NUMERICAL INVESTIGATION OF FRACTAL OSCILLATOR FOR A PENDULUM WITH A ROLLING WHEEL

Shaoqing Zheng (), Yingjun Lou (), Shaowei Shen () and Junfeng Lu
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Shaoqing Zheng: Department of Statistics, Zhejiang Gongshang University, Hangzhou College of Commerce, Hangzhou 311508, P. R. China
Yingjun Lou: Department of Finance, Zhejiang Gongshang University, Hangzhou College of Commerce, Hangzhou 311508, P. R. China
Shaowei Shen: School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, P. R. China
Junfeng Lu: Department of Statistics, Zhejiang Gongshang University, Hangzhou College of Commerce, Hangzhou 311508, P. R. China

FRACTALS (fractals), 2025, vol. 33, issue 09, 1-11

Abstract: In this paper, we consider a fractal oscillator for modeling the motion of a pendulum attached to a rolling wheel. It is a fractal oscillation system defined by He’s fractal derivative. The difficulty for solving this fractal differential equation results from its nonlinear parts and fractal operators. By using Taylor series approximation, an approximated fractal oscillation equation is obtained. A combined technique based upon two-scale fractal theory and harmonic method is proposed for solving the corresponding approximated system. By applying the fractal complex transformation, the fractal equation is approximately transformed as an ordinary second-order differential equation. The fractal or conventional approximations are given with the help of the spreading harmonic balance method. Numerical comparisons with Runge–Kutta method and sensitivity analysis of the approximated solutions and frequencies are presented to confirm the stability and efficiency of the proposed approach.

Keywords: Oscillator; Frequency; Approximation; Fractal Complex Transformation; Spreading Residue Harmonic Balance Method (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X2550077X

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