Clarifying Chaos: Examples and Counterexamples

Over the past fifteen years there have been various attempts to define chaos. In an effort to find a universally acceptable definition we began constructing new examples of chaotic systems in the hope that the salient features of chaos could be captured. Our efforts to date have failed and the examples we have constructed seem to suggest that no such definition exists. However, these examples have proved to be valuable in spite of our inability to hone a universal definition of chaos from them. Consequently, we present this list of examples and their significance. Some interesting conclusions that we can draw from them are: It is possible to construct simple closed form solutions of chaotic one-dimensional maps; sensitive dependence on initial conditions, the most widely used definition of chaos, has many counterexamples; there are invertible chaotic dynamical systems defined by simple differential equations that do not have horseshoes; three important properties that are thought to characterize chaos, continuous power spectral density, exponentially sensitive dependence on initial conditions, and exponential loss of information (Chaitin’s concept of algorithmic complexity), are independent. Chaos seems to be tied to our notion of rates of divergence of orbits or degradation of information such as is found in systems with positive Lyapunov exponents. The reliance on rates seems to open the door to a pandora’s box of rates, both higher and lower than exponential. The intuitive notion of pseudo-randomness, a practical feature of chaos, is present in examples that do not have positive Lyapunov exponents. And in general, nonlinear polynomial rates of degradation of information are also quite “unpredictable”. We conclude that it appears that for any given definition of chaos, there may always be some “clearly” chaotic systems which do not fall under that definition, thus making chaos a cousin to Gödel’s undecidability.