The dynamical properties of many univariate chaotic maps are well understood and have been the basis of several chaos-based communications models over recent years. Associated statistical results are not so readily available and have been the focus of the Birmingham Group for the past few years. Several important statistical results and clarification of various statistical issues concerning independence and non-linear dependence of their chaotic sequences have been obtained, (Kohda et al., 2000; Lawrance and Balakrishna, 2001). A comprehensive review of the use of chaos in telecommunications is given by Kennedy et al., (2000).

More sophisticated chaos-based communication systems involve bivariate or higher order chaotic maps about which less is known. There are more complex issues of stability, pre-image regional structure, synchronization, dynamic behavior and joint statistical behavior, which deserve further understanding. In the past, continuous chaotic flows, rather than discrete maps, have mainly been investigated. Current experimental communication implementations of bivariate and higher order maps involving synchronization appear to have outreached their theoretical foundations in many instances. It is the purpose of this talk to address these issues focusing on the statistical behavior.

The relevance of bivariate and higher order maps to embryonic communication systems will be discussed, one aspect being that they can produce marginal sequences with low dependence and hence high security. A second issue is the necessity of simultaneously knowing the chaotic sequence at both the transmitting and receiving stations. When noise is present in the transmission channel recovery of the message may be inaccurate. The talk will discuss how a series of transformations can be applied so that the noise reduces the probability of the so called bit error rate and has a reduced impact on the underlying invariant distribution of the received signal

References:

T. Kohda, A. Tsundea & A. J. Lawrance, "Correlational Properties of Chebyshev Chaotic Sequences", Journal of Time Series Analysis, vol. 21, 2, 181-192, 2000.

A. J. Lawrance & N. Balakrishna, "Statistical Aspects of Chaotic Maps with Negative Dependency in a Communications Setting", Journal of the Royal Statistical Society B, to appear 2002.

M. P. Kennedy, R. Rovatti & G. Setti, Eds., "Chaotic Elecronics in Telecommunications", CRC Press, Boca Raton, 2000.

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