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By Edwin Newbolt Bradley

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We see that y and (and also z and Z) are less similar to each other than x and 2. Synchronization in Two Coupled Chaotic Systems 44 12 I 10 - 8 - 6 - 4 - -2 I I 0 Fig. 22 -3 1 0 10 20 30 40 50 time (s) 60 70 I 80 90 100 B as a function of time, with c = -20a, 6 = 1 and p(x) = sgn(x)(elzl - 1). I 10 20 30 40 50 60 70 80 90 100 time (s) Fig. 23 x and f as functions of time. c = -2a, 6 = 10, p(x) = sgn(x). In Fig. 26 we show Li as a function of time. We see that 6 does not converge towards a, but fluctuates around a value below a = 10.

Both schemes can also be considered as synchronization schemes for two systems which are not identical. There have been many approaches to implementing communication systems based on chaotic synchronization [29, 37, 14, 15, 38, 39, 40, 411. In these systems, the input signal is scrambled or converted to a chaotic signal in the transmitter and this chaotic signal is transmitted to the receiver. Nearly all of them utilize an autonomous chaotic system and the information signal does not play a significant role in generating the chaos.

Let j be the index such that maxi [hi[= Ihj 1. Then Note that lp(hk)l 5 Ip(hj)l since p is odd-symmetric and increasing. 0 The following theorem gives theoretical justifications for the adaptive scheme in Eq. 14 Consider the adaptive controllers for &ij given by Eq. 39) where p is a piecewise-continuous, odd-symmetric and increasing function in class V such that x p ( x ) is continuous. W e also assume that Adaptive synchronization 39 t i ' s and hi's are linear functions of the form L ~ ( x )= kixi, hi(^) = hixi and adapt the coeficients using the following controllers: + where nt K!

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