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

Alena Kirsanova; Vladimir Sobolev; Alena Kirsanova; Vladimir Sobolev

doi:10.3934/mine.2025009

Mathematics in Engineering

2025, Volume 7, Issue 3: 194-207. doi: 10.3934/mine.2025009

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

Alena Kirsanova ,
Vladimir Sobolev ^,

Samara National Research University, Moskovskoye Shosse, 34, Samara 443086, Russia

Received: 22 December 2024 Revised: 29 April 2025 Accepted: 03 May 2025 Published: 08 May 2025

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.
- singular perturbations,
- invariant manifolds,
- canards,
- asymptotic expansions,
- Padéapproximants,
- wind energy,
- Darrieus wind turbine
Citation: Alena Kirsanova, Vladimir Sobolev. Padé approximants of canards and critical regimes of Darrieus wind turbine model[J]. Mathematics in Engineering, 2025, 7(3): 194-207. doi: 10.3934/mine.2025009

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Abstract

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.

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