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Fractal Fract
Volume 10
Issue 1
10.3390/fractalfract10010017
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Modeling Phase Transitions in Starling Flocks Using Fractal Dimension of Self-Affine Functions

by
Kunyuan Li
1,
Xiongwei Zhang
1,
Kui Yao
2,*,
Kai Zhang
3,
Meng Sun
1,
Ming He
3,4,
Kefeng Liu
5 and
Yangjun Wang
5
1
Laboratory of Intelligent Information Processing, Army Engineering University of PLA, Nanjing 210007, China
2
Department of General Education, Army Engineering University of PLA, Nanjing 210007, China
3
College of Command & Control Engineering, Army Engineering University of PLA, Nanjing 210007, China
4
National Key Laboratory on Near-Surface Detection, Beijing 100072, China
5
College of Advanced Interdisciplinary Studies, National University of Defense Technology, Nanjing 211101, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2026, 10(1), 17; https://doi.org/10.3390/fractalfract10010017 (registering DOI)
Submission received: 29 October 2025 / Revised: 18 December 2025 / Accepted: 23 December 2025 / Published: 27 December 2025
Versions Notes

Abstract

This paper uses the theory of self-affine fractal functions to model the dynamic flight graphs of starling flocks, integrating the fractional calculus of self-affine fractal functions to quantitatively characterize the intrinsic nonlinear dynamics and memory effects within the system, employing statistical inference methods to find the fractal fit for the images. The changes in box dimensions over time could characterize the phase transition process of the starling flight flocks. By analyzing the rate of change of fractal dimensions, we identify critical points corresponding to phase transitions during collective flight behavior. During the flight of the starling flocks, a real-time phase transition process for evading attacks and effective advancement has been identified. Experimental data confirms the effectiveness of controlling the phase transition.
Keywords: self-affine fractal functions; fractional calculus; box dimension; differential equation; statistical inference; phase transition self-affine fractal functions; fractional calculus; box dimension; differential equation; statistical inference; phase transition

Share and Cite

MDPI and ACS Style

Li, K.; Zhang, X.; Yao, K.; Zhang, K.; Sun, M.; He, M.; Liu, K.; Wang, Y. Modeling Phase Transitions in Starling Flocks Using Fractal Dimension of Self-Affine Functions. Fractal Fract. 2026, 10, 17. https://doi.org/10.3390/fractalfract10010017

AMA Style

Li K, Zhang X, Yao K, Zhang K, Sun M, He M, Liu K, Wang Y. Modeling Phase Transitions in Starling Flocks Using Fractal Dimension of Self-Affine Functions. Fractal and Fractional. 2026; 10(1):17. https://doi.org/10.3390/fractalfract10010017

Chicago/Turabian Style

Li, Kunyuan, Xiongwei Zhang, Kui Yao, Kai Zhang, Meng Sun, Ming He, Kefeng Liu, and Yangjun Wang. 2026. "Modeling Phase Transitions in Starling Flocks Using Fractal Dimension of Self-Affine Functions" Fractal and Fractional 10, no. 1: 17. https://doi.org/10.3390/fractalfract10010017

APA Style

Li, K., Zhang, X., Yao, K., Zhang, K., Sun, M., He, M., Liu, K., & Wang, Y. (2026). Modeling Phase Transitions in Starling Flocks Using Fractal Dimension of Self-Affine Functions. Fractal and Fractional, 10(1), 17. https://doi.org/10.3390/fractalfract10010017

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