A New Hamiltonian Chaotic System with Coexisting Chaotic Orbits and its Dynamical Analysis

  • Authors

    • Sundarapandian Vaidyanathan Vel Tech University
    • Aceng Sambas Universitas Muhammadiyah Tasikmalaya
    • Sen Zhang Xiangtan University
    • Mohamad Afendee Mohamed Universiti Sultan Zainal Abidin
    • Mustafa Mamat Universiti Sultan Zainal Abidin
    2018-09-20
    https://doi.org/10.14419/ijet.v7i4.16826
  • Chaos, Chaotic systems, conservative systems, Hamiltonian systems, Lyapunov exponents.

  • Hamiltonian chaotic systems are conservative chaotic systems which arise in many applications in Classical Mechanics. A famous Hamiltonian chaotic system is the H´enon-Heiles system (1964), which was modeled by H´enon and Heiles, describing the nonlinear motion of a star around a galactic centre with the motion restricted to a plane. In this research work, by modifying the dynamics of the H´enon-Heiles system (1964), we obtain a new Hamiltonian chaotic system with coexisting chaotic orbits. We describe the dynamical properties of the new Hamiltonian chaotic system.

  • References

    1. [1] S. Vaidyanathan and C. Volos, Advances and Applications in Chaotic Systems, Springer, Berlin, (2017).

      [2] A.T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Berlin, (2017).

      [3] O.I. Tacha, C.K. Volos, I.M. Kyprianidis, I.N. Stouboulos, S. Vaidyanathan and V.T. Pham, “Analysis, adaptive control and circuit simulation of a novel nonlinear finance systemâ€, Applied Mathematics and Computation, Vol. 276, (2016), pp. 200–217.

      [4] X. Zhao, Z. Li and S. Li. Synchronization of a chaotic finance system. Applied Mathematics and Computation, Vol. 217, No. 13, (2011), 6031-6039.

      [5] 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 designâ€, Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 271–295.

      [6] X. J. Tong, M. Zhang, Z.Wang, Y. Liu and J. Ma, “An image encryption scheme based on a new hyperchaotic finance systemâ€, Optik, Vol. 126, No. 20, (2015), pp. 2445–2452.

      [7] I. Klioutchnikov, M. Sigova and N. Beizerova, “Chaos theory in financeâ€, Procedia Computer Science, Vol. 119, (2017), pp. 368–375.

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

      [9] 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 encryptionâ€, International Journal of Simulation and Process Modelling, Vol. 13, No. 3, (2018), pp. 281–296.

      [10] F. P. Russell, P. D. D¨uben, X. Niu, W. Luk and T. N. Palmer, “Exploiting the chaotic behaviour of atmospheric models with reconfigurable architecturesâ€, Computer Physics Communications, Vol. 221, (2017), pp. 160–173.

      [11] S. Vaidyanathan, “Output regulation of the forced Van der Pol chaotic oscillator via adaptive control methodâ€, International Journal of PharmTech Research, Vol. 8, No. 6, (2015), pp. 106–116.

      [12] T. Akaishi, T. Takahashi and I. Nakashima, “Chaos theory for clinical manifestations in multiple sclerosisâ€, Medical Hypotheses, Vol. 115, (2018), pp. 87–93.

      [13] S. Vaidyanathan, “Global chaos synchronization of the forced Van der Pol chaotic oscillators via adaptive control methodâ€, International Journal of PharmTech Research, Vol. 8, No. 6, (2015), pp. 156-166.

      [14] S. Vaidyanathan, “Adaptive control of the FitzHugh-Nagumo chaotic neuron modelâ€, International Journal of PharmTech Research, Vol. 8, No. 6, (2015), pp. 117–127.

      [15] M.H. Wang, S.D. Lu and M.J. Hsieh, “Application of extension neural network algorithm and chaos synchronization detection method to partial discharge diagnosis of power capacitorâ€, Measurement, Vol. 129, (2018), pp. 227–235.

