Information Flow in Chaos Synchronization: Fundamental Tradeoffs in Precision, Delay, and Anticipation

We use symbolic dynamics to examine the flow of information in unidirectionally coupled chaotic oscillators exhibiting synchronization. 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 we explore so-called achronal synchronization, in which the response lags or leads the drive by a fixed amount of time. We find fundamental tradeoffs between the precision to which the drive state is detected, the quality of synchronization attained, and the delay or anticipation exhibited by the response system. To illustrate these tradeoffs, we provide a physical example using electronic circuits.

Figure 1

Drive-response configuration for investigating synchronization with unidirectional coupling.

Figure 2

Experimental configuration for demonstrating synchronization via a symbol channel. Oscillators $D$ and $R$ are identical chaotic oscillators, $v_S$ is a low-level sine wave to phase lock the circuits, $N$ represents nonlinear and active components, and $D_{\mathrm{L}\mathrm{I}\mathrm{M}}$ is an ideal diode limiter implemented using an active circuit.

Figure 3

Return map of successive peak voltages for $D$ (black) and $R$ (gray), including generating partitions (vertical lines). The inset shows the attractor for $D$.

Figure 4

Drive ($v_D$) and response ($v_R$) waveforms showing synchronization with an 8-cycle lag ($\Delta t=6.7$ ms). $D$ is detected with $n=1$ bit of precision, and $R$ is controlled at the $m=8$th future bit.

Figure 5

Drive ($v_D$) and response ($v_R$) waveforms showing synchronization with 2-cycle anticipation ($\Delta t=1.7$ ms). $D$ is detected with $n=9$, and $R$ is controlled at $m=6$.