Detecting chaos requires careful analysis of nearly periodic data

We show that models fitted to data in many cases fit unstable periodic solutions in attracting periodic solutions of the ‘true model’ that generated the data. An attracting solution containing the neighborhood of the fitted unstable solution in its domain of attraction may possess entirely different dynamical properties. Thus, an attracting chaotic solution with positive Lyapunov exponent may describe periodic solutions with negative Lyapunov exponents and vice versa. These problems can in principle be remedied, if the fitted models would be allowed to contain an arbitrary complexity and if an infinite amount of data would be available. We claim that we stay far from such limits in ecology, for instance. Therefore, we think our approach is essential to bear in mind when making data-based predictions concerning dynamical behavior. Our general conclusion is that less data is required in nearly periodic cases than in chaotic cases for rejecting models not allowing complex behavior.