Numerical simulation of chaotic dynamics in a fractional-order vibration model with Grünwald-Letnikov fractional derivative

Jiaxin Zhang; Wei Zhang; Xiaoyu Li; Jiaxin Zhang; Wei Zhang; Xiaoyu Li

doi:10.3934/nhm.2025027

Networks and Heterogeneous Media

2025, Volume 20, Issue 2: 625-647. doi: 10.3934/nhm.2025027

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Research article

Numerical simulation of chaotic dynamics in a fractional-order vibration model with Grünwald-Letnikov fractional derivative

Jiaxin Zhang ¹,
Wei Zhang ^{2
,
,},
Xiaoyu Li ^{3
,
,}

1.
School of Information and Network Engineering, Anhui Science and Technology University, Bengbu 233030, China
2.
Institute of Economics and Management, Jining Normal University, Ulanqab 012000, China
3.
College of Date Science and Application, Inner Mongolia University of Technology, Hohhot 010080, China

Received: 01 March 2025 Revised: 01 May 2025 Accepted: 16 May 2025 Published: 03 June 2025

This paper investigates the chaotic dynamics in a fractional-order vocal fold vibration (VCV) model based on the Grünwald-Letnikov fractional derivative (GLFD). Studying the characteristics of vocal fold vibration is of great significance for revealing its vibration mechanism, the etiology of abnormal vibrations, and natural speech synthesis. Traditional vocal fold vibration models are based on integer-order systems and are unable to describe the memory effects present in real physical systems. To overcome this limitation, this paper introduces fractional derivatives and develops a high-precision numerical method to simulate the fractional-order VCV model. By incorporating nonlinear elastic and damping forces, the model can more accurately describe the complex dynamic characteristics of vocal fold vibrations, including memory effects and non-locality. The numerical simulation results reveal novel chaotic behaviors in the fractional-order VCV model, which have not been observed in integer-order models. These findings provide new insights into the possible dynamic states of vocal fold vibrations and lay the foundation for further theoretical and experimental studies on the vocal cord vibration mechanism.
- chaotic dynamic behavior,
- high precision scheme,
- Grünwald-Letnikov fractional derivative,
- vocal cord vibration model
Citation: Jiaxin Zhang, Wei Zhang, Xiaoyu Li. Numerical simulation of chaotic dynamics in a fractional-order vibration model with Grünwald-Letnikov fractional derivative[J]. Networks and Heterogeneous Media, 2025, 20(2): 625-647. doi: 10.3934/nhm.2025027

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Abstract

This paper investigates the chaotic dynamics in a fractional-order vocal fold vibration (VCV) model based on the Grünwald-Letnikov fractional derivative (GLFD). Studying the characteristics of vocal fold vibration is of great significance for revealing its vibration mechanism, the etiology of abnormal vibrations, and natural speech synthesis. Traditional vocal fold vibration models are based on integer-order systems and are unable to describe the memory effects present in real physical systems. To overcome this limitation, this paper introduces fractional derivatives and develops a high-precision numerical method to simulate the fractional-order VCV model. By incorporating nonlinear elastic and damping forces, the model can more accurately describe the complex dynamic characteristics of vocal fold vibrations, including memory effects and non-locality. The numerical simulation results reveal novel chaotic behaviors in the fractional-order VCV model, which have not been observed in integer-order models. These findings provide new insights into the possible dynamic states of vocal fold vibrations and lay the foundation for further theoretical and experimental studies on the vocal cord vibration mechanism.

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