A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis
Abstract
:1. Introduction
2. Formulation of a New 3D Chaotic Model and the Main Results
Numerical Result
3. Hopf Bifurcation and Proofs of Propositions 4 and 5
4. Existence of Heteroclinic Orbit and Proofs of Propositions 6–9
- (i)
- If ∃, such that and , then is one of the equilibrium points of system (1).
- (ii)
- If and , then , .
- (i)
- If ∃, such that and , then is one of the equilibrium points of system (1).
- (ii)
- If (or , or ), and (or , or ), then (or , or ) , .
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Zhang, X.; Chen, G. Constructing an autonomous system with infinitely many chaotic attractors. Chaos Interdiscip. J. Nonlinear Sci. 2017, 27, 071101. [Google Scholar]
- Yang, T.; Yang, Q. A 3D autonomous system with infinitely many chaotic attractors. Int. J. Bifurc. Chaos 2019, 29, 1950166. [Google Scholar]
- Zhang, X. Boundedness of a class of complex Lorenz systems. Int. J. Bifurc. Chaos 2021, 31, 2150101. [Google Scholar] [CrossRef]
- Wang, H.; Ke, G.; Pan, J.; Su, Q. Conjoined Lorenz-Like Attractors Coined. Miskolc Mathematical Notes, Code: MMN-4489. 2023. Available online: http://mat76.mat.uni-miskolc.hu/mnotes/forthcoming?volume=0&number=0 (accessed on 19 March 2025).
- Wang, H.; Pan, J.; Ke, G. Revealing more hidden attractors from a new sub-quadratic Lorenz-like system of degree 65. Int. J. Bifurc. Chaos 2024, 34, 2450071. [Google Scholar] [CrossRef]
- Kuate, P.; Fotsin, H. Complex dynamics induced by a sine nonlinearity in a five-term chaotic system FPGA hardware design and synchronization. Chaos 2020, 30, 123107. [Google Scholar] [PubMed]
- Sahoo, S.; Roy, B.K. Design of multi-wing chaotic systems with higher largest Lyapunov exponent. Chaos Solitons Fractals 2022, 157, 111926. [Google Scholar]
- Olkhov, V. Expectations, price fluctuations and Lorenz attractor. MPRA Pap. 2018, 1–29. Available online: https://mpra.ub.uni-muenchen.de/89222/ (accessed on 19 March 2025).
- Zhang, X.; Chen, G. Impulsive systems with growing numbers of chaotic attractors. Chaos Interdiscip. J. Nonlinear Sci. 2022, 32, 071102. [Google Scholar] [CrossRef] [PubMed]
- Zhou, Z.; Tigan, G.; Yu, Z. Hopf bifurcations in an extended Lorenz system. Adv. Diff. Eqs. 2017, 28, 1–10. [Google Scholar] [CrossRef]
- Kuzenetsov, Y.A. Elements of Applied Bifurcation Theory, 3rd ed.; Springer: New York, NY, USA, 2004; Volume 112. [Google Scholar]
- Sotomayor, J.; Mello, L.F.; Braga, D.C. Lyapunov coefficients for degenerate Hopf bifurcations. arXiv 2007, arXiv:0709.3949. [Google Scholar]
- Wang, H.; Pan, J.; Hu, F.; Ke, G. Asymmetric singularly degenerate heteroclinic cycles. Int. J. Bifurc. Chaos 2025, 2550072. [Google Scholar] [CrossRef]
- Pan, J.; Wang, H.; Hu, F. Revealing asymmetric homoclinic and heteroclinic orbits. Electron. Res. Arch. 2025, 33, 1337–1350. [Google Scholar]
- Li, T.; Chen, G.; Chen, G. On homoclinic and heteroclinic orbits of the Chen’s system. Int. J. Bifurc. Chaos 2006, 16, 3035–3041. [Google Scholar] [CrossRef]
- Tigan, G.; Llibre, J. Heteroclinic, homoclinic and closed orbits in the Chen system. Int. J. Bifurc. Chaos 2016, 26, 1650072. [Google Scholar] [CrossRef]
