Covariant Lyapunov Vectors from Reconstructed Dynamics: The Geometry behind True and Spurious Lyapunov Exponents

The estimation of Lyapunov exponents from time series suffers from the appearance of spurious Lyapunov exponents due to the necessary embedding procedure. Separating true from spurious exponents poses a fundamental problem which is not yet solved satisfactorily. We show, in this Letter, analytically and numerically that covariant Lyapunov vectors associated with true exponents lie in the tangent space of the reconstructed attractor. Therefore, we use the angle between the covariant Lyapunov vectors and the tangent space of the reconstructed attractor to identify the true Lyapunov exponents. The usefulness of our method, also for noisy situations, is demonstrated by applications to data from model systems and a NMR laser experiment.

Figure 1

Estimated Lyapunov exponents ${\widehat{\lambda }}_i$ and the averaged angle $⟨{\theta }_i⟩$, $i=1,\dots ,m$, versus the neighborhood radius $r$ for a time delay embedding with dimension $m=5$ and noise level $\delta =0$. Dashed and dotted lines in (a) correspond to true (${\lambda }_1=0.42$, ${\lambda }_2=-1.62$) and spurious Lyapunov exponents ($2{\lambda }_1$, ${\lambda }_1+{\lambda }_2$, $2{\lambda }_2$), see Ref. 6. True Lyapunov exponents always correspond to a near-zero averaged angle $⟨\theta ⟩$. Vertical dot-dashed lines indicate avoided crossings.

Figure 2

Estimated Lyapunov exponents ${\widehat{\lambda }}_i$ and the averaged angles $⟨{\theta }_i⟩$ versus the noise level $\delta$ for the case $m=4$ and $r=0.001$. Dashed and dotted lines indicate the values of the true Lyapunov exponents and their linear combinations, respectively.

Figure 3

The averaged angles $⟨{\theta }_i⟩$ versus the estimated Lyapunov exponents ${\widehat{\lambda }}_i$. Symbols are for different parameter settings of the embedding: (○) $m=5$, $\delta =0$, ${10}^{-4}\le r\le {10}^{-1}$; (□) $m=5$, $\delta ={10}^{-10}$, ${10}^{-4}\le r\le {10}^{-1}$; (◊) $m=5$, $\delta ={10}^{-8}$, ${10}^{-4}\le r\le {10}^{-1}$; (▵) $m=5$, $r={10}^{-3}$, ${10}^{-14}\le \delta \le {10}^{-8}$; (▿) $m=4$, $\delta =0$, ${10}^{-4}\le r\le {10}^{-1}$; (◃) $m=4$, $r={10}^{-3}$, ${10}^{-14}\le \delta \le {10}^{-8}$. Colors (gray levels) indicate different exponents with the same parameter setting. The two true Lyapunov exponents lie at ${\lambda }_1\approx 0.42$ and ${\lambda }_2\approx -1.62$.

Figure 4

(a) The estimated Lyapunov exponents ${\widehat{\lambda }}_i$ and (b) the averaged angles $⟨{\theta }_i⟩$ versus neighborhood radius $r$ for the time delay embedding of the experimental data of a NMR laser [19]. The embedding dimension $m$ increases from 3 to 7 from left to right. The angles $⟨{\theta }_i⟩$ are trivially zero for $m=3$ (not shown).