      [16] K. Bouallegue, “A new class of neural networks and its applicationsâ€, Neurocomputing, Vol. 249, (2017), pp. 28–47.

      [17] S. Vaidyanathan, “Synchronization of 3-cells cellular neural network (CNN) attractors via adaptive control method â€, International Journal of PharmTech Research, Vol. 8, No. 5, (2015), pp. 946–955.

      [18] S. Vaidyanathan, “A novel chemical chaotic reactor system and its adaptive controlâ€, International Journal of ChemTech Research, Vol. 8, No. 7, (2015), pp. 146–158.

      [19] V.K. Yadav, 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 uncertaintiesâ€, Chinese Journal of Physics, Vol. 55, No. 3, (2017), pp. 594–605.

      [20] S. Vaidyanathan, “Adaptive synchronization of novel 3-D chemical chaotic reactor systemsâ€, International Journal of ChemTech Research, Vol. 8, No. 7, (2015), pp. 159–171.

      [21] N. I. Kol’tsov and V. K. Fedotov, “Two-Dimentional Chaos in Chemical Reactionsâ€, Russian Journal of Physical Chemistry B, Vol. 12, No. 3, (2018), pp.590-592.

      [22] J. Chattopadhyay, N. Pal, S. Samanta, E. Venturino and Q.J.A. Khan, “Chaos control via feeding switching in an omnivory systemâ€, Biosystems, Vol. 138, (2015), pp. 18–24.

      [23] V. Voorsluijs and Y.D. Decker, “Emergence of chaos in a spatially confined reactive systemâ€, Physica D: Nonlinear Phenomena, Vol. 335, (2016), pp. 1–9.

      [24] S. Vaidyanathan, “Lotka-Volterra population biology models with negative feedback and their ecological monitoringâ€, International Journal of PharmTech Research, Vol. 8, No. 5, (2015), pp. 974–981.

      [25] S. Vaidyanathan, “Adaptive controller and synchronization design for the Qi-Chen chaotic systemâ€, Lecture Notes of the Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, Vol. 85, (2012), pp. 124–133.

      [26] D. Ghosh, A. Mukherjee, N.R. Das and B.N. Biswas, “Generation & control of chaos in a single loop optoelectronic oscillatorâ€, Optik, Vol. 165, (2018), pp. 275–287.

      [27] S. Vaidyanathan and S. Rasappan, “Hybrid synchronization of hyperchaotic Qi and L¨u systems by nonlinear controlâ€, Communications in Computer and Information Science, Vol. 131, (2011), pp. 585–593.

      [28] S. Mishra and R. D. S. Yadava, “A Method for Chaotic Self-Modulation in Nonlinear Colpitts Oscillator and its Potential Applicationsâ€, Circuits, Systems, and Signal Processing, Vol. 37, No. 2, (2018), pp. 532–552.

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

      [30] J. Jin, “Programmable multi-direction fully integrated chaotic oscillatorâ€, Microelectronics Journal, Vol. 75, (2018), pp. 27–34.

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

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

      [33] X. Wang, S. Vaidyanathan, C. Volos, V.T. Pham and T. Kapitaniak, “Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractorsâ€, Nonlinear Dynamics, Vol. 89, No. 3, (2017), pp. 1673–1687.

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

      [35] A. Nguomkam Negou and J. Kengne, “Dynamic analysis of a unique jerk system with a smoothly adjustable symmetry and nonlinearity: Reversals of period doubling, offset boosting and coexisting bifurcationsâ€, AEU-International Journal of Electronics and Communications, Vol. 90, (2018), pp. 1–19.

      [36] Y.R. Bai, D. Baleanu and G.C. Wu, “A novel shuffling technique based on fractional chaotic mapsâ€, Optik, Vol. 168, (2018), pp. 553–562.

      [37] G.C.Wu, D. Baleanu and Z.X. Lin, “Image encryption technique based on fractional chaotic time seriesâ€, Journal of Vibration and Control, Vol. 22, No. 8 (2014), pp. 2092–2099.

