Chaos modeling of a symmetrical Jerk system via analog circuit design and radial basis function neural network

Aceng Sambas; Hatem E. Semary; Abdullah Gokyildirim; Sadam Hussain; Rameshbabu Ramar; Sulaiman M. Ibrahim; A. S. Al-Moisheer; Rabiu Bashir Yunus; Aceng Sambas; Hatem E. Semary; Abdullah Gokyildirim; Sadam Hussain; Rameshbabu Ramar; Sulaiman M. Ibrahim; A. S. Al-Moisheer; Rabiu Bashir Yunus

doi:10.3934/math.2026067

AIMS Mathematics

2026, Volume 11, Issue 1: 1616-1636. doi: 10.3934/math.2026067

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Chaos modeling of a symmetrical Jerk system via analog circuit design and radial basis function neural network

1.
Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, 46196, Indonesia
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
3.
Department of Electrical and Electronics Engineering, Faculty of Engineering and Natural Sciences, Bandirma Onyedi Eylul University, Bandirma 10200, Balikesir, Turkey
4.
School of Science, Harbin Institute of Technology, Shenzhen 518055, China
5.
Department of Electronics and Communication Engineering, V.S.B. Engineering College, Karur, 639111, Tamil Nadu, India
6.
College of Applied and Health Sciences, A'Sharqiyah University, Post Box No. 42, Post Code No. 400 Ibra, Sultanate of Oman
7.
Department of Mathematics, Faculty of Computing and Mathematical Sciences, Aliko Dangote University of Science and Technology Wudil, Kano, 713101, Nigeria

Received: 14 September 2025 Revised: 21 November 2025 Accepted: 25 November 2025 Published: 19 January 2026
MSC : 34C28, 37C10, 37D45, 94C05, 68T07

In this paper, we investigated the chaotic behavior of a Jerk system proposed by Sambas et al. (2024), which features symmetrical attractors arising from the interplay of sinusoidal, hyperbolic, and absolute nonlinearities. The system's complex dynamics were analyzed using established numerical methods such as phase portraits, stability analysis, bifurcation diagrams, and Lyapunov exponents. Furthermore, through amplitude modulation, we showed that the control parameter δ can enhance or attenuate signal amplitudes without disrupting the system's stability or chaotic nature. The theoretical findings were further validated through Multisim circuit simulations, with experimental attractors closely matching the numerical results. In addition, a Radial Basis Function Neural Network (RBFNN) was implemented to approximate the chaotic trajectories of the system. The model was trained using simulated data and optimized via the least squares method. Network performance was evaluated using Root Mean Square Error (RMSE) and relative error. The results showed that the RBFNN accurately predicts the system's state variables, achieving MSE values on the order of 10^-10–10^-9 and relative error below 1.1 × 10^-8.
- chaotic system,
- Jerk circuit,
- circuit design,
- RBFNN
Citation: Aceng Sambas, Hatem E. Semary, Abdullah Gokyildirim, Sadam Hussain, Rameshbabu Ramar, Sulaiman M. Ibrahim, A. S. Al-Moisheer, Rabiu Bashir Yunus. Chaos modeling of a symmetrical Jerk system via analog circuit design and radial basis function neural network[J]. AIMS Mathematics, 2026, 11(1): 1616-1636. doi: 10.3934/math.2026067

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Abstract

In this paper, we investigated the chaotic behavior of a Jerk system proposed by Sambas et al. (2024), which features symmetrical attractors arising from the interplay of sinusoidal, hyperbolic, and absolute nonlinearities. The system's complex dynamics were analyzed using established numerical methods such as phase portraits, stability analysis, bifurcation diagrams, and Lyapunov exponents. Furthermore, through amplitude modulation, we showed that the control parameter δ can enhance or attenuate signal amplitudes without disrupting the system's stability or chaotic nature. The theoretical findings were further validated through Multisim circuit simulations, with experimental attractors closely matching the numerical results. In addition, a Radial Basis Function Neural Network (RBFNN) was implemented to approximate the chaotic trajectories of the system. The model was trained using simulated data and optimized via the least squares method. Network performance was evaluated using Root Mean Square Error (RMSE) and relative error. The results showed that the RBFNN accurately predicts the system's state variables, achieving MSE values on the order of 10^-10–10^-9 and relative error below 1.1 × 10^-8.

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