Robust Chaos in Polynomial unimodal Maps

Robust chaos occurs when a dynamical system has a neighborhood in parameter space such that there is one unique chaotic attractor, and no periodic attractors are present [Barreto et al., 1997; Banerjee et al., 19981. This behavior, expected to be relevant for any practical applications of chaos, was shown to exist in a general family of piecewise-smooth twodimensional maps, but conjectured to be impossible for smooth unimodal maps [Banerjee et al., 19981. In fact, the two best known examples of this last type of maps exhibit clearly the rationale for this conjecture: the bifurcation diagram of the (smooth) logistic map is densely populated by periodic windows [Collet & Eckmann, 19801; while on the other hand, a continuously chaotic bifurcation diagram is obtained for the tent map, which is nonsmooth. Recently, however, Andrecut and Ali [2001a] have found an example, and later a full family [2001b], of smooth unimodal maps with continuously chaotic bifurcation diagrams.' These maps have negative Schwarzian derivative, and therefore a unique attractor with the critical point in its basin of attraction. The maps are constructed in such a way that the critical point gets mapped into an unstable fixed point, and so it is not in the basin of attraction of a periodic attractor. This means, therefore, that no periodic attractors are present. The first example has the form [Andrecut & Ali, 2001al: