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The Regularized Visible Fold Revisited
The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter ϵ → 0 . Alternatively, these singularly perturbed systems can be thought of as r...
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| Published in: | Journal of nonlinear science 2020-12, Vol.30 (6), p.2463-2511 |
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| Language: | English |
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| container_issue | 6 |
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| container_title | Journal of nonlinear science |
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| creator | Kristiansen, K. Uldall |
| description | The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter
ϵ
→
0
. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit
ϵ
→
0
, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law. |
| doi_str_mv | 10.1007/s00332-020-09627-8 |
| format | article |
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ϵ
→
0
. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit
ϵ
→
0
, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.</description><identifier>ISSN: 0938-8974</identifier><identifier>EISSN: 1432-1467</identifier><identifier>DOI: 10.1007/s00332-020-09627-8</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Analysis ; Belt conveyors ; Bifurcations ; Classical Mechanics ; Economic Theory/Quantitative Economics/Mathematical Methods ; Mass-spring systems ; Mathematical and Computational Engineering ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Singular perturbation ; Theoretical</subject><ispartof>Journal of nonlinear science, 2020-12, Vol.30 (6), p.2463-2511</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-5a9cc773c7eac5db7d185641a43f00815d7d4635984b5b3e86310c766a39c6713</citedby><cites>FETCH-LOGICAL-c319t-5a9cc773c7eac5db7d185641a43f00815d7d4635984b5b3e86310c766a39c6713</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Kristiansen, K. Uldall</creatorcontrib><title>The Regularized Visible Fold Revisited</title><title>Journal of nonlinear science</title><addtitle>J Nonlinear Sci</addtitle><description>The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter
ϵ
→
0
. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit
ϵ
→
0
, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.</description><subject>Analysis</subject><subject>Belt conveyors</subject><subject>Bifurcations</subject><subject>Classical Mechanics</subject><subject>Economic Theory/Quantitative Economics/Mathematical Methods</subject><subject>Mass-spring systems</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Singular perturbation</subject><subject>Theoretical</subject><issn>0938-8974</issn><issn>1432-1467</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWKt_wFNB8Bad7GQzyVGKVaEgSPUasklat6xdTVpBf73RFbx5Gmbm_YCHsVMBFwKALjMAYsWhAg5GVcT1HhsJWU5CKtpnIzCouTYkD9lRzmsAQTVWI3a-eI6Th7jadS61nzFMntrcNl2czPoulMd7WbcxHLODpetyPPmdY_Y4u15Mb_n8_uZuejXnHoXZ8toZ74nQU3S-Dg0FoWslhZO4BNCiDhSkwtpo2dQNRq1QgCelHBqvSOCYnQ25r6l_28W8tet-lzal0laSkAxIlEVVDSqf-pxTXNrX1L649GEF2G8eduBhCw_7w8PqYsLBlIt4s4rpL_of1xdkqWA2</recordid><startdate>20201201</startdate><enddate>20201201</enddate><creator>Kristiansen, K. Uldall</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20201201</creationdate><title>The Regularized Visible Fold Revisited</title><author>Kristiansen, K. Uldall</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-5a9cc773c7eac5db7d185641a43f00815d7d4635984b5b3e86310c766a39c6713</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Analysis</topic><topic>Belt conveyors</topic><topic>Bifurcations</topic><topic>Classical Mechanics</topic><topic>Economic Theory/Quantitative Economics/Mathematical Methods</topic><topic>Mass-spring systems</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Singular perturbation</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kristiansen, K. Uldall</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of nonlinear science</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kristiansen, K. Uldall</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Regularized Visible Fold Revisited</atitle><jtitle>Journal of nonlinear science</jtitle><stitle>J Nonlinear Sci</stitle><date>2020-12-01</date><risdate>2020</risdate><volume>30</volume><issue>6</issue><spage>2463</spage><epage>2511</epage><pages>2463-2511</pages><issn>0938-8974</issn><eissn>1432-1467</eissn><abstract>The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter
ϵ
→
0
. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit
ϵ
→
0
, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00332-020-09627-8</doi><tpages>49</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Journal of nonlinear science, 2020-12, Vol.30 (6), p.2463-2511 |
| issn | 0938-8974 1432-1467 |
| language | eng |
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| source | Springer Nature |
| subjects | Analysis Belt conveyors Bifurcations Classical Mechanics Economic Theory/Quantitative Economics/Mathematical Methods Mass-spring systems Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Singular perturbation Theoretical |
| title | The Regularized Visible Fold Revisited |
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