A new three-dimensional chaotic system

a single four-scroll attractor

Authors

  • Amir Hosein Refahi Sheikhani Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
  • Peyman Gholamin Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran https://orcid.org/0000-0001-8599-9981
  • Alireza Ansari https://orcid.org/0000-0003-4387-3779

Keywords:

Chaos, New chaotic system, Lyapunov exponents, Poincar´e map, Compound structure.

Abstract

In this paper, a new three-dimensional autonomous chaotic system is presented with nine terms including three multipliers, which is different from the Lorenz system and other existing systems. Basic dynamical properties of the new system are analyzed via equilibria, Lyapunov exponent spectrum, a dissipative system, phase portraits, the Poincar map and bifurcation diagrams. The compound structures of this new system are also analyzed. The theoretical analysis and numerical simulation validate the main results of this paper.

References

AMINIKHAH, H.; REFAHI SHEIKHANI, A. H.; REZAZADEH, H. Stability Analysis of Distributed Order fractional Chen System, The Scientific World Journal, v. 2013, ID:645080, p. 1-14, 2013.

AMINIKHAH, H.; REFAHI SHEIKHANI, A. H.; REZAZADEH, H. Stability analysis of linear distributed order system with multiple time delays, U.P.B. Sci. Bull. Series A, v. 77, i. 2, p. 207-218, 2015a.

AMINIKHAH, H.; REFAHI SHEIKHANI, A. H.; REZAZADEH, H. Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Boletim da Sociedade Paranaense de Matemática, v. 34, i. 2, p. 213-229, 2015b.

AMINIKHAH, H.; REFAHI SHEIKHANI, A. H.; REZAZADEH, H. Approximate analytical solutions of distributed order fractional Riccati differential equation, Ain Shams Engineering Journal, http://dx.doi.org/10.1016/j.asej.2016.03.007, 2016a

AMINIKHAH, H.; REFAHI SHEIKHANI, A. H.; REZAZADEH, H. Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, Scientia Iranica, Transaction B, Mechanical Engineering, v. 23, i. 3, p. 1048-1054, 2016b.

ANSARI, A.; REFAHI SHEIKHANI, A. H. New identities for the wright and the mittag-leffler functions using the laplace transform, Asian-European Journal of Mathematics, v. 7, i. 3, ID:1450038, p. 1-8, 2014.

BUSCARINO, A.; FORTUNA, L.; FRASCA, M.; GAMBUZZA, L. V. A Chaotic Circuit Based on Hewlett-Packard Memristor, Chaos, v. 22, i. 2: 023136, p. 1-9, 2012.

CHEN, G.; DONG, X. From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific, Singapore, 1998.

CHEN, G.; LAI, D. Feedback anticontrol of discrete chaos, Int. J. Bifurc. Chaos, v. 8, p. 1585-1590, 1998.

CHEN, G.; UETA, T. Yet another chaotic attractor, Internat. J. Bifur. Chaos, v. 9, i. 7, p. 1465-1466, 1999.

GUAN, Z. H.; LIU, N. Generating chaos for discrete time delayed systems via impulsive control, Chaos, v. 20, i. 1 :013135, p. 1-6, 2010.

DAFTARDAR-GEJJI, V.; BHALEKAR, S. Chaos in fractional ordered Liu system, Comput. Math. Appl., v. 59, p. 1117-1127, 2010.

KHALIL, H. K. Nonlinear Systems, Prentice-Hall, 3nd edition, 1992.

KILIÇ, R.; SARAÇOGLU, Ö. G.; YILDIRIM, F. A new nonautonomous version of Chua's circuit: Experimental observations, Journal of the Franklin Institute, v. 343, p. 191-203, 2006.

LIU, Y. J. Analysis of global dynamics in an unusual 3D chaotic system, Nonlinear Dyn., v. 70, p. 2203-2212, 2012.

LIU, N.; GUAN, Z. H. Chaotification for a class of cellular neural networks with distributed delays, Phys. Lett. A, v. 375, p. 463-467, 2011.

LIU, C.; LIU, T.; LIU, L.; LIU, K. A new chaotic attractor, Chaos, Solitons and Fractals, v. 22, i. 5, p. 1031-1038, 2004.

LORENZ, E. N. Deterministic nonperiodic flow, J. Atmos. Sci., v. 20, i. 1, p. 130-141, 1963.

