Padé approximants of canards and critical regimes of Darrieus wind turbine model

This paper presents a theoretical approach to studying so-called canards (or duck trajectories) and their possible approximations using Padé approximations for the Darier wind turbine model. One of the central issues arising when applying the theory of canards to solve specific practical problems is the challenge of calculating the so-called canard values of the parameters. To demonstrate the advantages of Padé approximations in the study of canards, both a mathematical example and the van der Pol equation are considered. Subsequently, a model of wind turbine dynamics under varying external loads is examined. It is shown that the model can experience an Andronov-Hopf bifurcation followed by a canard explosion, i.e., a sharp increase in the cycle amplitude when one of the parameters changes in a very narrow interval. It is the fact that this phenomenon is characterized by an exponentially small change of a parameter that was the motivation for increasing the accuracy of the applied asymptotic methods without additional cumbersome calculations. Numerical experiments demonstrate a good agreement of numerical data with the results of asymptotic analysis and a noticeable advantage of fractional-rational approximations Padé over the commonly used approximations based on Maclaurin series with expansions by powers of a small parameter.