- Chen, Y.; Yang, Q. Dynamics of a hyperchaotic Lorenz-type system. Nonlinear Dyn. 2014, 77, 569–581. [Google Scholar] [CrossRef]
- Tigan, G.; Constantinescu, D. Heteroclinic orbits in the T and the Lü system. Chaos Solitons Fractals 2009, 42, 20–23. [Google Scholar] [CrossRef]
- Liu, Y.; Yang, Q. Dynamics of a new Lorenz-like chaotic system. Nonl. Anal. RWA 2010, 11, 2563–2572. [Google Scholar]
- Liu, Y.; Pang, W. Dynamics of the general Lorenz family. Nonlinear Dyn. 2012, 67, 1595–1611. [Google Scholar] [CrossRef]
- Li, X.; Ou, Q. Dynamical properties and simulation of a new Lorenz-like chaotic system. Nonlinear Dyn. 2011, 65, 255–270. [Google Scholar] [CrossRef]
- Li, X.; Wang, P. Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system. Nonlinear Dyn. 2013, 73, 621–632. [Google Scholar]
n | Property of | |
---|---|---|
>0 | <0 | A 2D and a 1D |
=0 | A 1D and a 2D | |
>0 | A 1D , a 1D , and a 1D | |
=0 | <0 | A 3D |
=0 | A 3D | |
>0 | A 1D , a 1D , and a 1D | |
<0 | <0 | A 1D and a 2D |
=0 | A 2D and a 1D | |
>0 | A 1D , a 1D , and a 1D |
n | Property of | |
---|---|---|
>0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 2D and a 1D | |
>0 | A 1D and a 2D | |
=0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 3D | |
>0 | A 3D | |
<0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 2D and a 1D | |
>0 | A 1D and a 2D |
Property of | ||
---|---|---|
>0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 1D and a 2D | |
>0 | A 2D and a 1D | |
=0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 3D | |
>0 | A 3D | |
<0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 2D and a 1D | |
>0 | A 1D and a 2D |
Property of | ||
---|---|---|
>0 | <0 | A 1D and a 2D |
=0 | A 1D and a 2D | |
>0 | A 1D , a 1D , and a 1D | |
=0 | <0 | A 3D |
=0 | A 3D | |
>0 | A 1D , a 1D , and a 1D | |
<0 | <0 | A 1D and a 2D |
=0 | A 2D and a 1D | |
>0 | A 1D , a 1D , and a 1D |
Property of | ||
---|---|---|
>0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 1D and a 2D | |
>0 | A 2D and a 1D | |
=0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 3D | |
>0 | A 3D | |
<0 | <0 | A 1D , a 1D , and a 1D |
=0 | A 2D and a 1D | |
>0 | A 1D and a 2D |
Property of | ||
---|---|---|
>0 | <0 | A 1D and a 2D |
=0 | A 1D and a 2D | |
>0 | A 1D , a 1D , and a 1D | |
=0 | <0 | A 3D |
=0 | A 3D | |
>0 | A 1D , a 1D , and a 1D | |
<0 | <0 | A 1D and a 2D |
=0 | A 2D and a 1D | |
>0 | A 1D , a 1D , and a 1D |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Pan, J.; Wang, H.; Ke, G.; Hu, F. A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis. Axioms 2025, 14, 264. https://doi.org/10.3390/axioms14040264
Pan J, Wang H, Ke G, Hu F. A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis. Axioms. 2025; 14(4):264. https://doi.org/10.3390/axioms14040264
Chicago/Turabian StylePan, Jun, Haijun Wang, Guiyao Ke, and Feiyu Hu. 2025. "A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis" Axioms 14, no. 4: 264. https://doi.org/10.3390/axioms14040264
APA StylePan, J., Wang, H., Ke, G., & Hu, F. (2025). A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis. Axioms, 14(4), 264. https://doi.org/10.3390/axioms14040264