Behavior of a Logistic Map Driven by White Noise

In the real world, every nonlinear system is inevitably affected by noise. As an example, a logistic map driven by white noise is studied. Unlike previous studies which focused on the behavior under local parameters to find analytical results, we investigate the whole driven logistic map. For a white noise driven logistic map, its non-divergent interval decreases with increasing white noise. The white noise does not change the equilibrium point and two-cycle intervals in statistics, if the driven logistic map is kept non-divergent. In particular, chaos can be excited by white noise only after the four-cycle bifurcation begins. The latest result is a necessary condition which has not been given in the literature Int. J. Bifur. Chaos 18(2008)509, and it can be deduced from Sharkovsky's theorem. Numerical simulations prove these analytical results.