Mathematics and Applied Mathematics
December 29, 2019
logistic map, generalized logistic map, complex dynamical systems, chaos
The well known standard logistic map and its generalizations have found numerous applications in modelling behaviors of non-linear system dynamics. Some characteristics and behaviors of a sparsely discussed special case, here called LM(1/2,1/2,1), are developed by computer simulations and then supported by rigorous analysis. The LM(1/2,1/2,1) inherits many important behaviors of the standard logistic map. However, the simulations show that some significant and potentially useful differences exist. Not surprisingly, the LM(1/2,1/2,1) exhibits the 2n-period double route to chaos. Other n-cycles and possible m2^n-period doubling regions are identified with Sharkovskii’s theorem holding. However, fixed point x1* = 0 is not stable and an alternating exponentially decreasing autocorrelation function that does not vanish for the first few lags is found. Notably, the cumulative distribution function contains flat regions. Formal statements of important behaviors along with proofs are provided in an Appendix. Based on these results, the existence of two chaotic regimes after a double cascading, almost periodic chaos (with stability islands) and developed chaos (with stability islands), is hypothesized.