Coding, Channel Capacity, and Noise Resistance in Communicating with Chaos

Abstract

Recent work has considered the possibility of utilizing symbolic representations of controlled chaotic orbits for digital communication. We argue that dynamically a coding scheme usually leads to trajectories that live on a nonattracting but noise-resisting chaotic saddle embedded in the chaotic attractor. We present analyses and numerical computation which indicate that the channel capacity of the chaotic saddle has a devil-staircase-like behavior as a function of the noise-resisting strength. The implication is that nonlinear digital communication using chaos can yield a substantial channel capacity even in a noisy environment. [S0031-9007(97)04462-1]

The control of chaos: theory and applications

“…is increased from zero [56,57], as shown in Fig. 28b for 04s(s +0.42, where the dashed horizontal line at h 2 "0.466 de"nes the minimum gap size s above which the generating partition for the chaotic saddle is simply y"0.…”

“…In so far as the noise amplitude is smaller than the size of the gap across the partition point x A , the symbolic dynamics is immune to noise. This is of tremendous value to practical implementation of communication with chaos [56,57].…”

“…The key implication is that utilizing chaotic saddles with gap size smaller than s A but larger than s seems to be practically bene"cial in communication applications: the speci"cation of the symbolic dynamics is straightforward, yet the channel capacity is close to that obtained when one utilizes the original chaotic attractor. It was conjectured [125] that the function of h 2 versus s is a devil's staircase, a statement that can be made rigorous for certain one-dimensional maps [56,57].…”

“…One can choose the chaotic saddles so that the symbolic dynamics are robust against small random noise. Thus, utilizing hyperbolic chaotic saddles for communication also provides a solution to overcome the in#uence of noise [56,57].…”

“…In general, when choosing an optimal chaotic repeller for digital encoding, there is a trade-o! between the degree of di$culty of generating a trajectory near the chaotic repeller and the noise immunity [56,57].…”