Research on discrete differential solution methods for derivatives of chaotic systems

Xinyu Pan; Xinyu Pan

doi:10.3934/math.20241621

AIMS Mathematics

2024, Volume 9, Issue 12: 33995-34012. doi: 10.3934/math.20241621

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Research on discrete differential solution methods for derivatives of chaotic systems

Xinyu Pan ^,

School of Electronics & Information Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu, 215009, China

Received: 15 July 2024 Revised: 21 November 2024 Accepted: 22 November 2024 Published: 02 December 2024
MSC : 39A12, 39B12, 65P20

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.
- chaotic systems,
- continuous differential derivative,
- discrete differential derivative
Citation: Xinyu Pan. Research on discrete differential solution methods for derivatives of chaotic systems[J]. AIMS Mathematics, 2024, 9(12): 33995-34012. doi: 10.3934/math.20241621

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Abstract

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.

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