Canards Existence in the Hindmarsh–Rose model
In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J...
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| Veröffentlicht in: | Mathematical modelling of natural phenomena 2019, Vol.14 (4), p.409 |
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| creator | Ginoux, Jean-Marc Llibre, Jaume Tchizawa, Kiyoyuki |
| description | In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381–431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model. |
| doi_str_mv | 10.1051/mmnp/2019012 |
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T. ; Mortell, M.P. ; Sobolev, V.A. ; Korobeinikov, A. ; Korobeinikov, A. ; Lázaro, J. T. ; Sobolev, V.A. ; Mortell, M.P.</contributor><creatorcontrib>Ginoux, Jean-Marc ; Llibre, Jaume ; Tchizawa, Kiyoyuki ; Lázaro, J. T. ; Mortell, M.P. ; Sobolev, V.A. ; Korobeinikov, A. ; Korobeinikov, A. ; Lázaro, J. T. ; Sobolev, V.A. ; Mortell, M.P.</creatorcontrib><description>In two previous papers we have proposed a new method for proving the existence of “canard solutions” on one hand for three and four-dimensional singularly perturbed systems with only one fast variable and, on the other hand for four-dimensional singularly perturbed systems with two fast variables [J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2016) 381–431; J.M. Ginoux and J. Llibre, Qual. Theory Dyn. Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.</description><identifier>ISSN: 0973-5348</identifier><identifier>EISSN: 1760-6101</identifier><identifier>DOI: 10.1051/mmnp/2019012</identifier><language>eng</language><publisher>Les Ulis: EDP Sciences</publisher><subject>34C23 ; 34C25 ; 37G10 ; canard solutions ; Dimensional stability ; Hindmarsh–Rose model ; Linear algebra ; Singularity (mathematics) ; singularly perturbed dynamical systems</subject><ispartof>Mathematical modelling of natural phenomena, 2019, Vol.14 (4), p.409</ispartof><rights>2019. 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Syst. 15 (2015) 342010]. The aim of this work is to extend this method which improves the classical ones used till now to the case of three-dimensional singularly perturbed systems with two fast variables. This method enables to state a unique generic condition for the existence of “canard solutions” for such three-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. Applications of this method to a famous neuronal bursting model enables to show the existence of “canard solutions” in the Hindmarsh-Rose model.</description><subject>34C23</subject><subject>34C25</subject><subject>37G10</subject><subject>canard solutions</subject><subject>Dimensional stability</subject><subject>Hindmarsh–Rose model</subject><subject>Linear algebra</subject><subject>Singularity (mathematics)</subject><subject>singularly perturbed dynamical systems</subject><issn>0973-5348</issn><issn>1760-6101</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNo9kMFKAzEQhoMoWGpvPsCCV9dOMtlkc5TSWqUgSqXHkCZZurW7W5Mt1FvfwTf0SdzS4lz-y8fMPx8htxQeKGR0WFX1dsiAKqDsgvSoFJAKCvSS9EBJTDPk-TUZxLiGbpByBOiR4cjUJriYjPdlbH1tfVLWSbvyybSsXWVCXP0eft6b6JOqcX5zQ64Ks4l-cM4--ZiM56NpOnt9eh49zlKLqNqUgRfGSM4UiNwWvKDKZMCMUN5a5zm3UlLuCwuZsN45CUtnl8glOsxFLrFP7k57t6H52vnY6nWzC3V3UjOWI8spZqqj7k-UDU2MwRd6G8qu9LemoI9W9NGKPlvp8PSEH1_d_7MmfGohUWY6h4Weq8nihb-hBvwDrDBjkg</recordid><startdate>2019</startdate><enddate>2019</enddate><creator>Ginoux, Jean-Marc</creator><creator>Llibre, Jaume</creator><creator>Tchizawa, Kiyoyuki</creator><general>EDP Sciences</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7QF</scope><scope>7QO</scope><scope>7QQ</scope><scope>7SC</scope><scope>7SE</scope><scope>7SP</scope><scope>7SR</scope><scope>7TA</scope><scope>7TB</scope><scope>7TK</scope><scope>7U5</scope><scope>8BQ</scope><scope>8FD</scope><scope>F28</scope><scope>FR3</scope><scope>H8D</scope><scope>H8G</scope><scope>JG9</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>P64</scope><orcidid>https://orcid.org/0000-0002-9511-5999</orcidid></search><sort><creationdate>2019</creationdate><title>Canards Existence in the Hindmarsh–Rose model</title><author>Ginoux, Jean-Marc ; Llibre, Jaume ; Tchizawa, Kiyoyuki</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c339t-20e6aa7429068cf4f19a502a69eccde44c7714efc056cedd70bdcb3473d386873</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>34C23</topic><topic>34C25</topic><topic>37G10</topic><topic>canard solutions</topic><topic>Dimensional stability</topic><topic>Hindmarsh–Rose model</topic><topic>Linear algebra</topic><topic>Singularity (mathematics)</topic><topic>singularly perturbed dynamical systems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ginoux, Jean-Marc</creatorcontrib><creatorcontrib>Llibre, Jaume</creatorcontrib><creatorcontrib>Tchizawa, Kiyoyuki</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Aluminium Industry Abstracts</collection><collection>Biotechnology Research Abstracts</collection><collection>Ceramic Abstracts</collection><collection>Computer and Information Systems Abstracts</collection><collection>Corrosion Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>Engineered Materials Abstracts</collection><collection>Materials Business File</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Neurosciences Abstracts</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>METADEX</collection><collection>Technology Research Database</collection><collection>ANTE: Abstracts in New Technology & Engineering</collection><collection>Engineering Research Database</collection><collection>Aerospace Database</collection><collection>Copper Technical Reference Library</collection><collection>Materials 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>Biotechnology and BioEngineering Abstracts</collection><jtitle>Mathematical modelling of natural phenomena</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ginoux, Jean-Marc</au><au>Llibre, Jaume</au><au>Tchizawa, Kiyoyuki</au><au>Lázaro, J. 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| subjects | 34C23 34C25 37G10 canard solutions Dimensional stability Hindmarsh–Rose model Linear algebra Singularity (mathematics) singularly perturbed dynamical systems |
| title | Canards Existence in the Hindmarsh–Rose model |
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