New approach to study the van der Pol equation for large damping

We present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hyperbolic limit cycle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line...

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Main Author:	Klaus Schneider
Format:	Article
Language:	English
Published:	University of Szeged 2018-02-01
Series:	Electronic Journal of Qualitative Theory of Differential Equations
Subjects:	relaxation oscillations dulac–cherkas function rotated vector field
Online Access:	http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6286

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record_format	Article
spelling	doaj-d2582d24051741bfabd6fba63b0934012021-07-14T07:21:30ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38751417-38752018-02-012018811010.14232/ejqtde.2018.1.86286New approach to study the van der Pol equation for large dampingKlaus Schneider0Weierstrass Institute for Applied Analysis and Stochastics, Berlin, GermanyWe present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hyperbolic limit cycle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line forming continua of equilibria. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation in previous approaches), on a Dulac–Cherkas function and on the property of rotating vector fields.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6286relaxation oscillationsdulac–cherkas functionrotated vector field
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Klaus Schneider
spellingShingle	Klaus Schneider New approach to study the van der Pol equation for large damping Electronic Journal of Qualitative Theory of Differential Equations relaxation oscillations dulac–cherkas function rotated vector field
author_facet	Klaus Schneider
author_sort	Klaus Schneider
title	New approach to study the van der Pol equation for large damping
title_short	New approach to study the van der Pol equation for large damping
title_full	New approach to study the van der Pol equation for large damping
title_fullStr	New approach to study the van der Pol equation for large damping
title_full_unstemmed	New approach to study the van der Pol equation for large damping
title_sort	new approach to study the van der pol equation for large damping
publisher	University of Szeged
series	Electronic Journal of Qualitative Theory of Differential Equations
issn	1417-3875 1417-3875
publishDate	2018-02-01
description	We present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hyperbolic limit cycle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line forming continua of equilibria. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation in previous approaches), on a Dulac–Cherkas function and on the property of rotating vector fields.
topic	relaxation oscillations dulac–cherkas function rotated vector field
url	http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6286
work_keys_str_mv	AT klausschneider newapproachtostudythevanderpolequationforlargedamping
_version_	1721303522223325184

New approach to study the van der Pol equation for large damping

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