Unstable dimension variability in coupled chaotic systems

Systems of coupled chaotic maps and flows arise in many situations of physical and biological interest. The aim of this paper is to analyze and to present numerical evidence for a common type of nonhyperbolic behavior in these systems: unstable dimension variability. We show that unstable periodic orbits embedded in the dynamical invariant set of such a system can typically have different numbers of unstable directions. The consequence of this may be severe: the system cannot be modeled deterministically in the sense that no trajectory of the model can be realized by the natural chaotic system that the model is supposed to describe and quantify. We argue that unstable dimension variability can arise for small values of the coupling parameter. Severe modeling difficulties, nonetheless, occur only for reasonable coupling when the unstable dimension variability is appreciable. We speculate about the possible physical consequences in this case.