Algorithm for Generating Bifurcation Diagrams Using Poincaré Intersection Plane

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 () and José Luis Echenausía-Monroy ()
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Luis Javier Ontañón-García: Coordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, Mexico
Juan Gonzalo Barajas-Ramírez: División 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, Mexico
Eric Campos-Cantón: División 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, Mexico
Daniel Alejandro Magallón-García: Coordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, Mexico
César Arturo Guerra-García: Coordinación Académica Región Altiplano Oeste, Universidad Autónoma de San Luis Potosí, Carretera a Santo Domingo 200, Salinas de Hidalgo 78600, SLP, Mexico
José Ricardo Cuesta-García: Applied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, Mexico
Jonatan Pena-Ramirez: Applied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, Mexico
José Luis Echenausía-Monroy: Applied Physics Division, Department of Electronics and Telecommunications, CICESE Research Center, Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Ensenada 22860, BC, Mexico

Mathematics, 2025, vol. 13, issue 11, 1-20

Abstract: 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 Parallel Computing Toolbox 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.

Keywords: bifurcation diagram; Feigenbaum diagram; dynamical systems; chaos; parallel computing; HPC (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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