LÜ, J.; CHEN, G. A new chaotic attractor coined, Internat. J. Bifur. Chaos, v. 12, i. 3, p. 659-661, 2002.

LÜ, J.; CHEN, G.; ZHANG, S. The compound structure of a new chaotic attractor, Chaos, Solitons and Fractals, v. 14, p. 669-672, 2002.

MASHOOF, M.; REFAHI SHEIKHANI, A. H. Simulating the solution of the distributed order fractional differential equations by block-pulse wavelets, U.P.B. Sci. Bull. Series A, v. 79 p. 193-206, 2017a

MASHOOF, M.; REFAHI SHEIKHANI, A. H. A Collocation Method for Solving Fractional Order Linear System, Journal of the Indonesian Mathematical Society, v. 23, i. 1, p. 27-42, 2017b.

MASHOOF, M.; REFAHI SHEIKHANI, A. H.; SABERI NAJAFI, H. Stability analysis of distributed order Hilfer-Prabhakar differential equations, Hacettepe Journal of Mathematics and Statistics, accepted, 2017.

REFAHI SHEIKHANI, A. H.; ANSARI, A.; SABERI NAJAFI, H.; MEHRDOUST, F. Analytic study on linear systems of distributed order fractional differential equations, Le Matematiche, v. 67, i. 2, p. 3-13, 2012.

REFAHI SHEIKHANI, A. H.; MASHOOF, M. Numerical solution of Fractional differential equation by Wavelets and Hybrid functions, Boletim da Sociedade Paranaense de Matemática, v. 36, i. 2, p. 231-244, 2017.

REZAZADEH, H.; AMINIKHAH, H.; REFAHI SHEIKHANI, A. H. Stability analysis of Hilfer fractional differential systems, Mathematical Communications v. 21, p. 45-64, 2016.

RÖSSLER, O. E. An equation for continuous chaos, Phys. Lett. A, v. 57, i. 5, p. 397-398, 1976.

SABERI NAJAFI, H.; EDALATPANAH, S. A.; REFAHI SHEIKHANI, A. H. Convergence Analysis of Modified Iterative Methods to Solve Linear Systems, Mediterranean Journal of Mathematics, v. 11, i. 3, p. 1019-1032, 2014.

SCHIFF, S. J.; JERGER, D. H.; DUONG, T. Controlling chaos in the brain, Nature, v. 370, p. 615-620, 1994.

SLOTINE, E. J. J.; LI, W. Applied Nonlinear Control, Prentice-Hallm, International editions, 1991.

STROGATZ, S. nonlinear dynamics and chaos, Perseus Books Publishing, L.L.C. 1994.

WANG, S.; KUANG, J.; LI, J.; LUO, Y.; LU, H.; HU, G. Chaos-Based Secure Communications in a Large Community, Phys. Rev. E, v. 66: 065202, p. 1-4, 2002.

WANG, J.; TANG, W.; ZHANG, J. Approximate Synchronization of Two Non-linear Systems via Impulsive Control, Proc. Inst. Mech. Eng. Part I, v. 226, p. 338-347, 2012.

WOLF, A.; SWIFT, J. B.; SWINNEY, H. L.; VASTANO, J. A. Determining Lyapunov exponents from a time series, Physica D, v. 16, p. 285-317, 1985.

WU, X.; LU, J. A.; CHI, K. T.; WANG, J.; LIU, J. Impulsive Control and Synchronization of the Lorenz Systems Family, Chaos, Solitons and Fractals, v. 31, p. 631-638, 2007.

ZHANG, W.; WONG, K. W.; YU, H.; ZHU, Z. L. An Image Encryption Scheme Using Reverse 2-Dimensional Chaotic Map and Dependent Diffusion, Commun. Nonlinear Sci. Numer. Simul., v. 18, p. 147-148, 2013.

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Published

22-08-2017

How to Cite

Refahi Sheikhani, A. H., Gholamin, P., & Ansari, A. (2017). A new three-dimensional chaotic system: a single four-scroll attractor. Sigmae, 6(1), 1–14. Retrieved from https://publicacoes.unifal-mg.edu.br/revistas/index.php/sigmae/article/view/496

Issue

Vol. 6 No. 1 (2017): "Good day sunshine"

Section

Pure Mathematics

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