On the angle between the first and the second Lyapunov vectors in spatio-temporal chaos

In a dynamical system the first Lyapunov vector (LV) is associated with the largest Lyapunov exponent and indicates ---at some point on the attractor--- the direction of maximal growth in tangent space. The LV corresponding to the second largest Lyapunov exponent generally points at a different direction, but tangencies between both vectors can in principle occur. Here we find that the probability density function (PDF) of the angle \psi spanned by the first and the second LVs should be expected approximately symmetric around \pi/4 and peaked at 0 and \pi/2. Moreover, for small angles we uncover a scaling law for the PDF Q of \psi_l=\ln\psi with the system size L: Q(\psi_l)=L^{-1/2} f(\psi_l L^{-1/2}). We give a theoretical argument that justifies this scaling form and also explains why it should be universal (irrespective of the system details) for spatio-temporal chaos in one spatial dimension.