Lyapunov Exponents

The analysis of potentially chaotic behavior in biological and biomedical phenomena has attracted great interest in recent years (1–6). Although no universally accepted mathematical definition of the term chaos exists, Strogatz (7) provides a working definition as ‘‘aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.’’ Aperiodic long-term behavior means that trajectories do not converge to fixed points, periodic orbits, or quasi-periodic orbits as t N, but instead exhibit irregular and unpredictable behavior. ‘‘Deterministic’’ means that this unpredictable, aperiodic behavior derives from the inherent nonlinearities in the system itself (which can be expressed by deterministic equations of motion), and are not because of noise or other stochastic elements in the system. ‘‘Sensitive dependence on initial conditions’’ (SDIC) means that trajectories that start arbitrarily near to each other will separate exponentially fast, an example of which is given in Fig. 1 that shows the effects of SDIC for the classic Lorenz attractor: