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

Journal: Journal of Physics A: Mathematical and Theoretical
Year: 2013   Volume: 46
Initial page: 254014  
Status: Published
Link to PDF: http://iopscience.iop.org/1751-8121/46/25/254014/pdf/1751-8121_46_25_254014.pdf
DOI: 10.1088/1751-8113/46/25/254014
Pazó, D., López, J.M.,

In a dynamical system the first Lyapunov vector (LV) is associated with the largest Lyapunov exponent and indicates—at any point on the attractor— the direction of maximal growth in tangent space. The LV corresponding to the second largest Lyapunov exponent generally points in a different direction, but tangencies between both vectors can in principle occur. Here we find that the probability density function (PDF) of the angle ψ spanned by the first and second LVs should be expected to be approximately symmetric around pi/4 and to peak at 0 and pi/2. Moreover, for small angles we uncover a scaling law for the PDF Q of ψ*=ln ψ with the system size L: Q(ψ*)=L^(−1/2)f(ψ*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.