A parameter-space orientation ridge in finite-time sensitivity maps of the Lorenz system and its local eigendirection explanation

We report a narrow ridge in maps of a directional finite-time separation measure for the Lorenz system. The maps are taken over the (sigma, rho) parameter plane, with the initial condition held in a fixed direction (x0, y0, z0) = (a, ma, 0). Near the origin an elementary property of the linearised dynamics predicts where the ridge lies. It follows the locus on which the initial direction m = y0/x0 coincides with the stable eigendirection of the (x, y) block of the Jacobian at the origin. That locus is rho = sigma*m*(m-1) + m. The eigendirection calculation uses standard linear algebra; the connection between it and a directly observed parameter-plane ridge, together with a predictive test of that connection, is the subject of this paper. For four values of m held back from the earlier measurements, the relation gives the measured ridge slope to between 0.007% and 0.023%, with no coefficient fitted. We then characterise the regime - horizon, timestep, perturbation size and direction, integrator, grid resolution and beta - over which the ridge can be extracted reliably. Separately, the accuracy of the local relation degrades as the initial condition moves away from the equilibrium. The result is local to the equilibrium; we make no claim about global structure.