Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects
Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the mileston...
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MDPI AG
2024-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/22 |
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| author | José Luis Echenausía-Monroy Luis Alberto Quezada-Tellez Hector Eduardo Gilardi-Velázquez Omar Fernando Ruíz-Martínez María del Carmen Heras-Sánchez Jose E. Lozano-Rizk José Ricardo Cuesta-García Luis Alejandro Márquez-Martínez Raúl Rivera-Rodríguez Jonatan Pena Ramirez Joaquín Álvarez |
| author_facet | José Luis Echenausía-Monroy Luis Alberto Quezada-Tellez Hector Eduardo Gilardi-Velázquez Omar Fernando Ruíz-Martínez María del Carmen Heras-Sánchez Jose E. Lozano-Rizk José Ricardo Cuesta-García Luis Alejandro Márquez-Martínez Raúl Rivera-Rodríguez Jonatan Pena Ramirez Joaquín Álvarez |
| author_sort | José Luis Echenausía-Monroy |
| collection | DOAJ |
| description | Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems. |
| format | Article |
| id | doaj-art-f49cc78b054945eaa1833ddd3cecaccc |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-f49cc78b054945eaa1833ddd3cecaccc2025-01-24T13:33:24ZengMDPI AGFractal and Fractional2504-31102024-12-01912210.3390/fractalfract9010022Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic EffectsJosé Luis Echenausía-Monroy0Luis Alberto Quezada-Tellez1Hector Eduardo Gilardi-Velázquez2Omar Fernando Ruíz-Martínez3María del Carmen Heras-Sánchez4Jose E. Lozano-Rizk5José Ricardo Cuesta-García6Luis Alejandro Márquez-Martínez7Raúl Rivera-Rodríguez8Jonatan Pena Ramirez9Joaquín Álvarez10Department of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoEscuela Superior de Apan, UAEH, Carretera Apan-Calpulalpan Km. 8, Colonia Chimalpa Tlalayote, Apan 43900, HD, MexicoFacultad de Ingeniería, Universidad Panamericana, Josemaría Escrivá de Balaguer 101, Aguascalientes 20296, AG, MexicoFacultad de Ingeniería, Universidad Panamericana, Josemaría Escrivá de Balaguer 101, Aguascalientes 20296, AG, MexicoDepartamento de Matemáticas, Universidad de Sonora, Rosales y Blvd. Luis Encinas S/N, Hermosillo 83000, SO, MexicoDivision of Telematics, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoDepartment of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoDepartment of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoDepartment of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoDepartment of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoDepartment of Electronics and Telecommunications, Applied Physics Division, CICESE Research Center, Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, BC, MexicoFractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems.https://www.mdpi.com/2504-3110/9/1/22chaosfractional-order derivativedynamical systemsfractional-order dynamicsfractional-derivativesLyapunov exponent |
| spellingShingle | José Luis Echenausía-Monroy Luis Alberto Quezada-Tellez Hector Eduardo Gilardi-Velázquez Omar Fernando Ruíz-Martínez María del Carmen Heras-Sánchez Jose E. Lozano-Rizk José Ricardo Cuesta-García Luis Alejandro Márquez-Martínez Raúl Rivera-Rodríguez Jonatan Pena Ramirez Joaquín Álvarez Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects Fractal and Fractional chaos fractional-order derivative dynamical systems fractional-order dynamics fractional-derivatives Lyapunov exponent |
| title | Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects |
| title_full | Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects |
| title_fullStr | Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects |
| title_full_unstemmed | Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects |
| title_short | Beyond Chaos in Fractional-Order Systems: Keen Insight in the Dynamic Effects |
| title_sort | beyond chaos in fractional order systems keen insight in the dynamic effects |
| topic | chaos fractional-order derivative dynamical systems fractional-order dynamics fractional-derivatives Lyapunov exponent |
| url | https://www.mdpi.com/2504-3110/9/1/22 |
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