Numerical bifurcation analysis of a class of nonlinear renewal equations
We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits tra...
| 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 |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5273 |
| _version_ | 1797830562569781248 |
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| author | Dimitri Breda Odo Diekmann Davide Liessi Francesca Scarabel |
| author_facet | Dimitri Breda Odo Diekmann Davide Liessi Francesca Scarabel |
| author_sort | Dimitri Breda |
| collection | DOAJ |
| description | We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits transcritical, Hopf and period doubling bifurcations. The reliability is demonstrated by comparing the results to those obtained by a reduction to a Hamiltonian Kaplan-Yorke system and to those obtained by direct application of collocation methods (the latter also yield estimates for positive Lyapunov exponents in the chaotic regime). We conclude that the methodology described here works well for a class of delay equations for which currently no tailor-made tools exist (and for which it is doubtful that these will ever be constructed). |
| first_indexed | 2024-04-09T13:38:04Z |
| format | Article |
| id | doaj.art-29ede9867c364990abde1c55747d149f |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:04Z |
| publishDate | 2016-09-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-29ede9867c364990abde1c55747d149f2023-05-09T07:53:06ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752016-09-0120166512410.14232/ejqtde.2016.1.655273Numerical bifurcation analysis of a class of nonlinear renewal equationsDimitri Breda0Odo Diekmann1Davide Liessi2Francesca Scarabel3University of Udine, ItalyDepartment of Mathematics, Utrecht University, Utrecht, The NetherlandsDepartment of Mathematics, Computer Science and Physics, University of Udine, Udine, ItalyDepartment of Mathematics and Statistics, University of Helsinki, Helsinki, FinlandWe show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits transcritical, Hopf and period doubling bifurcations. The reliability is demonstrated by comparing the results to those obtained by a reduction to a Hamiltonian Kaplan-Yorke system and to those obtained by direct application of collocation methods (the latter also yield estimates for positive Lyapunov exponents in the chaotic regime). We conclude that the methodology described here works well for a class of delay equations for which currently no tailor-made tools exist (and for which it is doubtful that these will ever be constructed).http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5273renewal equationsstructured populationsstability of periodic solutionsperiod doubling cascadenumerical continuation and bifurcationpseudospectral and collocation methodskaplan-yorke periodic orbits |
| spellingShingle | Dimitri Breda Odo Diekmann Davide Liessi Francesca Scarabel Numerical bifurcation analysis of a class of nonlinear renewal equations Electronic Journal of Qualitative Theory of Differential Equations renewal equations structured populations stability of periodic solutions period doubling cascade numerical continuation and bifurcation pseudospectral and collocation methods kaplan-yorke periodic orbits |
| title | Numerical bifurcation analysis of a class of nonlinear renewal equations |
| title_full | Numerical bifurcation analysis of a class of nonlinear renewal equations |
| title_fullStr | Numerical bifurcation analysis of a class of nonlinear renewal equations |
| title_full_unstemmed | Numerical bifurcation analysis of a class of nonlinear renewal equations |
| title_short | Numerical bifurcation analysis of a class of nonlinear renewal equations |
| title_sort | numerical bifurcation analysis of a class of nonlinear renewal equations |
| topic | renewal equations structured populations stability of periodic solutions period doubling cascade numerical continuation and bifurcation pseudospectral and collocation methods kaplan-yorke periodic orbits |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5273 |
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