Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation
The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp.,...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/197914 |
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| _version_ | 1832567134792712192 |
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| author | Fengjie Geng Song Li |
| author_facet | Fengjie Geng Song Li |
| author_sort | Fengjie Geng |
| collection | DOAJ |
| description | The bifurcation of a
nongeneric homoclinic orbit (i.e., the orbit comes from the
equilibrium along the unstable manifold instead of the center
manifold) connecting a nonhyperbolic equilibrium is
investigated, and the nonhyperbolic equilibrium undergoes a
pitchfork bifurcation. The existence (resp., nonexistence) of a
homoclinic orbit and an 1-periodic orbit are established when the
pitchfork bifurcation does not happen, while as the nonhyperbolic
equilibrium undergoes a pitchfork bifurcation, we obtain the
sufficient conditions for the existence of homoclinic orbit and two
or three heteroclinic orbits, and so forth. Moreover, we explore the
difference between the bifurcation of the nongeneric homoclinic
orbit and the generic one. |
| format | Article |
| id | doaj-art-da1f0c2dee294eaba2d888fb6930760a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-da1f0c2dee294eaba2d888fb6930760a2025-02-03T01:02:19ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/197914197914Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork BifurcationFengjie Geng0Song Li1School of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaSchool of Science, China University of Geosciences (Beijing), Beijing 100083, ChinaThe bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.http://dx.doi.org/10.1155/2014/197914 |
| spellingShingle | Fengjie Geng Song Li Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation Abstract and Applied Analysis |
| title | Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation |
| title_full | Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation |
| title_fullStr | Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation |
| title_full_unstemmed | Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation |
| title_short | Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation |
| title_sort | bifurcation of nongeneric homoclinic orbit accompanied by pitchfork bifurcation |
| url | http://dx.doi.org/10.1155/2014/197914 |
| work_keys_str_mv | AT fengjiegeng bifurcationofnongenerichomoclinicorbitaccompaniedbypitchforkbifurcation AT songli bifurcationofnongenerichomoclinicorbitaccompaniedbypitchforkbifurcation |