Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation
This paper concerns the quadratically-damped Mathieu equation: x ̈ +(δ+ε cos t)x+ x ̇ | x ̇ |=0. Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In δ– ε parameter space, this secondary b...
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| Veröffentlicht in: | International journal of non-linear mechanics 2004-04, Vol.39 (3), p.491-502 |
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| container_title | International journal of non-linear mechanics |
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| creator | Ramani, Deepak V Keith, William L Rand, Richard H |
| description | This paper concerns the quadratically-damped Mathieu equation:
x
̈
+(δ+ε
cos
t)x+
x
̇
|
x
̇
|=0.
Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In
δ–
ε parameter space, this secondary bifurcation appears as a curve which emanates from one of the
transition curves of the linear Mathieu equation for
ε≈1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines. |
| doi_str_mv | 10.1016/S0020-7462(02)00218-4 |
| format | Article |
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x
̈
+(δ+ε
cos
t)x+
x
̇
|
x
̇
|=0.
Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In
δ–
ε parameter space, this secondary bifurcation appears as a curve which emanates from one of the
transition curves of the linear Mathieu equation for
ε≈1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.</description><identifier>ISSN: 0020-7462</identifier><identifier>EISSN: 1878-5638</identifier><identifier>DOI: 10.1016/S0020-7462(02)00218-4</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Bifurcation ; Mathieu ; Nonlinear ; Perturbation ; Resonance</subject><ispartof>International journal of non-linear mechanics, 2004-04, Vol.39 (3), p.491-502</ispartof><rights>2003</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c385t-c0af949f407e1f151648b008340885c1cfcec34dd0b045525dd3f2dee9f73c563</citedby><cites>FETCH-LOGICAL-c385t-c0af949f407e1f151648b008340885c1cfcec34dd0b045525dd3f2dee9f73c563</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0020746202002184$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3536,27903,27904,65309</link.rule.ids></links><search><creatorcontrib>Ramani, Deepak V</creatorcontrib><creatorcontrib>Keith, William L</creatorcontrib><creatorcontrib>Rand, Richard H</creatorcontrib><title>Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation</title><title>International journal of non-linear mechanics</title><description>This paper concerns the quadratically-damped Mathieu equation:
x
̈
+(δ+ε
cos
t)x+
x
̇
|
x
̇
|=0.
Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In
δ–
ε parameter space, this secondary bifurcation appears as a curve which emanates from one of the
transition curves of the linear Mathieu equation for
ε≈1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.</description><subject>Bifurcation</subject><subject>Mathieu</subject><subject>Nonlinear</subject><subject>Perturbation</subject><subject>Resonance</subject><issn>0020-7462</issn><issn>1878-5638</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2004</creationdate><recordtype>article</recordtype><recordid>eNqFkE9LxDAQxYMouK5-BKEn0UN1kibb7Elk8R-sKKg3IaTJhI10292kFfbbm1rxKgSSmfzeMO8RckrhkgKdXb0CMMhLPmPnwC5SQWXO98iEylLmYlbIfTL5Qw7JUYyfkHQcygn5eMHQ9aHSnW-bLLZ1__NwbcgimraxOuyyyrs-mBHxTdatMNv22obUMbqud7nV6w3a7El3K499hul3gI_JgdN1xJPfe0re727fFg_58vn-cXGzzE0hRZcb0G7O5y4thNRRkVaTFYAsOEgpDDXOoCm4tVABF4IJawvHLOLclYVJBqfkbJy7Ce22x9iptY8G61o32PZRMVmwks1ZAsUImtDGGNCpTfDrZFFRUEOW6idLNQSlIJ0hS8WT7nrUYXLx5TGoaDw2Bq0PaDplW__PhG-ITX1K</recordid><startdate>20040401</startdate><enddate>20040401</enddate><creator>Ramani, Deepak V</creator><creator>Keith, William L</creator><creator>Rand, Richard H</creator><general>Elsevier Ltd</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SM</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>KR7</scope></search><sort><creationdate>20040401</creationdate><title>Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation</title><author>Ramani, Deepak V ; Keith, William L ; Rand, Richard H</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c385t-c0af949f407e1f151648b008340885c1cfcec34dd0b045525dd3f2dee9f73c563</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2004</creationdate><topic>Bifurcation</topic><topic>Mathieu</topic><topic>Nonlinear</topic><topic>Perturbation</topic><topic>Resonance</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ramani, Deepak V</creatorcontrib><creatorcontrib>Keith, William L</creatorcontrib><creatorcontrib>Rand, Richard H</creatorcontrib><collection>CrossRef</collection><collection>Earthquake Engineering Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Civil Engineering Abstracts</collection><jtitle>International journal of non-linear mechanics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ramani, Deepak V</au><au>Keith, William L</au><au>Rand, Richard H</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation</atitle><jtitle>International journal of non-linear mechanics</jtitle><date>2004-04-01</date><risdate>2004</risdate><volume>39</volume><issue>3</issue><spage>491</spage><epage>502</epage><pages>491-502</pages><issn>0020-7462</issn><eissn>1878-5638</eissn><abstract>This paper concerns the quadratically-damped Mathieu equation:
x
̈
+(δ+ε
cos
t)x+
x
̇
|
x
̇
|=0.
Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In
δ–
ε parameter space, this secondary bifurcation appears as a curve which emanates from one of the
transition curves of the linear Mathieu equation for
ε≈1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/S0020-7462(02)00218-4</doi><tpages>12</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | International journal of non-linear mechanics, 2004-04, Vol.39 (3), p.491-502 |
| issn | 0020-7462 1878-5638 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_28327292 |
| source | Elsevier ScienceDirect Journals |
| subjects | Bifurcation Mathieu Nonlinear Perturbation Resonance |
| title | Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation |
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