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Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation

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Abstract

This paper reports the results of the analytical, numerical and analogical analyses of integer- and fractional-order chaotic systems with hyperbolic sine nonlinearity (HSN). By varying a parameter, the integer order of the system displays transcritical bifurcation and new complex shapes of bistable double-scroll chaotic attractors and four-scroll chaotic attractors. The coexistence among four-scroll chaotic attractors, a pair of double-scroll chaotic attractors and a pair of point attractors is also reported for specific parameter values. Numerical results indicate that commensurate and incommensurate fractional orders of the systems display bistable double-scroll chaotic attractors, four-scroll chaotic attractors and coexisting attractors between a pair of double-scroll chaotic attractors and a pair of point attractors. Moreover, the physical existence of chaotic attractors and coexisting attractors found in the integer-order and commensurate fractional-order chaotic systems with HSN is verified using PSIM software. Numerical simulations and PSIM results have a good qualitative agreement. The results obtained in this work have not been reported previously in three-dimensional autonomous system with HSN and thus represent an enriching contribution to the understanding of the dynamics of this class of systems. Finally, combination synchronisation of such three-coupled identical commensurate fractional-order chaotic systems is analysed using the active backstepping method.

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Authors and Affiliations

  1. Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  2. Department of Electrical Engineering, Fotso Victor University Institute of Technology (IUT-FV), University of Dschang, P.O. Box 134, Bandjoun, Cameroon

    Justin Roger Mboupda Pone

  3. Department of Physics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bamenda, Cameroon

    Gaetan Fautso Kuiate

  4. Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Yen Nghia, Ha Dong District, Hanoi,  100000, Vietnam

    Viet-Thanh Pham

  5. Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, 167 Hoang Ngan, Hanoi,  100000, Vietnam

    Viet-Thanh Pham

Authors
  1. Sifeu Takougang Kingni
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  2. Justin Roger Mboupda Pone
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  3. Gaetan Fautso Kuiate
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  4. Viet-Thanh Pham
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Correspondence to Justin Roger Mboupda Pone.

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Kingni, S.T., Pone, J.R.M., Kuiate, G.F. et al. Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation. Pramana - J Phys 93, 12 (2019). https://doi.org/10.1007/s12043-019-1786-3

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  • DOI: https://doi.org/10.1007/s12043-019-1786-3

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