Statistical properties of flip-flop processes associated to the chaotic behavior of systems with strange attractors

The chaotic behavior of systems with strange attractors can be discussed by examining the flip-flop process associated to the system dynamics. This was already shown by Lorenz (1963) in his first seminal paper. A somewhat surprising result was obtained by Aizawa (1982), who, studying the same Lorenz attractor at the parameter value r = 28, reached the conclusion that the associated flip-flop was a typical Markov process. Since the process is generated in a deterministic way, one may wonder if the Aizawa result is accidental, depending on the particular parameter value, or if a similar conclusion can be extended to other systems, with different attractors. Our conclusions are that the Aizawa result is mostly accidental, because for other parameter values and for other attractors there are sharp deviations from the Markovian process.