Identification of parameters in amplitude equations describing coupled wakes
We study the flow behind an array of equally spaced parallel cylinders. A system of Stuart-Landau equations with complex parameters is used to model the oscillating wakes. Our purpose is to identify the 6 scalar parameters which most accurately reproduce the experimental data of Chauve and Le Gal [{Physica D {\bf 58}}, pp 407--413, (1992)]. To do so, we perform a computational search for the minimum of a distance $\calj$. We define $\calj$ as the sum-square difference of the data and amplitudes reconstructed using coupled equations. The search algorithm is made more efficient through the use of a partially analytical expression for the gradient $\nabla \cal J$. Indeed $\nabla \cal J$ can be obtained by the integration of a dynamical system propagating backwards in time (a backpropagation equation for the Lagrange multipliers). Using the parameters computed via the backpropagation method, the coupled Stuart-Landau equations accurately predicted the experimental data from Chauve and Le Gal over a correlation time of the system. Our method turns out to be quite robust as evidenced by using noisy synthetic data obtained from integrations of the coupled Stuart-Landau equations. However, a difficulty remains with experimental data: in that case the several sets of identified parameters are shown to yield equivalent predictions. This is due to a strong discretization or ``round-off" error arising from the digitalization of the video images in the experiment. This ambiguity in parameter identification has been reproduced with synthetic data subjected to the same kind of discretization.