Pulses of chaos synchronization in coupled map chains with delayed transmission

Pulses of synchronization in chaotic coupled map lattices are discussed in the context of transmission of information. Synchronization and desynchronization propagate along the chain with different velocities which are calculated analytically from the spectrum of convective Lyapunov exponents. Since the front of synchronization travels slower than the front of desynchronization, the maximal possible chain length for which information can be transmitted by modulating the first unit of the chain is bounded.

Figure 1

Scheme of the examined chain: each unit is driven by its predecessor.

Figure 2

(Color online) Absolute value of the difference ${\delta }_t^n$ between neighboring units: for $t<0$, $100<t$ the first unit is untuned $\left(r_0=r^{\prime }=3.95\right)$. For $0<t<100$ the chain is tuned $\left(r_0=r=4\right)$, thus, creating a synchronized pulse. $\epsilon =0.8,{\tau }_0=10$.

Figure 3

(Color online) Propagation velocity of synchronization in units of the maximum velocity $1/{\tau }_0$ for different coupling parameters $\epsilon$ at ${\tau }_0=10$. Here the critical coupling is ${\epsilon }_c\approx 0.35$.

Figure 4

(Color online) The additional delay $\gamma =1/v_s-{\tau }_0$ as a function of the coupling delay ${\tau }_0$ with $\epsilon =0.6$: for large ${\tau }_0$ there is a saturation effect.

Figure 5

(Color online) The convective Lyapunov exponent $\Lambda \left(v\right)$, Eq. (8), for ${\lambda }_0=0.3$ and $\epsilon =0.6$.

Figure 6

(Color online) $v_s$ and $v_d$ obtained by simulations are compared to the predictions made by $\Lambda \left(v_i\right)=0$ for ${\tau }_0=1$.