Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane
In the study of dynamic systems, bifurcation diagrams are a very popular graphical tool for studying stability and nonlinear changes in behavior. They are instrumental in analyzing the system’s response to parameter changes. These diagrams show the system’s various dynamic patterns and phase transit...
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MDPI AG
2025-05-01
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| author | Luis Javier Ontañón-García Juan Gonzalo Barajas-Ramírez Eric Campos-Cantón Daniel Alejandro Magallón-García César Arturo Guerra-García José Ricardo Cuesta-García Jonatan Pena-Ramirez José Luis Echenausía-Monroy |
| author_facet | Luis Javier Ontañón-García Juan Gonzalo Barajas-Ramírez Eric Campos-Cantón Daniel Alejandro Magallón-García César Arturo Guerra-García José Ricardo Cuesta-García Jonatan Pena-Ramirez José Luis Echenausía-Monroy |
| author_sort | Luis Javier Ontañón-García |
| collection | DOAJ |
| description | In the study of dynamic systems, bifurcation diagrams are a very popular graphical tool for studying stability and nonlinear changes in behavior. They are instrumental in analyzing the system’s response to parameter changes. These diagrams show the system’s various dynamic patterns and phase transitions by plotting the relationship between the system response and the parameters. This paper presents a computational algorithm for studying bifurcations in dynamic systems, especially for systems with chaotic behavior that depends on parameter changes. Depending on the type of system to be analyzed, the following two strategies for computing bifurcation diagrams are described: (i) detecting crossing points through the Poincaré plane and (ii) the identification of local maxima (or minima) in one of the system solutions. In addition, this paper presents a method for implementing parallel computation in MATLAB using the <i>Parallel Computing Toolbox</i> from MathWorks, which significantly reduces the computational time required to generate bifurcation diagrams. This work contributes to the study of dynamic systems by providing the reader with accessible tools for studying any dynamic system under established standards and reducing the computational time required for these types of studies by implementing these algorithms utilizing the multi-core processors found in modern computers. |
| format | Article |
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| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-99f9b7d7f5a5411b8dda64c1524a48072025-08-20T03:46:43ZengMDPI AGMathematics2227-73902025-05-011311181810.3390/math13111818Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection PlaneLuis Javier Ontañón-García0Juan Gonzalo Barajas-Ramírez1Eric Campos-Cantón2Daniel Alejandro Magallón-García3César Arturo Guerra-García4José Ricardo Cuesta-García5Jonatan Pena-Ramirez6José Luis Echenausía-Monroy7Coordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, MexicoDivisión de Control y Sistemas Dinámicos, Instituto Potosino de Investigación Científica y Tecnológica A.C. (IPICyT), Camino a la Presa San José 2255, Lomas 4ta. Sección, San Luis Potosí 78216, SLP, MexicoDivisión de Control y Sistemas Dinámicos, Instituto Potosino de Investigación Científica y Tecnológica A.C. (IPICyT), Camino a la Presa San José 2255, Lomas 4ta. Sección, San Luis Potosí 78216, SLP, MexicoCoordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, MexicoCoordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, MexicoApplied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, MexicoApplied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, MexicoApplied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, MexicoIn the study of dynamic systems, bifurcation diagrams are a very popular graphical tool for studying stability and nonlinear changes in behavior. They are instrumental in analyzing the system’s response to parameter changes. These diagrams show the system’s various dynamic patterns and phase transitions by plotting the relationship between the system response and the parameters. This paper presents a computational algorithm for studying bifurcations in dynamic systems, especially for systems with chaotic behavior that depends on parameter changes. Depending on the type of system to be analyzed, the following two strategies for computing bifurcation diagrams are described: (i) detecting crossing points through the Poincaré plane and (ii) the identification of local maxima (or minima) in one of the system solutions. In addition, this paper presents a method for implementing parallel computation in MATLAB using the <i>Parallel Computing Toolbox</i> from MathWorks, which significantly reduces the computational time required to generate bifurcation diagrams. This work contributes to the study of dynamic systems by providing the reader with accessible tools for studying any dynamic system under established standards and reducing the computational time required for these types of studies by implementing these algorithms utilizing the multi-core processors found in modern computers.https://www.mdpi.com/2227-7390/13/11/1818bifurcation diagramFeigenbaum diagramdynamical systemschaosparallel computingHPC |
| spellingShingle | Luis Javier Ontañón-García Juan Gonzalo Barajas-Ramírez Eric Campos-Cantón Daniel Alejandro Magallón-García César Arturo Guerra-García José Ricardo Cuesta-García Jonatan Pena-Ramirez José Luis Echenausía-Monroy Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane Mathematics bifurcation diagram Feigenbaum diagram dynamical systems chaos parallel computing HPC |
| title | Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane |
| title_full | Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane |
| title_fullStr | Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane |
| title_full_unstemmed | Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane |
| title_short | Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane |
| title_sort | algorithm for generating bifurcation diagrams using poincare intersection plane |
| topic | bifurcation diagram Feigenbaum diagram dynamical systems chaos parallel computing HPC |
| url | https://www.mdpi.com/2227-7390/13/11/1818 |
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