Organized structures of two bidirectionally coupled logistic maps.

Abstract

We report some organized structures of two linearly coupled logistic maps with different harvesting. The coupled system exhibits chaos via period-bubbling and quasiperiodic routes for identical and weak coupling strength, in contrast to conventional period-doubling route for a simple logistic map. Studies reveal the existence of infinite families of periodic Arnold tongues and self-similar shrimp-shaped structures with period-adding sequences for periodic windows embedded in quasiperiodic and chaotic regions, respectively. Different Fibonacci-like sequences are formed leading to the Golden Mean. The shrimp-shaped structures maintain period 3-times self-similarity scaling. The quasiperiodicity route is the necessary condition for the occurrence of periodic Arnold tongues in this coupled system resulting in the appearance of shrimps in the chaotic region near the tongues. It is also revealed that the existence of shrimp implies the period-bubbling cascade but the reverse is not true. The bifurcation-induced hysteresis is born in a certain parameter range resulting in the birth of coexisting multiple attractors of different kinds. Basin sets of the coexisting attractors have either self-similar or intertwining fractal basin boundaries.