A New 3-D Chaotic System with Conch-Shaped Equilibrium Curve and its Circuit Implementation

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    This paper reports the finding a new chaotic system with a conch-shaped equilibrium curve. The proposed system is a new addition to existing chaotic systems with closed curves of equilibrium points in the literature. Lyapunov exponents of the new chaotic system are studied

    for verifying chaos properties and phase portraits of the new system via MATLAB are unveiled. An electronic circuit simulation of the new chaotic system with conch-shaped equilibrium curve is shown using MultiSIM to check the model feasibility.


  • Keywords


    Chaos, chaotic systems, circuit simulation, hidden attractors, Lyapunov exponents

  • References


      [1] S. Vaidyanathan, “Synchronization of Tokamak systems with symmetric and magnetically confined plasma via adaptive control”, International Journal of ChemTech Research, Vol. 8, No. 6, (2015), pp.818–827.

      [2] B. A. Idowu, S. Vaidyanathan, A. Sambas, O. I. Olusola and O. S. Onma, “A New Chaotic Finance System: Its Analysis, Control, Synchronization and Circuit Design”, Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 271–295.

      [3] O.I. Massoud, J. Huisman, E. Beninca, M.C. Dietze, W. Bouten, and J.A. Vrugt, “Probing the limits of predictability: data assimilation of chaotic dynamics in complex food webs”, Ecology Letters, Vol.21, No. 1, (2018), pp. 93–103.

      [4] M. Kim, U. Ha, K.J. Lee, Y. Lee, and H.J. Yoo, “A 82-nW Chaotic Map True Random Number Generator Based on a Sub-Ranging SAR ADC”, IEEE Journal of Solid-State Circuits, Vol. 52, No. 7, (2017), pp. 1953–1965.

      [5] S. Vaidyanathan, “Adaptive control design for the anti-synchronization of novel 3-D chemical chaotic reactor systems”, International Journal of ChemTech Research, Vol. 8, No. 11, (2015), pp. 654-668.

      [6] S. Sarwar and S. Iqbal, “Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction”, Chinese Journal of Physics, Vol. 56, No. 1, (2018), pp. 374-384.

      [7] C. H. Miwadinou, A. V. Monwanou, J. Yovogan, L. A. Hinvi, P. N. Tuwa and J. C. Orou, “Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions”, Chinese Journal of Physics, Vol. 56, No. 3, (2018), pp. 1089–1104.

      [8] S. Vaidyanathan, “Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method”, International Journal of ChemTech Research, Vol. 8, No. 7, (2015), pp. 209–221.

      [9] V. K. Yadav, S. Das, B. S. Bhadauria, A. K. Singh and M. Srivastava, “Stability analysis, chaos control of a fractional order chaotic chemical reactor system and its function projective synchronization with parametric uncertainties”, Chinese Journal of Physics, Vol. 55, No. 3, (2017), pp. 594–605.

      [10] S. Vaidyanathan, A. Sambas, M. Mamat, and W. S. M. Sanjaya, “A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot”, Archives of Control Sciences, Vol. 27, No. 4, (2017), pp. 541-554.

      [11] A. Sambas, S. Vaidyanathan, M. Mamat, W. S. M. Sanjaya and D. S. Rahayu, “A 3-D novel jerk chaotic system and its application in secure communication system and mobile robot navigation”, Studies in Computational Intelligence, Vol. 636, (2016), pp. 283-310.

      [12] S. Rasappan and S. Vaidyanathan, “Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control”, Far East Journal of Mathematical Sciences, Vol. 67, No. 2, (2012), pp. 265–287.

      [13] S. Vaidyanathan, C.K. Volos, K. Rajagopal, I.M. Kyprianidis and I. N. Stouboulos, “Adaptive backstepping controller design for the antisynchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation”, Journal of Engineering Science and Technology Review, Vol. 8, No. 2, (2015), pp. 74–82.

      [14] S. Vaidyanathan, A.T. Azar, K. Rajagopal, A. Sambas, S. Kacar and U. Cavusoglu, “A new hyperchaotic temperature fluctuations model, its circuit simulation, FPGA implementation and an application to image encryption”, International Journal of Simulation and Process Modelling, Vol. 13, No. 3, (2018), pp. 281–296.

      [15] V.T. Pham, C.K. Volos and S. Vaidyanathan, “Multi-scroll chaotic oscillator based on a first-order delay differential equation”, Studies in Computational Intelligence, Vol. 581, (2015), pp. 59–72.

      [16] S. Vaidyanathan, C.K. Volos and V.T. Pham, “Global chaos control of a novel nine-term chaotic system via sliding mode control”, Studies in Computational Intelligence, Vol. 576, (2015), pp. 571–590.

