Lyapunov Exponents

Abstract The analysis of potentially chaotic behavior in biological and biomedical phenomena has attracted increased attention in recent years. Chaotic systems exhibit a trademark “sensitive dependence on initial conditions” (SDIC), meaning that very small perturbations will cause the resulting trajectories to separate exponentially fast. Lyapunov exponents directly measure SDIC by quantifying the exponential rates at which neighboring orbits on an attractor diverge (or converge) as the system evolves in time. Dissipative deterministic systems that exhibit at least one positive Lyapunov exponent are by definition “chaotic.” Hence, Lyapunov exponents are considered one of the most useful diagnostic tools available for analyzing potentially chaotic dynamical systems. However, Lyapunov exponents are also notoriously difficult to estimate reliably from experimental data and one should always be very skeptical about accepting any claims of “chaos” based solely on findings of positive Lyapunov exponents alone. This article reviews the basic definitions of Lyapunov exponents and outlines several of the more widely disseminated algorithms available to estimate them from experimental time series data. Critical issues related to finding spurious exponents, analyzing noisy signals, and validating against surrogate data are reviewed. A brief discussion of how Lyapunov exponents have been applied to study a wide variety of biomedical and biological phenomena is also presented. Finally, some thoughts on appropriate interpretations of such findings and the future directions for using Lyapunov exponents to study potentially chaotic systems are presented.