*2.3. Lyapunov Exponents for ABNs*

One of the most reliable tools to demonstrate chaotic behavior is computing the Lyapunov exponent's spectrum [1–29]. A positive Lyapunov exponent is a signature of chaos [40]. It is defined as the exponential divergence of trajectories with nearly identical initial conditions. For the ABNs case, since the states are discrete, indicating a phase space composed just by 2*<sup>N</sup>* states, the Lyapunov exponent's needs to be computed from distance measures tailored for Boolean systems [41].

Zhang et al. proposed a method to estimate the largest Lyapunov exponent using the Boolean distance definition [30]. The approach works as follows. (i) Acquire experimentally a long time series from an output voltage of the ABN. (ii) Convert that voltage to a Boolean variable *x*(*t*). (iii) Given any two segments of starting at times *ta* and *tb*, define a Boolean distance with *d*(*s*) = <sup>1</sup> *T <sup>s</sup>*+*<sup>T</sup> <sup>s</sup> x*(*t* + *ta*) ⊕ *x*(*t* + *tb*)*dt* , where T is a fixed parameter, ⊕ is the XOR logic operation, and the Boolean distance *d*(*s*) evolves as a function of the time *s*. (iv) Search in *x*(*t*) for all the pairs *ta* and *tb* corresponding to the earliest times in each interval *T* over which *d*(0) < 0.01. *v*. Finally, *v*) compute ln*d*(*s*), where  means an average over all matching (*ta*, *tb*) pairs.

As a conclusion, the divergence ln*d*(*s*) increases exponentially, as expected for an adequate definition of distance between trajectories in a chaotic system [40].