      [17] G. D. Leutcho and J. Kengne, “A unique chaotic snap system with a smoothly adjustable symmetry and nonlinearity: Chaos, offset boosting, antimonotonicity, and coexisting multiple attractors”, Chaos, Solitons and Fractals, Vol. 113, (2018), pp. 275–293.

      [18] A. Sambas, Mujiarto, M. Mamat, and W. S. M. Sanjaya, “Numerical simulation and circuit implementation for a Sprott chaotic system with one hyperbolic sinusoidal nonlinearity”, Far East Journal of Mathematical Sciences, Vol. 102, No. 6, (2017), pp. 1165–1177.

      [19] S. Vaidyanathan, “Hyperchaos, qualitative analysis, control and synchronization of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities”, International Journal of Modelling, Identification and Control, Vol. 23, No. 4, (2015), pp. 380–392.

      [20] S. Vaidyanathan, “Analysis, control, and synchronization of a 3-D novel jerk chaotic system with two quadratic nonlinearities”, Kyungpook Mathematical Journal, Vol. 55, No. 3, (2015), pp. 563–586.

      [21] S. Pakiriswamy and S. Vaidyanathan, “Generalized projective synchronization of three-scroll chaotic systems via active control”, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, Vol. 85, (2012), pp. 146–155.

      [22] S. Vaidyanathan, “Analysis and synchronization of the hyperchaotic Yujun systems via sliding mode control”, Advances in Intelligent Systems and Computing, Vol. 176, (2012), pp. 329–337.

      [23] J. Wang and G. Peng, “Bifurcation and chaos in discrete-time BVP oscillator”, International Journal of Non-Linear Mechanics, Vol. 45, No. 6, (2010), pp. 608–620.

      [24] S. Vaidyanathan and S. Pakiriswamy, “The design of active feedback controllers for the generalized projective synchronization of hyperchaotic Qi and hyperchaotic Lorenz systems”, Communications in Computer and Information Sciences, Vol. 245, (2011), pp. 231–238.

      [25] Y. Dai, H. Wang, and H. Sun, “Cyclic-shift chaotic medical image encryption algorithm based on plain text key-stream”, International Journal of Simulation: Systems, Science and Technology, Vol. 17, No. 27, (2016), pp. 24.1-24.8.

      [26] A. Ullah, S. S. Jamal and T. Shah, “A novel scheme for image encryption using substitution box and chaotic system”, Nonlinear Dynamics, Vol. 91, No. 1, (2018), pp. 359-370.

      [27] S. Vaidyanathan, A. Sambas, M. Mamat, and W. S. M. Sanjaya, “Analysis, synchronisation and circuit implementation of a novel jerk chaotic system and its application for voice encryption”, International Journal of Modelling, Identification and Control, Vol. 28, No. 2, (2017), pp. 153-166.

      [28] C.K. Volos, V.T. Pham, S. Vaidyanathan, I.M. Kyprianidis and I. N. Stouboulos, “Synchronization phenomena in coupled Colpitts circuits”, Journal of Engineering Science and Technology Review, Vol. 8, No. 2, (2015), pp. 142–151.

      [29] C. Jayawickrama, S. Kumar and H. Song, “Novel wideband chaotic approach LNA with microcontroller compatibility for 5G wireless secure communication”, Microwave and Optical Technology Letters, Vol. 60, No. 2, (2018), pp. 48-494.

      [30] A. Sambas, W. S. M. Sanjaya, and M. Mamat, “Design and numerical simulation of unidirectional chaotic synchronization and its application in secure communication system”, Journal of Engineering Science and Technology Review, Vol. 6, No. 4, (2013), pp. 66-73.

      [31] A. Sambas, W. S. M. Sanjaya, M. Mamat and R. P. Prastio, “Mathematical modelling of chaotic Jerk circuit and its application in secure communication System”, Studies in Fuzziness and Soft Computing, Vol. 337, (2016), pp. 133-153.

      [32] N. V. Kutnetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE”, Communications in Nonlinear Science and Numerical Simulation, Vol. 51, (2017), pp. 34-49.

      [33] G. A. Leonov, N. V. Kutnetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity”, Communications in Nonlinear Science and Numerical Simulation, Vol. 28, No. 3, (2017), pp. 166-174.

      [34] D. S. Dudkowski, S. jafari, T. Kapitaniak, N. V. Kutnetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems”, Physics Reports, Vol. 637, (2016), pp. 1–50.

      [35] V. T. Pham, Ch. K. Volos, S. Jafari and T. Kapitaniak, “Coexistence of hidden chaotic attractors in a novel no-equilibrium system”, Nonlinear Dynamics, Vol. 87, No. 3, (2017), pp. 2001–2010.

