Boolean networks have been studied in many different contexts . They simplify the treatment of highly nonlinear systems, where the response of elements has a switch-like behavior. The state space of deterministic Boolean networks with synchronous (clocked) update is, however, finite, and thus, they cannot exhibit deterministic chaos, as defined by an exponential sensitivity to initial conditions. One way of recovering non-periodic behavior is to update the state of the Boolean elements in a random order , but then, the randomness does not come from the dynamics of the system itself, but rather from the updating process. Another way to have non-periodic behavior is to take away the clock and let the Boolean elements update their outputs autonomously, as soon as they can process the changes occurring in their inputs . The information propagates between logic elements with time delays, introduced by the signal transit time or processing time by the logic elements.
The evolution of autonomous Boolean systems without a clock can be described in terms of Boolean Delay Equations. We use such equations as a starting point to model and to inspire our electronic circuits, but we have found that they cannot produce chaotic solutions with constant time delays . Chaos appear, however, when the equations are modified to include memory effects in the logic processing elements [5,6].
In the next sections we present an introductory review about Boolean delay equations, describe the electronic circuits we designed to produce chaos at high speed and discuss the obtained results.
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