Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation
Identifieur interne : 000817 ( Main/Exploration ); précédent : 000816; suivant : 000818Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation
Auteurs : Gregor Kova I [États-Unis] ; Stephen Wiggins [États-Unis]Source :
- Physica D: Nonlinear Phenomena [ 0167-2789 ] ; 1992.
Abstract
In this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two-degree-of-freedom Hamiltonian systems. Our methods are fundamentally different than other global perturbation methods (e.g. standard Melnikov theory) in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation. Our methods combine the higher dimensional Melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds. We apply our methods to a modified model of the forced and damped sine-Gordon equation developed by Bishop et al. We give explicit conditions (in terms of the system parameters) for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle-focus type; chaotic dynamics follow from a theorem of Silnikov. This provides a mechanism for chaotic dynamics geometrically similar to that observed by Bishop et al.; namely, a random “jumping” between two spatially dependent states with an intermediate passage through a spatially independent state. However, in order for this type of Silnikov dynamics to exist we require a different, and unphysical, type of damping compared to that used by Bishop et al.
Url:
DOI: 10.1016/0167-2789(92)90092-2
Affiliations:
Links toward previous steps (curation, corpus...)
Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation</title>
<author><name sortKey="Kova I, Gregor" sort="Kova I, Gregor" uniqKey="Kova I G" first="Gregor" last="Kova I">Gregor Kova I</name>
</author>
<author><name sortKey="Wiggins, Stephen" sort="Wiggins, Stephen" uniqKey="Wiggins S" first="Stephen" last="Wiggins">Stephen Wiggins</name>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">ISTEX</idno>
<idno type="RBID">ISTEX:2B009AFA0E75216D666F56F9CDA12DEA1DC26346</idno>
<date when="1992" year="1992">1992</date>
<idno type="doi">10.1016/0167-2789(92)90092-2</idno>
<idno type="url">https://api.istex.fr/document/2B009AFA0E75216D666F56F9CDA12DEA1DC26346/fulltext/pdf</idno>
<idno type="wicri:Area/Main/Corpus">000379</idno>
<idno type="wicri:Area/Main/Curation">000379</idno>
<idno type="wicri:Area/Main/Exploration">000817</idno>
<idno type="wicri:explorRef" wicri:stream="Main" wicri:step="Exploration">000817</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title level="a">Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation</title>
<author><name sortKey="Kova I, Gregor" sort="Kova I, Gregor" uniqKey="Kova I G" first="Gregor" last="Kova I">Gregor Kova I</name>
<affiliation wicri:level="1"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, NY 12180</wicri:regionArea>
<wicri:noRegion>NY 12180</wicri:noRegion>
</affiliation>
</author>
<author><name sortKey="Wiggins, Stephen" sort="Wiggins, Stephen" uniqKey="Wiggins S" first="Stephen" last="Wiggins">Stephen Wiggins</name>
<affiliation></affiliation>
<affiliation wicri:level="1"><country xml:lang="fr">États-Unis</country>
<wicri:regionArea>Applied Mechanics 104-44, CALTECH, Pasadena CA 91125</wicri:regionArea>
<wicri:noRegion>Pasadena CA 91125</wicri:noRegion>
</affiliation>
</author>
</analytic>
<monogr></monogr>
<series><title level="j">Physica D: Nonlinear Phenomena</title>
<title level="j" type="abbrev">PHYSD</title>
<idno type="ISSN">0167-2789</idno>
<imprint><publisher>ELSEVIER</publisher>
<date type="published" when="1992">1992</date>
<biblScope unit="volume">57</biblScope>
<biblScope unit="issue">1–2</biblScope>
<biblScope unit="page" from="185">185</biblScope>
<biblScope unit="page" to="225">225</biblScope>
</imprint>
<idno type="ISSN">0167-2789</idno>
</series>
<idno type="istex">2B009AFA0E75216D666F56F9CDA12DEA1DC26346</idno>
<idno type="DOI">10.1016/0167-2789(92)90092-2</idno>
<idno type="PII">0167-2789(92)90092-2</idno>
</biblStruct>
</sourceDesc>
<seriesStmt><idno type="ISSN">0167-2789</idno>
</seriesStmt>
</fileDesc>
<profileDesc><textClass></textClass>
<langUsage><language ident="en">en</language>
</langUsage>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">In this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two-degree-of-freedom Hamiltonian systems. Our methods are fundamentally different than other global perturbation methods (e.g. standard Melnikov theory) in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation. Our methods combine the higher dimensional Melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds. We apply our methods to a modified model of the forced and damped sine-Gordon equation developed by Bishop et al. We give explicit conditions (in terms of the system parameters) for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle-focus type; chaotic dynamics follow from a theorem of Silnikov. This provides a mechanism for chaotic dynamics geometrically similar to that observed by Bishop et al.; namely, a random “jumping” between two spatially dependent states with an intermediate passage through a spatially independent state. However, in order for this type of Silnikov dynamics to exist we require a different, and unphysical, type of damping compared to that used by Bishop et al.</div>
</front>
</TEI>
<affiliations><list><country><li>États-Unis</li>
</country>
</list>
<tree><country name="États-Unis"><noRegion><name sortKey="Kova I, Gregor" sort="Kova I, Gregor" uniqKey="Kova I G" first="Gregor" last="Kova I">Gregor Kova I</name>
</noRegion>
<name sortKey="Wiggins, Stephen" sort="Wiggins, Stephen" uniqKey="Wiggins S" first="Stephen" last="Wiggins">Stephen Wiggins</name>
</country>
</tree>
</affiliations>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Amerique/explor/CaltechV1/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000817 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 000817 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Amerique |area= CaltechV1 |flux= Main |étape= Exploration |type= RBID |clé= ISTEX:2B009AFA0E75216D666F56F9CDA12DEA1DC26346 |texte= Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation }}
This area was generated with Dilib version V0.6.32. |