A statistical perspective on the dynamics of bivariate chaotic maps for communications modelling

Statistical and dynamical properties of bivariate (two-dimensional) maps, are less understood than their univariate counterparts. This paper will give a synthesis of extended results with exemplifications by the contrasting bivariate logistic and Arnold cat maps. The use of synchronization from bivariate maps in communication modelling is also described. I. Issues Concerning Bivariate Maps The dynamical and statistical properties of many univariate chaotic maps are well understood and have been the basis of several chaos-based communications models over recent years. Associated research has also produced important statistical results and clarified several statistical issues concerning independence and non-linear dependence of their chaotic sequences [1, 2]. A comprehensive review of the use of chaos in telecommunications is given by [3]. More sophisticated chaos-based communication systems involve bivariate chaotic maps for which less is known. There are more complex issues of stability, pre-image regional structure, synchronization, dynamic behaviour and joint statistical behaviour, which deserve further understanding. In the past, continuous chaotic flows, rather than discrete maps, have mainly been investigated. Current experimental communication implementations of bivariate maps involving synchronization appear to have outreached their theoretical foundations in many instances. It is the purpose of this paper to address these issues by theoretical synthesization and exploration of two contrasting bivariate chaotic maps; the focus will be on their dynamical and statistical behaviour and their relevance to embryonic communication systems. These maps are the bivariate logistic map and a bivariate Arnold cat map. For the latter there is surprising independence, not obtainable with one-dimensional maps, both in the individual sequences and between them. II. Mathematical Basis of Bivariate Maps A general bivariate map will be taken in the form {xt, yt} = {τx(xt−1, yt−1), τy(xt−1, yt−1)} , (1) for t = 1, 2, . . . using a function τ = (τx, τy) over a region A. Perhaps the most fundamental aspect of any map is its fixed points. As in one dimension, the existence of fixed points, satisfying τ(x, y) = (x, y), is relevant to the dynamical behaviour; secondly, the behaviour at these points is determined by eigen analysis of the Jacobian of the map at these points. The chaotic aspect, as in one dimension, is concerned with the idea of divergence after close initial conditions, and is now determined by a Lyapunov exponent matrix and its eigen values. The component-wise divergence at time t, ∆(xt, yt), of a process started from two nearby points (x0, y0) and (x0, y ′ 0), where ∆(x0, y0) = (x0 − x0, y0 − y′ 0) is ∆(xt, yt) ≈ DT (x0, y0)∆(x0, y0), t = 0, 1, . . . (2) where DT t = ∏t i=1 DT (xi, yi) and DT (x, y) is the Jacobian of the map at the point (x, y). As t → ∞, it may be shown that ∆(xt, yt) ≈ exp(λt)∆(x0, y0) where λ is obtained from the largest eigenvalue q(t, x, y) of DT (x, y) as λ = limt→∞ 1t [log |q(t, x, y)|]. WhenDT (x, y) is constant in (x, y), λ is simply the log of the absolute value of the largest eigenvalue of DT . This does not necessarily mean that if the sequence is divergent the individual components will also be divergent, since the directions of convergence or divergence lie along the eigenvectors and not the x− and y-axes. A consequence is that chaotic bivariate maps can be used to transmit messages using synchronisation, since a chaotic bivariate map can be non-divergent in the x and y directions whilst the (x, y) sequence is divergent. Conditional Lyapunov exponents of the map, to be considered in Section VII, are central to determining synchronization. A crucial aspect of a bivariate map is its pre-image structure. Most simply a map can have unique preimage structure g = (gx, gy), such that {xt−1, yt−1} = {gx(xt, yt), gy(xt, yt)} t = 0, 1, . . . although often it may be more complicated with multiple pre-images. The multiple pre-image structure of bivariate maps is governed by pre-image curves, PIC, determined by the determinant of the Jacobian of the map being zero. Applying the map to all positions on these curves then gives the critical curves, CC, of the map. The PIC and CC curves enable the pre-image regional structure to be determined, identifying the number and multiplicity of pre-images at any position and in particular on regional boundaries. That the number of pre-images of a point usually depends on its position is also true for one-dimensional maps with ‘curtailed’ branches, but not so for the standard one-dimensional maps. Statistical aspects of bivariate maps begin with a bivariate invariant distribution, satisfying a PerronFrobenius type equation, but usually mathematically intractable for interesting non-invertible bivariate maps such as the bivariate logistic map of Section III. It follows that their dependency structure which includes joint dependencies is similarly intractable. Recourse to numerical simulation is thus inevitable, but can produce illuminating results. There can be analytical tractability for invertible maps as shown for the bivariate Arnold cat map in Section IV. III. Bivariate Logistic Maps Dynamical