Long tailed maps as a representation of mixed mode oscillatory systems

Mixed mode oscillatory (MMO) systems are known to exhibit generic features such as the reversal of period doubling sequences and crossover to period adding sequences as bifurcation parameters are varied. In addition, they exhibit a nearly one dimensional unimodal Poincare map with a long tail. The numerical results of a map with a unique critical point (map-L) show that these dynamical features are reproduced. We show that a few generic conditions extracted from the map-L are adequate to explain the reversal of period doubling sequences and crossover to period adding sequences. We derive scaling relations that determine the parameter widths of the dominant windows of periodic orbits sandwiched between two successive states of RL k sequence and verify the same with the map-L. As the conditions used to derive the scaling relations do not depend on the form of map, we suggest that the analysis is applicable to a family of two parameter one dimensional maps that satisfy these conditons.