A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

Muhamad Deni Johansyah

doi:10.46336/ijqrm.v2i1.126

A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

Muhamad Deni Johansyah

Abstract

This work introduces a new 3-D chaotic system with a line of equilibrium points. We carry out a detailed dynamic analysis of the proposed chaotic system with five nonlinear terms. We show that the chaotic system exhibits multistability with two coexisting chaotic attractors. We apply integral sliding mode control for the complete synchronization of the new chaotic system with itself as leader-follower systems.

Keywords

Chaos; chaotic systems; synchronization; line equilibrium

Full Text:

PDF

References

Abro, K. A. and Atangana, A.(2021). Numerical Study and Chaotic Analysis of Meminductor and Memcapacitor Through Fractal–Fractional Differential Operator.Arabian Journal for Science and Engineering,46, 857-871.

Baysal, V., Erkan, E., & Yilmaz, E. (2021). Impacts of autapse on chaotic resonance in single neurons and small-world neuronal networks. Philosophical Transactions of the Royal Society A, 379(2198), 20200237.

Belato, D., Weber, H. I., Balthazar, J. M., and Mook, D. T. (2001). Chaotic vibrations of a nonideal electro-mechanical system. International Journal of Solids and Structures, 38(10-13), 1699-1706.

Cai, G., and Tan, Z. (2007). Chaos synchronization of a new chaotic system via nonlinear control. Journal of Uncertain systems, 1(3), 235-240.

Chakraborty, P., and Poria, S. (2019). Extreme multistable synchronisation in coupled dynamical systems. Pramana, 93(2), 1-13.

Gao, S., Zhang, Y., Zhang, Y., and Zhang, G. (2020). Elman Neural Network Soft-Sensor Model of PVC Polymerization Process Optimized by Chaos Beetle Antennae Search Algorithm. IEEE Sensors Journal, 21(3), 3544-3551.

Jalal, A. A., Amen, A. I., and Sulaiman, N. A. (2020). Darboux integrability of the simple chaotic flow with a line equilibria differential system. Chaos, Solitons & Fractals, 135, 109712.

Jang, M. J., Chen, C. L., & Chen, C. O. K. (2002). Sliding mode control of chaos in the cubic Chua's circuit system. International Journal of Bifurcation and Chaos, 12(06), 1437-1449.

Moutsinga, C. R. B., Pindza, E., and Mare, E. (2020). A robust spectral integral method for solving chaotic finance systems. Alexandria Engineering Journal, 59(2), 601-611.

Ouannas, A., Karouma, A., Grassi, G., and Pham, V. T. (2021). A novel secure communications scheme based on chaotic modulation, recursive encryption and chaotic masking. Alexandria Engineering Journal, 60(1), 1873-1884.

Pan, J., Ding, Q., and Du, B. (2012). A new improved scheme of chaotic masking secure communication based on Lorenz system. International Journal of Bifurcation and Chaos, 22(05), 1250125.

Peng, C. C., and Chen, C. L. (2008). Robust chaotic control of Lorenz system by backstepping design. Chaos, Solitons & Fractals, 37(2), 598-608.

Sambas, A., Vaidyanathan, S., Zhang, S., Zeng, Y., Mohamed, M. A., and Mamat, M. (2019). A new double-wing chaotic system with coexisting attractors and line equilibrium: bifurcation analysis and electronic circuit simulation. IEEE Access, 7, 115454-115462.

Sambas, A., Vaidyanathan, S., Tlelo-Cuautle, E., Abd-El-Atty, B., Abd El-Latif, A. A., Guillén-Fernández, O., Hidayat, Y and Gundara, G. (2020). A 3-D multi-stable system with a peanut-shaped equilibrium curve: Circuit design, FPGA realization, and an application to image encryption. IEEE Access, 8, 137116-137132.

Sambas, A., Vaidyanathan, S., Zhang, S., Mohamed, M. A., Zeng, Y., and Azar, A. T. (2021a). A new 3-D chaotic jerk system with a saddle-focus rest point at the origin, its active backstepping control, and circuit realization. In Backstepping Control of Nonlinear Dynamical Systems (pp. 95-114). Academic Press.

Sambas, A., Vaidyanathan, S., Bonny, T., Zhang, S., Hidayat, Y., Gundara, G., and Mamat, M. (2021b). Mathematical Model and FPGA Realization of a Multi-Stable Chaotic Dynamical System with a Closed Butterfly-Like Curve of Equilibrium Points. Applied Sciences, 11(2), 788.

Sukono, Sambas, A., He, S., Liu, H., Vaidyanathan, S., Hidayat, Y., and Saputra, J. (2020). Dynamical analysis and adaptive fuzzy control for the fractional-order financial risk chaotic system. Advances in Difference Equations, 674(1), 1-12.

Qi, G., Chen, G., Du, S., Chen, Z., and Yuan, Z. (2005). Analysis of a new chaotic system. Physica A: Statistical Mechanics and its Applications, 352(2-4), 295-308.

Qin, C., Sun, K., and He, S. (2021). Characteristic Analysis of Fractional-Order Memristor-Based Hypogenetic Jerk System and Its DSP Implementation. Electronics, 10(7), 841.

Tlelo-Cuautle, E., de la Fraga, L. G., Pham, V. T., Volos, C., Jafari, S., and de Jesus Quintas-Valles, A. (2017). Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points. Nonlinear Dynamics, 89(2), 1129-1139.

Vaidyanathan, S. (2015). Adaptive control of the FitzHugh-Nagumo chaotic neuron model. International Journal of PharmTech Research, 8(6), 117-127.

Vaidyanathan, S., Sambas, A., Mamat, M., and Sanjaya, W. S. M. (2017). A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Archives of Control Sciences, 27(4), 541-554.

Vaidyanathan, S., Dolvis, L. G., Jacques, K., Lien, C. H., and Sambas, A. (2019). A new five-dimensional four-wing hyperchaotic system with hidden attractor, its electronic circuit realisation and synchronisation via integral sliding mode control. International Journal of Modelling, Identification and Control, 32(1), 30-45.

Vaidyanathan, S., Tlelo-Cuautle, E., Anand, P. G., Sambas, A., Guillén-Fernández, O., & Zhang, S. (2021). A new conservative chaotic dynamical system with lemniscate equilibrium, its circuit model and FPGA implementation. International Journal of Automation and Control, 15(2), 128-148.

Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3), 285-317.

Xia, Y., Wang, J., Meng, B., and Chen, X. (2020). Further results on fuzzy sampled-data stabilization of chaotic nonlinear systems. Applied Mathematics and Computation, 379, 125225.

Yassen, M. T. (2006). Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons & Fractals, 27(2), 537-548.

Zhou, L., Tan, F., Li, X., & Zhou, L. (2021). A fixed-time synchronization-based secure communication scheme for two-layer hybrid coupled networks. Neurocomputing, 433, 131-141.

DOI: https://doi.org/10.46336/ijqrm.v2i1.126