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

      [39] A. Sambas, M. Mamat and W. S. M. Sanjaya, “Bidirectional coupling scheme of chaotic systems and its application in secure communication systemâ€, Journal of Engineering Science and Technology Review, Vol. 8, No. 2, (2015), pp. 89–95.

      [40] B.K. Patle, D.R.K. Parhi, A. Jagadeesh and S.K. Kashyap, “Matrix- Binary Codes based Genetic Algorithm for path planning of mobile robotâ€, Computers & Electrical Engineering, Vol. 67, (2018), pp. 708–728.

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

      [42] A. Sambas, S. Vaidyanathan, M. Mamat and W.S. Mada Sanjaya, “A six-term novel chaotic system with hidden attractor and its circuit designâ€, Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 365–373.

      [43] S. Vaidyanathan, V.T. Pham, C. Volos and A. Sambas, “A novel 4- D hyperchaotic Rikitake dynamo system with hidden attractor, its properties, synchronization and circuit designâ€, Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 345–364.

      [44] 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 designâ€, Studies in Systems, Decision and Control, Vol. 133, (2018), pp. 271–295.

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

      [46] M. Mamat, S. Vaidyanathan, A. Sambas, Mujiarto,W.S.M. Sanjaya and Subiyanto, “A novel double-convection chaotic attractor, its adaptive control and circuit simulationâ€, IOP Conference Series: Materials Science and Engineering, Vol. 332, No. 1, (2018), Article ID 012033.

      [47] S. Vaidyanathan, A. Sambas, Sukono, M. Mamat, G. Gundara, W.S. Mada Sanjaya and Subiyanto, “A new chaotic attractor with two quadratic nonlinearities, its synchronization and circuit implementationâ€, IOP Conference Series: Materials Science and Engineering, Vol. 332, No. 1, (2018), Article ID 012048.

      [48] C.H. Lien, S. Vaidyanathan, A. Sambas, Sukono, M. Mamat, W.S.M. Sanjaya and Subiyanto, “A new two-scroll chaotic attractor with three quadratic nonlinearities, its adaptive control and circuit designâ€, IOP Conference Series: Materials Science and Engineering, Vol. 332, No.1, (2018), Article ID 012010.

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

      [50] M. Mamat, S. Vaidyanathan, A. Sambas, M.A. Mohamed, S. Sampath and W.S. Mada Sanjaya, “A new 3-D chaotic system with conch shaped equilibrium curve and its circuit implementationâ€, International Journal of Engineering & Technology, Vol. 7, No. 3, (2018), pp. 1410–1414.

      [51] S. Vaidyanathan, A. Sambas, M.A. Mohamed, M. Mamat and W.S. Mada Sanjaya, “A new hyperchaotic hyperjerk system with three nonlinear terms, its synchronization and circuit simulationâ€, International Journal of Engineering & Technology, Vol. 7, No. 3, (2018), pp. 1585–1592.

      [52] A. Sambas, M. Mamat, S. Viadyanathan, M.A. Mohamed, W.S. Mada Sanjaya and Mujiarto, “A novel chaotic hidden attractor, its synchronization and circuit implementationâ€, WSEAS Transactions on Systems and Control, Vol. 13, (2018), pp. 345–352.

      [53] V.T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, N.V. Kuznetsov and T.M. Hoang, “A novel memristive time-delay chaotic system without equilibrium pointsâ€, European Physical Journal: Special Topics, Vol. 225, (2016), pp. 127–136.

      [54] J.C. Sprott, Elegant Chaos, World Scientific, Singapore, (2010).

      [55] S. Nos´e, “A molecular dynamics method for simulations in the canonical ensembleâ€, Molecular Physics, Vol. 52, No. 2, (1984), pp. 255–268.

      [56] M. H´enon and C. Heiles, “The applicability of the third integral of motion: Some numerical experimentsâ€, The Astronomical Journal, Vol. 69 (1964), pp. 73–79.

