Simulating memory effects with discrete dynamical systems

We propose using discrete dynamical systems to model non-Markovian processes. This implies a whole hierarchy of dynamical systems with dimensionality increasing proportional to the memory. Specific long-range memory effects are investigated for a quadratic, a cubic and a quartic map. Markovian processes transform the quadratic map into a normal form of the logistic equation; the H6non map corresponds to non-Markovian "first-generation" memory effects. Higher memories imply new high-dimensional systems. Non-Markovian processes imply new "memory-routes" to chaos. For the three polynomial maps discussed here it is possible to define "critical memory lengths" above which the systems essentially lose memory.