 The other worry is that the definitions we have been considering may only hold for our mathematical models, but may not be applicable to actual target systems. The formal definitions seek to fully characterize chaotic behavior in mathematical models, but we are also interested in capturing chaotic behavior in physical and biological systems as well. Phenomenologically, the kinds of chaotic behaviors we see in actual-world systems exhibit features such as SDIC, aperiodicity, unpredictability, instability under small perturbations and apparent randomness. However, given that target systems run for only a finite amount of time and that the uncertainties are always larger than infinitesimal, such systems violate the assumptions necessary for deriving SD. In other words, even if we have good statistical measures that yield on average exponential growth in uncertainties for a physical data set, what guarantee do we have that this corresponds with the exponential growth of SD? After all, any growth in uncertainties (alternatively, any growth in distance between neighboring trajectories) can be fitted with an exponential. If there is no physical significance to global Lyapunov exponents (because they only apply to infinitesimal uncertainties), then one is free to choose any parameter to fit an exponential for the growth in uncertainties. 