Logistic equation with memory.

Abstract

A statistical treatment of the macroscopic equation of motion leads to memory functions if Markovian-like approximations are not admissible. A nonlinear discrete model, which is nonlocal in time, may be obtained from the logistic map by replacing its linear term by a convolution with an exponentially decaying memory function xt+1=a[\ensuremath{\Sigma}t\ensuremath{'}=0t\ensuremath{\gamma}0\ensuremath{\epsilon}t-t\ensuremath{'}xt\ensuremath{'}-xt2]. This one-dimensional discrete map, R1\ensuremath{\rightarrow}R1: x0\ensuremath{\rightarrow}x1\ensuremath{\rightarrow}x2\ensuremath{\cdots}, possesses the same fixed point 1-1/a for all \ensuremath{\epsilon}, if we choose the normalization \ensuremath{\gamma}0=1-\ensuremath{\epsilon}. It is the goal of this paper to demonstrate the main features of the map induced by the memory term. Due to the complexity of the problem, most of the results have been derived by a numerical treatment. The range of \ensuremath{\epsilon} lies in the interval -0.1\ensuremath{\leqslant}\ensuremath{\epsilon}0.3. For fixed a,\ensuremath{\epsilon} generally two alternative attractors S and S\ensuremath{'} appear. For small |\ensuremath{\epsilon}|\ensuremath{\ll}0.02 only the usual Feigenbaum scenario S(a) exists, slightly modified by \ensuremath{\epsilon}. For \ensuremath{\epsilon}\ensuremath{\gtrsim}0.02 the scenario S(a,\ensuremath{\epsilon}) is shifted, and different windows with the same period coalesce. The \ensuremath{\epsilon}-induced scenarios S\ensuremath{'}(a,\ensuremath{\epsilon}) arise suddenly. In the (a,\ensuremath{\epsilon}) plane there exist islands of three-, six-, nine-, or tenfold period-doubling scenarios S\ensuremath{'}(a,\ensuremath{\epsilon}). The insertion of these S\ensuremath{'}(a,\ensuremath{\epsilon}) discontinuously depends on the starting value.