Structure and Dynamics of Lyapunov Vectors in Spatio-Temporal Chaos
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
Spatio-temporal chaos is found in a number of contexts ranging from
meteorological models to excitable media. The growth of disturbances
(errors) in this high-dimensional chaotic regime has attracted some
attention in the last years, due to specific properties that have no
analogy in low-dimensional systems. Particularly useful to understand
error growth dynamics is the mapping of the problem (via Hopf-Cole
transformation) to an equivalent interface growth process described
by the Kardar-Parisi-Zhang equation.
So far, the interest has focused on the largest Lyapunov exponent $łambda_1$
and the dynamics of the associated Lyapunov vector (LV), because any typical
disturbance approaches asymptotically the profile of that LV: $w_1(x,t)$.
However from practical (say ensemble forecasting in meteorology) and
fundamental point of views, it is interesting to understand the
spatio-temporal behavior of Lyapunov vectors associated to other
Lyapunov exponents ($łambda_n < łambda_n-1$, $n 2$).
In this communication we report on several results for the spatial and temporal
dynamics of the Lyapunov vectors.
The power spectra of the associated interfaces $h_n(x,t)=łog(|w_n|)$
are KPZ-like (i.e. $k^-2$) only up to a characteristic length scale,
approximately of order $L/n$. At large
scales the spectra behave as $1/k$ noise. The figure shows the results
for a lattice of coupled logistic maps
equation
u_i(t+1) = (1 -2 \epsilon) f(u_i(t)) + f(u_i+1(t)) +f(u_i-1(t))
equation
with $\epsilon=0.1$
but equivalent results are obtained for a multiplicative stochastic equation
$\partial_t w(x,t) = \xi(x,t) w(x,t) + \nabla^2 w(x,t)$,
and the Kuramoto-Sivashinsky equation.
A close inspection reveals that intervals of any interface $h_n$ are simply copies
of the same intervals of $h_1$. This approximate relation through a piecewise constant
(multiplateau-like) function explains the existence of a characteristic scale in
the power spectrum. Furthermore, the characteristic length scale decreases with $n$, such that
we propose a scaling relation for the power spectra of different vectors ($n$) and
system sizes ($L$):
equation
S_n(k) L^-1 k^2 = głeft(k L/(n-1/2)^\right)
equation
We note the difference between backward and characteristic
Lyapunov vectors for the scaling function $g$, see lower panels of the figure.
Whereas the backward LVs arise when computing the Lyapunov
exponents using the standard Gramm-Schmidt orthonormalization technique,
characteristic LVs are freely evolving disturbances that grow with
the corresponding Lyapunov exponent from the remote past to the remote future.