      [36] Z. L. Zuo and C. Li, “Multiple attractors and dynamic analysis of a no-equilibrium chaotic system”, Optik, Vol. 127, No. 19, (2016), pp. 7952–7957.

      [37] S. Vaidyanathan, V. T. Pham and Ch. K. Volos, “A 5-D hyperchaotic Rikitake dynamo system with hidden attractors”, European Physical Journal, Vol. 224, No. 8, (2015), pp. 1575–1592.

      [38] A. Sambas, M. Mamat, S. Vaidyanathan, M. A. Mohamed and W. S. Mada Sanjaya, “A new 4-D chaotic system with hidden attractor and its circuit implementation”, International Journal of Engineering & Technology, Vol. 7, No. 3, (2018), pp. 1245–1250.

      [39] J. Petrzela and T. Gotthans, “New chaotic dynamical system with a conic-shaped equilibrium located on the plane structure”, Applied Sciences, Vol. 7, No. 10, (2017), pp. 976–989.

      [40] V. T. Pham, S. Jafari Ch. K. Volos, S. Vaidyanathan and T. Kapitaniak, “A chaotic system with infinite equilibria located on a piecewise linear curve”, Optik, Vol. 127, No. 20, (2016), pp. 9111–9117.

      [41] V. T. Pham, Ch. K. Volos, S. Jafari and X.Wang, “Dynamics and circuit of a chaotic system with a curve of equilibrium points”, International Journal of Electronics, Vol. 105, No. 3, (2018), pp. 385–397.

      [42] Z. Wei and W. Zhang, “Hidden hyperchaotic attractors in a modified lorenz-stenflo system with only one stable equilibrium”, International Journal of Bifurcation and Chaos, Vol. 24, No. 10, (2014), Article ID 1450127.

      [43] Y. Zhao and R. Wu, “Chaos and synchronisation of a new fractional order system with only two stable equilibria”, International Journal of Dynamical Systems and Differential Equations, Vol. 6, No. 3, (2016), pp. 187–202.

      [44] S. Jafari and J. C. Sprott, “Simple chaotic flows with a line equilibrium”, Chaos, Solitons and Fractals, Vol. 57, (2013), pp. 79–84.

      [45] T. Gotthans and J. Petrzela, “New classes of chaotic systems with circular equilibrium,” Nonlinear Dynamics, Vol. 81, No. 3, pp. 1143-1149, 2015.

      [46] S. Kingni, V.T. Pham, S. Jafari and P. Woafo, “A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form,” Chaos, Solitons & Fractals, Vol. 99, No. 1, pp. 209-218, 2017.

      [47] X. Wang, V.T. Pham and C. Volos, “Dynamics, circuit design and synchronization of a new chaotic system with closed curve equilibrium,” Complexity, Vol. 2017, pp. 7138971.

      [48] V.T. Pham, S. Jafai, X. Wang and J. Ma, “A chaotic system with different shapes of equilibria,”International Journal of Bifurcation and Chaos, Vol. 206, pp. 1650069, 2016.

      [49] A. Sambas, S. Vaidyanathan, M. Mamat and W. S. M Sanjaya, “A Six-Term Novel Chaotic System with Hidden Attractor and Its Circuit Design,” Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 365-373.

      [50] A.T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, (2016).

      [51] S. Vaidyanathan and C. Volos, Advances and Applications in Nonlinear Control Systems, Springer, (2017).

      [52] S. Vaidyanathan and C. Volos, Advances and Applications in Chaotic Systems, Springer, (2017).

      [53] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, “Determining Lyapunov exponents from a time series”, Physica D: Nonlinear Phenomena, Vol. 16, No. 3, (1985), pp. 285–317.

      [54] A. Sambas, S. Vaidyanathan, M. Mamat, W. S. M. Sanjaya and R. P. Prastio, “Design, analysis of the Genesio-Tesi chaotic system and its electronic experimental implementation”, International Journal of Control Theory and Applications, Vol. 9, No. 1, (2016), pp. 141–149.

      [55] C. Li, W. J. C. Thio, J. C. Sprott, R. X. Zhang and T. A. Lu, “Linear synchronization and circuit implementation of chaotic system with complete amplitude control”, Chinese Physics B, Vol. 26, No. 12, (2017), Article ID 120501.

      [56] E. Tlelo-Cuautle, L. G. De La Fraga, V. T. Pham, Ch. K. Volos, S. Jafari and A. J. Quintas-Valles, “Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points”, Nonlinear Dynamics, Vol. 89, No. 2, (2017), pp. 1129–1139.


 

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Article ID: 12544
 
DOI: 10.14419/ijet.v7i3.12544




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