Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties
The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel...
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Akif AKGUL
2024-06-01
|
| Series: | Chaos Theory and Applications |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/en/download/article-file/3475412 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832590414254702592 |
|---|---|
| author | Joaquin Alvarez-gallegos Sishu Shankar Muni Hector Gilardi-velázquez Rıcardo Cuesta-garcía J. L. Echenausía-monroy |
| author_facet | Joaquin Alvarez-gallegos Sishu Shankar Muni Hector Gilardi-velázquez Rıcardo Cuesta-garcía J. L. Echenausía-monroy |
| author_sort | Joaquin Alvarez-gallegos |
| collection | DOAJ |
| description | The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system. |
| format | Article |
| id | doaj-art-a8bd280ac0674c728bc440a53532486f |
| institution | Kabale University |
| issn | 2687-4539 |
| language | English |
| publishDate | 2024-06-01 |
| publisher | Akif AKGUL |
| record_format | Article |
| series | Chaos Theory and Applications |
| spelling | doaj-art-a8bd280ac0674c728bc440a53532486f2025-01-23T18:19:49ZengAkif AKGULChaos Theory and Applications2687-45392024-06-0162738210.51537/chaos.13761231971Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators PropertiesJoaquin Alvarez-gallegos0https://orcid.org/0000-0002-0858-9142Sishu Shankar Muni1https://orcid.org/0000-0001-9545-8345Hector Gilardi-velázquez2https://orcid.org/0000-0002-4978-4526Rıcardo Cuesta-garcía3https://orcid.org/0000-0001-7074-5962J. L. Echenausía-monroy4https://orcid.org/0000-0001-5314-3935CICESE-Center for Scientific Research and Higher Education at EnsenadaDigital University KeralaUniversidad Panamericana-AguascalientesCICESE-Center for Scientific Research and Higher Education at EnsenadaCenter for Scientific Research and Higher Education at Ensenada-CICESEThe study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.https://dergipark.org.tr/en/download/article-file/3475412nonlinear dynamicschaotic systemmultistabilitypwl systembifurcationtipping point |
| spellingShingle | Joaquin Alvarez-gallegos Sishu Shankar Muni Hector Gilardi-velázquez Rıcardo Cuesta-garcía J. L. Echenausía-monroy Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties Chaos Theory and Applications nonlinear dynamics chaotic system multistability pwl system bifurcation tipping point |
| title | Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties |
| title_full | Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties |
| title_fullStr | Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties |
| title_full_unstemmed | Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties |
| title_short | Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties |
| title_sort | predicting tipping points in a family of pwl systems detecting multistability via linear operators properties |
| topic | nonlinear dynamics chaotic system multistability pwl system bifurcation tipping point |
| url | https://dergipark.org.tr/en/download/article-file/3475412 |
| work_keys_str_mv | AT joaquinalvarezgallegos predictingtippingpointsinafamilyofpwlsystemsdetectingmultistabilityvialinearoperatorsproperties AT sishushankarmuni predictingtippingpointsinafamilyofpwlsystemsdetectingmultistabilityvialinearoperatorsproperties AT hectorgilardivelazquez predictingtippingpointsinafamilyofpwlsystemsdetectingmultistabilityvialinearoperatorsproperties AT rıcardocuestagarcia predictingtippingpointsinafamilyofpwlsystemsdetectingmultistabilityvialinearoperatorsproperties AT jlechenausiamonroy predictingtippingpointsinafamilyofpwlsystemsdetectingmultistabilityvialinearoperatorsproperties |