Research on discrete differential solution methods for derivatives of chaotic systems
The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivot...
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| Language: | English |
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241621 |
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| author | Xinyu Pan |
| author_facet | Xinyu Pan |
| author_sort | Xinyu Pan |
| collection | DOAJ |
| description | The pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications. |
| format | Article |
| id | doaj-art-4a6563c2b1fa41af9dc0aed3c991e3c2 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-4a6563c2b1fa41af9dc0aed3c991e3c22025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912339953401210.3934/math.20241621Research on discrete differential solution methods for derivatives of chaotic systemsXinyu Pan0School of Electronics & Information Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu, 215009, ChinaThe pivotal differential parameters inherent in chaotic systems hold paramount significance across diverse disciplines. This study delves into the distinctive features of discrete differential parameters within three typical chaotic systems: the logistic map, the henon map, and the tent map. A pivotal discovery emerges: both the mean value of the first-order continuous and discrete derivatives in the logistic map coincide, mirroring a similar behavior observed in the henon map. Leveraging the insights gained from the first derivative formulations, we introduce the discrete n-order derivative formulas for both logistic and henon maps. This revelation underscores a discernible mathematical correlation linking the mean value of the derivative, the respective chaotic parameters, and the mean of the chaotic sequence. However, due to the discontinuous points in the tent map, its continuous differential parameter cannot characterize its derivative properties, but its discrete differential has a clear functional relationship with the parameter μ. This paper proposes the use of discrete differential derivatives as an alternative to traditional derivatives, and demonstrates that the mean value of discrete derivatives has a clear mathematical relationship with chaotic map parameters in a statistical sense, providing a new direction for subsequent in-depth research and applications.https://www.aimspress.com/article/doi/10.3934/math.20241621chaotic systemscontinuous differential derivativediscrete differential derivative |
| spellingShingle | Xinyu Pan Research on discrete differential solution methods for derivatives of chaotic systems AIMS Mathematics chaotic systems continuous differential derivative discrete differential derivative |
| title | Research on discrete differential solution methods for derivatives of chaotic systems |
| title_full | Research on discrete differential solution methods for derivatives of chaotic systems |
| title_fullStr | Research on discrete differential solution methods for derivatives of chaotic systems |
| title_full_unstemmed | Research on discrete differential solution methods for derivatives of chaotic systems |
| title_short | Research on discrete differential solution methods for derivatives of chaotic systems |
| title_sort | research on discrete differential solution methods for derivatives of chaotic systems |
| topic | chaotic systems continuous differential derivative discrete differential derivative |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241621 |
| work_keys_str_mv | AT xinyupan researchondiscretedifferentialsolutionmethodsforderivativesofchaoticsystems |