Symbolic dynamics-based error analysis on chaos synchronization via noisy channels

In this study, symbolic dynamics is used to research the error of chaos synchronization via noisy channels. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation, a coder-decoder scheme is proposed. A model for the relationship among word length, region number of a partition, and synchronization error is provided. According to the model, the fundamental trade-off between word length and region number can be optimized to minimize the synchronization error. Numerical simulations provide support for our results.

Figure 1

Block diagram for drive-response controlled synchronization.

Figure 2

Flow chart of the decoder algorithm.

Figure 3

Synchronization error of a logistic map with $m=4$ as a function of word length for a SNR of (a) ${\gamma }_{\mathrm{SNR}}=4\mathrm{dB}$ and (b) ${\gamma }_{\mathrm{SNR}}=6\mathrm{dB}$. The solid curves represent simulation results, and the dashed curves represent the exact theoretical solutions of Eq. (7).

Figure 4

Synchronization error of a logistic map with $m=2$ as a function of word length for a SNR of (a) ${\gamma }_{\mathrm{SNR}}=4\mathrm{dB}$ and (b) ${\gamma }_{\mathrm{SNR}}=6\mathrm{dB}$. The solid curves represent simulation results, and the dashed curves represent the exact theoretical solutions of Eq. (7).

Figure 5

Synchronization error of a Chua system with $m=16$ as a function of word length for a SNR of (a) ${\gamma }_{\mathrm{SNR}}=3\mathrm{dB}$ and (b) ${\gamma }_{\mathrm{SNR}}=4\mathrm{dB}$. The solid curves represent the simulation results, and the dashed curves represent the exact theoretical solutions of Eq. (7).

Figure 6

Synchronization error of a Chua system with $m=16$ as a function of word length for a SNR of (a) ${\gamma }_{\mathrm{SNR}}=15\mathrm{dB}$ and (b) ${\gamma }_{\mathrm{SNR}}=17\mathrm{dB}$. The solid curves represent the simulation results, and the dashed curves represent the exact theoretical solutions of Eq. (7).

Figure 7

Optimal word length with ${10}^{-5}\le P_m\le 0.1$ as a function of symbol error probability for (a) logistic map and (b) Chua system. The dots represent the simulation results, and the solid curves represent the exact theoretical solutions of Eq. (8).