Testing for non-chaoticity under noisy dynamics using the largest Lyapunov exponent
Résumé
In this paper, we introduce a robust procedure to test for non-chaoticity when data are contaminated by an additive noise. Under the Kalman filter framework, our procedure first amounts to compute the largest Lyapunov exponent of the extracted signal. The exponent describes the log-divergence of a dynamical system (Rosenstein et al. in Phys D 65(1–2):117–134, 1993. https://doi.org/10.1016/0167-2789(93)90009-p). Then, using the so-called simulation smoother, we generate a high number of trajectories of the state-vector, conditional on the observed series, and compute the empirical distribution of the largest Lyapunov exponent. The distribution allows for computing confidence intervals. We can thus test if the largest Lyapunov exponent is not significantly greater than zero. Using Monte Carlo simulations, we show the validity of such an approach. We provide an illustration using toy models, which depict several dynamical systems. Finally, we implement tests of non-chaoticity on financial time series. We find no empirical evidence of chaotic patterns. Our approach is simple, efficient, and tests for chaos when data are measured with errors (i.e., noisy dynamics)