The Two-Dimensional Border-Collision Normal Form with a Zero Determinant.
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| Title: | The Two-Dimensional Border-Collision Normal Form with a Zero Determinant. |
|---|---|
| Authors: | Simpson, David J. W.1 D.J.W.Simpson@massey.ac.nz |
| Source: | SIAM Journal on Applied Dynamical Systems. 2025, Vol. 24 Issue 3, p2205-2245. 41p. |
| Subjects: | Bifurcation theory, Attractors (Mathematics), Dynamical systems, Continuous time models |
| Abstract: | The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterize the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analyzed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also discuss the robustness of the dynamics and describe three novel bifurcation structures that remain to be explored. [ABSTRACT FROM AUTHOR] |
| Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 188706141 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Two-Dimensional Border-Collision Normal Form with a Zero Determinant. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Simpson%2C+David+J%2E+W%2E%22">Simpson, David J. W.</searchLink><relatesTo>1</relatesTo><i> D.J.W.Simpson@massey.ac.nz</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Applied+Dynamical+Systems%22">SIAM Journal on Applied Dynamical Systems</searchLink>. 2025, Vol. 24 Issue 3, p2205-2245. 41p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Bifurcation+theory%22">Bifurcation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Attractors+%28Mathematics%29%22">Attractors (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamical+systems%22">Dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22Continuous+time+models%22">Continuous time models</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterize the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analyzed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also discuss the robustness of the dynamics and describe three novel bifurcation structures that remain to be explored. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/24M1683548 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 41 StartPage: 2205 Subjects: – SubjectFull: Bifurcation theory Type: general – SubjectFull: Attractors (Mathematics) Type: general – SubjectFull: Dynamical systems Type: general – SubjectFull: Continuous time models Type: general Titles: – TitleFull: The Two-Dimensional Border-Collision Normal Form with a Zero Determinant. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Simpson, David J. W. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: 2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 15360040 Numbering: – Type: volume Value: 24 – Type: issue Value: 3 Titles: – TitleFull: SIAM Journal on Applied Dynamical Systems Type: main |
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