Extending the 3D Dong chaotic system to a 5D hyperchaotic system with three positive Lyapunov exponents: Dynamics, multistability, offset boosting, and PSO-DE based complexity optimization

Khaled Benkouider; Mohammed A. Saleh; Aceng Sambas; Sulaiman M. Ibrahim; Abdulgader Z. Almaymuni; Badr Almutairi; D. Iranian; Khaled Benkouider; Mohammed A. Saleh; Aceng Sambas; Sulaiman M. Ibrahim; Abdulgader Z. Almaymuni; Badr Almutairi; D. Iranian

doi:10.3934/math.2026595

AIMS Mathematics

2026, Volume 11, Issue 5: 14522-14546. doi: 10.3934/math.2026595

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Research article

Extending the 3D Dong chaotic system to a 5D hyperchaotic system with three positive Lyapunov exponents: Dynamics, multistability, offset boosting, and PSO-DE based complexity optimization

1.
Department of Sciences and Technology, Institute of Sciences, University Center of Barika M'doukal Road, 05001 Barika, Algeria
2.
Department of Cybersecurity, College of Computer, Qassim University, Saudi Arabia
3.
Artificial Intelligence Research Centre for Islam and Sustainability (AIRIS), Universiti Sultan Zainal Abidin, Gongbadak, Terengganu 21300, Malaysia
4.
Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu 602105, India
5.
Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya 46196, Indonesia
6.
College of Applied and Health Sciences, A'Sharqiyah University, Ibra 400, Sultanate of Oman
7.
College of Computer and Information Sciences, Majmaah University, Almajmaah, Saudi Arabia

Received: 26 January 2026 Revised: 07 May 2026 Accepted: 14 May 2026 Published: 22 May 2026
MSC : 37D45, 37M25, 37N35, 93C10

In this paper, we propose a new five-dimensional system that is capable of producing multistability and hyperchaos with three positive Lyapunov exponents (LEs) for specific parameter settings. The proposed system was derived by making an extension of Dong's chaotic system with changes that increase its dimension, complexity, and applicability. Using numerical simulations, we confirmed that the model produces hyperchaos with three positive LEs depending on parameter values, which means that its dynamics expands in three directions of the phase space. Regions of periodicity, chaos, hyperchaos with two positive LEs, and hyperchaos with three positive LEs were identified by LE spectra and verified using phase diagrams. The ability of the system to generate coexisting attractors is also shown, where the system demonstrates various behaviors for various initial conditions at fixed parameters. Furthermore, offset boosting control is presented to show that the attractors can be shifted without altering the internal dynamics of the system. Moreover, a topological complexity optimization framework is proposed to maximize the Kaplan–Yorke dimension (KYD) using Particle Swarm Optimization (PSO) and Differential Evolution (DE), followed by the 0–1 test and Approximate Entropy calculation. With its multistability, properties, hyperchaos with three positive LEs, controllable offset boosting, and optimized complexity, the novel model has excellent potential for practical applications.
- hyperchaos,
- Lyapunov exponent,
- multistability,
- coexisting attractors,
- offset boosting,
- optimization method
Citation: Khaled Benkouider, Mohammed A. Saleh, Aceng Sambas, Sulaiman M. Ibrahim, Abdulgader Z. Almaymuni, Badr Almutairi, D. Iranian. Extending the 3D Dong chaotic system to a 5D hyperchaotic system with three positive Lyapunov exponents: Dynamics, multistability, offset boosting, and PSO-DE based complexity optimization[J]. AIMS Mathematics, 2026, 11(5): 14522-14546. doi: 10.3934/math.2026595

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Abstract

In this paper, we propose a new five-dimensional system that is capable of producing multistability and hyperchaos with three positive Lyapunov exponents (LEs) for specific parameter settings. The proposed system was derived by making an extension of Dong's chaotic system with changes that increase its dimension, complexity, and applicability. Using numerical simulations, we confirmed that the model produces hyperchaos with three positive LEs depending on parameter values, which means that its dynamics expands in three directions of the phase space. Regions of periodicity, chaos, hyperchaos with two positive LEs, and hyperchaos with three positive LEs were identified by LE spectra and verified using phase diagrams. The ability of the system to generate coexisting attractors is also shown, where the system demonstrates various behaviors for various initial conditions at fixed parameters. Furthermore, offset boosting control is presented to show that the attractors can be shifted without altering the internal dynamics of the system. Moreover, a topological complexity optimization framework is proposed to maximize the Kaplan–Yorke dimension (KYD) using Particle Swarm Optimization (PSO) and Differential Evolution (DE), followed by the 0–1 test and Approximate Entropy calculation. With its multistability, properties, hyperchaos with three positive LEs, controllable offset boosting, and optimized complexity, the novel model has excellent potential for practical applications.

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