Algebraic techniques and perturbation methods to approach frequency response curves
The Algebra is exploited to study approximated responses of nonlinear dynamical systems leading to tracing solutions of approximated bifurcation diagrams associated with polynomial equations resulting from search of approximated periodic solutions of nonlinear ordinary differential equations. In det...
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| Veröffentlicht in: | International journal of non-linear mechanics 2022-09, Vol.144, p.104096, Article 104096 |
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| container_title | International journal of non-linear mechanics |
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| creator | Lamarque, C.-H. Ture Savadkoohi, A. |
| description | The Algebra is exploited to study approximated responses of nonlinear dynamical systems leading to tracing solutions of approximated bifurcation diagrams associated with polynomial equations resulting from search of approximated periodic solutions of nonlinear ordinary differential equations. In detail via using the Gröbner basis, a polynomial with the smallest degree in term of the approximated amplitude of the systems, here the L2 norm of coefficients of truncated Fourier series, is extracted where its coefficients are parameters of the systems such as the frequency. The presented methodology permits to detect maximal number of solutions even those which belong to isola of the frequency response curves of the system.
•The Algebra is exploited to study approximated periodic responses of nonlinear dynamical systems.•Searching these responses leads to polynomial equations.•Gröbner bases are built from the set of polynomial equations.•A polynomial with the smallest degree in term of the approximated amplitude of the response is extracted. |
| doi_str_mv | 10.1016/j.ijnonlinmec.2022.104096 |
| format | Article |
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•The Algebra is exploited to study approximated periodic responses of nonlinear dynamical systems.•Searching these responses leads to polynomial equations.•Gröbner bases are built from the set of polynomial equations.•A polynomial with the smallest degree in term of the approximated amplitude of the response is extracted.</description><identifier>ISSN: 0020-7462</identifier><identifier>EISSN: 1878-5638</identifier><identifier>DOI: 10.1016/j.ijnonlinmec.2022.104096</identifier><language>eng</language><publisher>New York: Elsevier Ltd</publisher><subject>[formula omitted] norm ; Approximation ; Dynamical systems ; Euclidean division ; Fourier series ; Frequency response ; Gröbner basis ; Mathematical analysis ; Mathematical Physics ; Mechanics ; Nonlinear differential equations ; Nonlinear Sciences ; Nonlinear systems ; Perturbation methods ; Physics ; Polynomial ; Polynomials</subject><ispartof>International journal of non-linear mechanics, 2022-09, Vol.144, p.104096, Article 104096</ispartof><rights>2022 Elsevier Ltd</rights><rights>Copyright Elsevier BV Sep 2022</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c434t-5fe5dd9d7229ae15cd9ae782c085a20e06b3870c14e217d0237c115afc658b063</citedby><cites>FETCH-LOGICAL-c434t-5fe5dd9d7229ae15cd9ae782c085a20e06b3870c14e217d0237c115afc658b063</cites><orcidid>0000-0002-5209-7356</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.ijnonlinmec.2022.104096$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>230,315,781,785,886,3551,27929,27930,46000</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03749838$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Lamarque, C.-H.</creatorcontrib><creatorcontrib>Ture Savadkoohi, A.</creatorcontrib><title>Algebraic techniques and perturbation methods to approach frequency response curves</title><title>International journal of non-linear mechanics</title><description>The Algebra is exploited to study approximated responses of nonlinear dynamical systems leading to tracing solutions of approximated bifurcation diagrams associated with polynomial equations resulting from search of approximated periodic solutions of nonlinear ordinary differential equations. In detail via using the Gröbner basis, a polynomial with the smallest degree in term of the approximated amplitude of the systems, here the L2 norm of coefficients of truncated Fourier series, is extracted where its coefficients are parameters of the systems such as the frequency. The presented methodology permits to detect maximal number of solutions even those which belong to isola of the frequency response curves of the system.
