Characterizing Dynamics with Covariant Lyapunov Vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.

Figure 1

PDF of the angle between stable and unstable manifold. (a) Hénon map $x_{n+1}=1-1.4x_n^2+0.3x_{n-1}$ (green light line), and Lozi map $x_{n+1}=1-1.4|x_n|+0.3x_{n-1}$ (black line, rescaled by a factor 10). (b) TM ($L=12$, black dotted line), SM ($L=10$, green dashed line), RM ($L=32$, red dot-dashed line), and FPU ($L=32$, blue full line).

Figure 2

Inverse participation ratio $Y_2$ (see text) of both CLV and GSV for different dynamics. (a)–(c) Log-log plot of $Y_2$ as a function of chain length $L$ at fixed spectrum position $h$. CLV results are shown in full symbols, while GSV by empty symbols. In the log-log scale insets: inverse of the localization length $\ell$ has been subtracted from $Y_2$ to better show the CVL behavior $Y_2(L)\sim 1/\ell +L^{-\gamma }$ (see text). The dashed black lines mark a decay as $\gamma =\frac{1}{2}$. (a) TM for $h=0.2$ (black circles) and $h=0.4$ (red squares). (b) SM for $h=0.2$ (black circles) and $h=0.4$ (red squares). (c) FPU ($h=0.2$, black circles) and RM ($h=0.2$, red squares). (d) Lin-log plot of the localization length $\ell$ of CLV vs $h$ for TM (black circles), SM (red squares), and RM (blue triangles).

Figure 3

Trajectory averaged power spectrum (as a function of the wave number $k=j2\pi /L$, $j=1,\dots ,L/2$) of the space components of CLV (a) and GSV (b) corresponding to the smallest positive LE. Solid (black), dashed (red), and dot-dashed lines (blue) refer to FPU, RM, and SM, respectively, ($L=512$). The dotted green line, corresponding to a $1/k$ behavior is plotted for comparison in panel (a).