Using Chaos to Understand Biological Dynamics

The application of nonlinear dynamics has begun to move beyond the problem of demonstrating the existence of nonlinearities (and chaos) in biological data. We are starting to see exciting cases where considerations of nonlinear dynamics can explain observed patterns, and give insight into the forces structuring biological phenomena. We present some examples of these, drawn from the analysis of measles epidemics and cardiac pathologies. We show that a phenomenon of many chaotic systems called transient periodicity can explain apparently qualitative shifts in the observed dynamics. Such shifts require no change of or perturbation to the system, but can be intrinsic features of purely deterministic dynamics. We show how the techniques of nonlinear forecasting can be used as analytical tools, both for quantifying the complexity of the time series in a biologically meaningful way and for determining how well a particular model accounts for the dynamics. We also post a warning for practitioners of “conventional” nonlinear time series analysis (such as dimension calculations): many biological time series are nonuniform, and this can seriously mislead the various algorithms in common use.