•The Algebra is exploited to study approximated periodic responses of nonlinear dynamical systems.•Searching these responses leads to polynomial equations.•Gröbner bases are built from the set of polynomial equations.•A polynomial with the smallest degree in term of the approximated amplitude of the response is extracted.</description><subject>[formula omitted] norm</subject><subject>Approximation</subject><subject>Dynamical systems</subject><subject>Euclidean division</subject><subject>Fourier series</subject><subject>Frequency response</subject><subject>Gröbner basis</subject><subject>Mathematical analysis</subject><subject>Mathematical Physics</subject><subject>Mechanics</subject><subject>Nonlinear differential equations</subject><subject>Nonlinear Sciences</subject><subject>Nonlinear systems</subject><subject>Perturbation methods</subject><subject>Physics</subject><subject>Polynomial</subject><subject>Polynomials</subject><issn>0020-7462</issn><issn>1878-5638</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNqNkE9rGzEQxUVooa7b76DQUw7rjLR_pD0ak8YBQw5tz0IrzdZabGkr7Rr87SuzofSYywwMv_eY9wi5Z7BhwJrHYeMGH_zJ-TOaDQfO872CtrkjKyaFLOqmlB_ICoBDIaqGfyKfUxogaysQK_Jje_qNXdTO0AnN0bs_MyaqvaUjxmmOnZ5c8PSM0zHYRKdA9TjGoM2R9hEz7M2VRkxj8AmpmeMF0xfysdenhF_f9pr8-v70c7cvDq_PL7vtoTBVWU1F3WNtbWsF561GVhubl5DcgKw1B4SmK6UAwyrkTFjgpTCM1bo3TS07aMo1eVh8j_qkxujOOl5V0E7ttwd1u0EpqlaW8sIy-21h8_O3iJMawhx9fk_xpmUyD1lnql0oE0NKEft_tgzUrW81qP_6Vre-1dJ31u4WLebIF4dRJeNyPWhdRDMpG9w7XP4Cqq-PVA</recordid><startdate>20220901</startdate><enddate>20220901</enddate><creator>Lamarque, C.-H.</creator><creator>Ture Savadkoohi, A.</creator><general>Elsevier Ltd</general><general>Elsevier BV</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0002-5209-7356</orcidid></search><sort><creationdate>20220901</creationdate><title>Algebraic techniques and perturbation methods to approach frequency response curves</title><author>Lamarque, C.-H. ; Ture Savadkoohi, A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c434t-5fe5dd9d7229ae15cd9ae782c085a20e06b3870c14e217d0237c115afc658b063</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>[formula omitted] norm</topic><topic>Approximation</topic><topic>Dynamical systems</topic><topic>Euclidean division</topic><topic>Fourier series</topic><topic>Frequency response</topic><topic>Gröbner basis</topic><topic>Mathematical analysis</topic><topic>Mathematical Physics</topic><topic>Mechanics</topic><topic>Nonlinear differential equations</topic><topic>Nonlinear Sciences</topic><topic>Nonlinear systems</topic><topic>Perturbation methods</topic><topic>Physics</topic><topic>Polynomial</topic><topic>Polynomials</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lamarque, C.-H.</creatorcontrib><creatorcontrib>Ture Savadkoohi, A.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>International journal of non-linear mechanics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lamarque, C.-H.</au><au>Ture Savadkoohi, A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Algebraic techniques and perturbation methods to approach frequency response curves</atitle><jtitle>International journal of non-linear mechanics</jtitle><date>2022-09-01</date><risdate>2022</risdate><volume>144</volume><spage>104096</spage><pages>104096-</pages><artnum>104096</artnum><issn>0020-7462</issn><eissn>1878-5638</eissn><abstract>The Algebra is exploited to study approximated responses of nonlinear dynamical systems leading to tracing solutions of approximated bifurcation diagrams associated with polynomial equations resulting from search of approximated periodic solutions of nonlinear ordinary differential equations. 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| ispartof | International journal of non-linear mechanics, 2022-09, Vol.144, p.104096, Article 104096 |
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| language | eng |
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| source | ScienceDirect Journals (5 years ago - present) |
| subjects | [formula omitted] norm Approximation Dynamical systems Euclidean division Fourier series Frequency response Gröbner basis Mathematical analysis Mathematical Physics Mechanics Nonlinear differential equations Nonlinear Sciences Nonlinear systems Perturbation methods Physics Polynomial Polynomials |
| title | Algebraic techniques and perturbation methods to approach frequency response curves |
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