}^2\rangle$ (pink curve) in a ring of three linear noisy oscillators coupled with two delays [Eq. (16)] as a function of the ratio of the delays. (b) Node-to-node correlation in a ring of three nodes with two delays as a function of the ratio of the delays. Parameters are ${\kappa }_1={\kappa }_2=2.25,{\tau }_1=50,{\omega }_0=0$, and $\langle {\xi }^2\left(t\right)\rangle =1$.