Stochastic Computing (SC) is a digital computation approach that operates on random bit streams to perform complex tasks with much smaller hardware footprints compared to conventional approaches that employ binary radix. For stochastic logic to work, the input random bit streams have to be independent, which is a challenge when implementing systems with feedback: outputs that are generated based on input bit streams would be correlated to those streams and cannot be readily combined as inputs to stochastic logic for another iteration of the function. We propose re-randomization techniques for stochastic computing and use the Logistic Map x → r x(1-x) as a case study for dynamical systems in general. We show that complex behaviors such as period-doubling and chaos do indeed occur in digital logic with only a few gates operating on a few 0's and 1's. We employ a number of techniques such as random number generator sharing and using table-lookup pre-computations to significantly reduce the total energy of the computation. Compared to the conventional binary approach, we achieve between 8% and 25% energy consumption.