Abstract
In this paper, an integer- and fractional-order form of a four-dimensional (4-D) chaotic system with hidden attractors is investigated using theoretical/numerical and analogue methods. The system is constructed not through the extension of a three-dimensional existing nonlinear system as in current approaches, but by modifying the well-known two-dimensional Lotka-Volterra system. The equilibrium point of the integer-order system is determined and its stability analysis is studied using Routh-Hurwitz criterion. When the selected bifurcation parameter is varied, the system exhibits various dynamical behaviors and features including intermittency route to chaos, chaotic bursting oscillations and offset boosting. Moreover, the fractional-order form of the system is examined through bifurcation analysis. It is revealed that chaotic behaviors still exist in the system with order less than four. To validate the numerical approaches, a corresponding electronic circuit for the model in its integer and fractional order form is designed and implemented in Orcard-Pspice software. The Pspice results are consistent with those from the numerical simulations.
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References
P. Gaspard, Physica A 263, 315 (1999)
M. Kyriazis, Exp. Gerontology 26, 569 (1991)
J.C. Sprott, J.A. Vano, J.C. Wildenberg, M.B. Anderson, J.K. Noel, Phys. Lett. A 335, 207(2005)
K. Aihira, T. Takabe, M. Toyoda, Phys. Lett. A 144, 333 (1990)
S. Lankalapalli, A. Ghosal, Int. J. Bifurc. Chaos 7, 707 (1997)
H.T. Yau, C.S. Shieh, Nonlinear Anal.: Real World Appl. 9, 1800 (2008)
A.E. Matouk, H.N. Agiza, J. Math. Anal. Appl. 341, 259 (2008)
T.I. Chien, T.L. Liao, Chaos Solitons Fractals 24, 241 (2005)
Q. Guoyuan, C. Guanrong, Phys. Lett. A 352, 386 (2006)
S. Bouali, Nonlinear Dyn. 70, 2375 (2012)
V.T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, Eur. Phys. J. Special Topics 224, 1507. (2015)
S. Bouali, Ann. Rev. Chaos Theory Bifurcations Dyn. Syst. 6, 48 (2016)
S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 79 (2013)
V.T. Pham, S. Jafari, C. Volos, T. Kapitaniak, Chaos Solitons Fractals 93, 58 (2016)
X. Wang, G. Chen, Nonlinear Dyn. 71, 429 (2013)
V.T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, S.T. Kingni, Optik 127, 3259 (2016)
V.T. Pham, S. Jafari, X. Wang, J. Ma, Int. J. Bifurc. Chaos 26, 1650069 (2016)
A. Bayani, K. Rajagopal, A.J.M. Khalaf, S. Jafari, G.D. Leutcho, J. Kengne, Phys. Lett. A 383, 1450 (2019)
W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical recipes (Cambridge University Press, 1992)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Wastano, Physica D 16, 285 (1985)
V.T. Pham, C. Volos, S. Jafari, X. Wang, S. Vaidyanathan, Optoelectron. Adv. Mater. Rapid Commun. 8, 1 (2014)
A.L. Fitch, D. Yu, H.H.C. Iu, V. Sreeram, Int. J. Bifurc. Chaos 22, 1250133 (2012)
C. Li, J.C. Sprott, Nonlinear Dyn. 73, 1335 (2013)
C. Li, J.C. Sprott, Optik 27, 10389 (2016)
V. Kamdoum Tamba, R. Karthikeyan, V.T. Pham, D.V. Hoang, Adv. Math. Phys.
A. Schmidt, L. Gaul, Nonlinear Dyn. 29, 37 (2002)
I. Schafer, K. Kruger, J. Phys. D 41, 1 (2008)
S. Faraji, M.S. Tavazoei, Cent. Eur. J. Phys. 11, 836 (2013)
G.S. Mbouna Ngueuteu, P. Woafo, Mech. Res. Commun. 46, 20 (2012)
I.S. Jesus, J.A.T. Machado, Nonlinear Dyn. 56, 45 (2009)
R. Karthikeyan, N. Fahimeh, J. Sajad, K. Anitha, Eur. Phys. J. Special Topics 226, 3827 (2017)
X. Wang, S. Takougang Kingni, C. Volos, V.T. Pham, D.V. Hoang, S. Jafari, Int. J. Electron. 106, 109 (2018)
B. Atiyeh, J. Mohammad Ali, R. Karthikeyan, J. Haibo, J. Sajad, Eur. Phys. J. Special Topics 226, 3729 (2017)
R. Karthikeyan, N. Fahimeh, G. Laarem, K. Anitha, S. Ashokkumar, J. Sajad, J. Circuits Syst. Comput. (2019)
I. Grigorenko, E. Grigorenko, Phys. Rev. Lett. 91, 034101 (2003)
I. Petras, Fractional-order nonlinear systems: modeling, analysis and simulation (Springer, 2011)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, Calif, USA, 1999)
C.F. Lorenzo, T.T. Hartley, Int. J. Appl. Math. 3, 249 (2000)
C.F. Lorenzo, T.T. Hartley, Nonlinear Dyn. 29, 57 (2002)
I. Podlubny, A.M.A. El-Sayed, On Two Definition of Fractional Calculus (Solvak Academy of science institute of Experimental Phys., 1996)
A.M. Concepcion, Y.Q. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-order Systems and Controls (Springer, London, 2010)
I. Podlubny, Fractional Differential Equations (NY, 1999).
E. Zambrano-Serrano, E. Campos-Canton, J.M. Munoz-Pacheco, Nonlinear Dyn. 83, 1629 (2016)
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Tamba, V.K., Kom, G.H., Kingni, S.T. et al. Analysis and electronic circuit implementation of an integer- and fractional-order four-dimensional chaotic system with offset boosting and hidden attractors. Eur. Phys. J. Spec. Top. 229, 1211–1230 (2020). https://doi.org/10.1140/epjst/e2020-900169-1
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DOI: https://doi.org/10.1140/epjst/e2020-900169-1