Bifurcation diagrams for singularly perturbed system: the multi-dimensional case.
We consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddl...
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| Format: | Article |
| Language: | English |
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University of Szeged
2013-08-01
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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=1880 |
| _version_ | 1797830635004362752 |
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| author | Matteo Franca |
| author_facet | Matteo Franca |
| author_sort | Matteo Franca |
| collection | DOAJ |
| description | We consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multi-dimensional case the results obtained in a previous paper where the slow-time system is $1$-dimensional. We prove the existence of a unique trajectory $(\breve{x}(t,\varepsilon,\lambda),\breve{y}(t,\varepsilon,\lambda))$ homoclinic to a centre manifold of the slow manifold. Then we construct curves in the $2$-dimensional parameters space, dividing it in different areas where $(\breve{x}(t,\varepsilon,\lambda),\breve{y}(t,\varepsilon,\lambda))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples. |
| first_indexed | 2024-04-09T13:39:13Z |
| format | Article |
| id | doaj.art-3671e0ef93e5428f9056e0d216772bc8 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:13Z |
| publishDate | 2013-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-3671e0ef93e5428f9056e0d216772bc82023-05-09T07:53:03ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752013-08-0120135213610.14232/ejqtde.2013.1.521880Bifurcation diagrams for singularly perturbed system: the multi-dimensional case.Matteo Franca0Università Politecnica delle Marche, Ancona, ItalyWe consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multi-dimensional case the results obtained in a previous paper where the slow-time system is $1$-dimensional. We prove the existence of a unique trajectory $(\breve{x}(t,\varepsilon,\lambda),\breve{y}(t,\varepsilon,\lambda))$ homoclinic to a centre manifold of the slow manifold. Then we construct curves in the $2$-dimensional parameters space, dividing it in different areas where $(\breve{x}(t,\varepsilon,\lambda),\breve{y}(t,\varepsilon,\lambda))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1880singular perturbationhomoclinic trajectorytranscritical bifurcation central manifold |
| spellingShingle | Matteo Franca Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. Electronic Journal of Qualitative Theory of Differential Equations singular perturbation homoclinic trajectory transcritical bifurcation central manifold |
| title | Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. |
| title_full | Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. |
| title_fullStr | Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. |
| title_full_unstemmed | Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. |
| title_short | Bifurcation diagrams for singularly perturbed system: the multi-dimensional case. |
| title_sort | bifurcation diagrams for singularly perturbed system the multi dimensional case |
| topic | singular perturbation homoclinic trajectory transcritical bifurcation central manifold |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1880 |
| work_keys_str_mv | AT matteofranca bifurcationdiagramsforsingularlyperturbedsystemthemultidimensionalcase |