Analytical estimations of limit cycle amplitude for delay-differential equations
The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach th...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2016-09-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5297 |
| _version_ | 1797830607087075328 |
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| author | Tamás Molnár Tamás Insperger Gábor Stépán |
| author_facet | Tamás Molnár Tamás Insperger Gábor Stépán |
| author_sort | Tamás Molnár |
| collection | DOAJ |
| description | The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing. |
| first_indexed | 2024-04-09T13:38:47Z |
| format | Article |
| id | doaj.art-d3a09db3e324472784caf1bb4675563d |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:47Z |
| publishDate | 2016-09-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-d3a09db3e324472784caf1bb4675563d2023-05-09T07:53:06ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752016-09-0120167711010.14232/ejqtde.2016.1.775297Analytical estimations of limit cycle amplitude for delay-differential equationsTamás Molnár0Tamás Insperger1Gábor Stépán2Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, HungaryBudapest University of Technology and Economics, HungaryDepartment of Applied Mechanics, Budapest University of Technology and Economics, Budapest, HungaryThe amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5297 delay-differential equationhopf bifurcationlimit cyclecenter manifold reductionnormal form theory |
| spellingShingle | Tamás Molnár Tamás Insperger Gábor Stépán Analytical estimations of limit cycle amplitude for delay-differential equations Electronic Journal of Qualitative Theory of Differential Equations delay-differential equation hopf bifurcation limit cycle center manifold reduction normal form theory |
| title | Analytical estimations of limit cycle amplitude for delay-differential equations |
| title_full | Analytical estimations of limit cycle amplitude for delay-differential equations |
| title_fullStr | Analytical estimations of limit cycle amplitude for delay-differential equations |
| title_full_unstemmed | Analytical estimations of limit cycle amplitude for delay-differential equations |
| title_short | Analytical estimations of limit cycle amplitude for delay-differential equations |
| title_sort | analytical estimations of limit cycle amplitude for delay differential equations |
| topic | delay-differential equation hopf bifurcation limit cycle center manifold reduction normal form theory |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5297 |
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