      [57] S. Vaidyanathan and C. Volos, “Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic systemâ€, Archives of Control Sciences, Vol. 25, No. 3, (2015), pp. 333–353.

      [58] S. Vaidyanathan and S. Pakiriswamy, “A 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive controlâ€, Journal of Engineering Science and Technology Review, Vol. 8, No. 2, (2015), pp. 52–60.

      [59] S. Vaidyanathan and S. Pakiriswamy, “A five-term 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive control methodâ€, International Journal of Control Theory and Applications, Vol. 9, No. 1, (2016), pp. 61–78.

      [60] S. Vaidyanathan and C.K. Volos, “A novel conservative jerk chaotic system with two cubic nonlinearities and its adaptive backstepping controlâ€, Studies in Computational Intelligence, Vol. 636, (2016), pp.85–108.

      [61] S. Vaidyanathan, “A conservative hyperchaotic hyperjerk system based on memristive deviceâ€, Studies in Computational Intelligence, Vol. 701, (2017), pp. 393–423.

      [62] S. Vaidyanathan, “A novel four-dimensional conservative chaotic system without linear term, its analysis and adaptive control via integral sliding mode controlâ€, International Journal of Modelling, Identification and Control, Vol. 30, No. 2, (2018), pp. 132–142.

      [63] Y. Zhu, Z. Zhong, M.V. Basin and D. Zhou, “A descriptor system approach to stability and stabilization of discrete-time switched PWA systemsâ€, IEEE Transactions on Automatic Control, (2018), DOI 10.1109/TAC.2018.2797173.

      [64] Y. Zhu, Z. Zhong, W.X. Zheng and D. Zhou, “HMM-based HÂ¥ filtering for discrete-time Markov jump LPV systems over unreliable communication channelsâ€, IEEE Transactions on Systems, Man, and Cybernetics: Systems, (2017), DOI: 10.1109/TSMC.2017.2723038.

      [65] Y. Zhu, L. Zhang and W.X. Zheng, “Distributed HÂ¥ filtering for a class of discrete-time Markov jump Lur’e systems with redundant channelsâ€, IEEE Transactions on Industrial Electronics, Vol. 63, No. 3, (2016), pp. 1876–1885.

      [66] L. Zhang, Y. Zhu, and W.X. Zheng, “State estimation of discrete-time switched neural networks with multiple communication channelsâ€, IEEE Transactions on Cybernetics, Vol. 47, No. 4, (2017), pp. 1028–1040.

      [67] L. Zhang, Y. Zhu, Z. Ning and X. Yin, “Resilient estimation for networked systems with variable communication capabilityâ€, IEEE Transactions on Automatic Control , Vol. 61, No. 12, (2016), pp. 4150–4156.

      [68] C.J. Song and Y. Zhang, “Conserved quantities for Hamiltonian systems on time scalesâ€, Applied Mathematics and Computation, Vol. 313, (2017), pp. 24–36.

      [69] D.S. Tourigny, “Networks of planar Hamiltonian systemsâ€, Communications in Nonlinear Science and Numerical Simulation, Vol. 53, (2017), pp. 263–277.

      [70] C. Deng and D.L. Wu, “Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systemsâ€, Boundary Value Problems, Vol. 2018, No. 1, (2018), pp. 56.

      [71] Z. Wang and T. Ma, ‘Infinitely many periodic solutions of planar Hamiltonian systems via the Poincare-Birkhoff theoremâ€, Boundary Value Problems, Vol. 2018, No. 1, (2018), pp. 102.

      [72] F. Gesztesy and M. Zinchenko, “Renormalized oscillation theory for Hamiltonian systemsâ€, Advances in Mathematics, Vol. 311, (2017), pp. 569–597.

      [73] S. Zhang, Y.C. Zeng, Z.J. Li, M.J. Wang and L. Xiong, “Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistibilityâ€, Chaos, Vol. 28, (2018), pp. 013113.

      [74] S. Zhang, Y. Zeng and Z. Li, “One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamicsâ€, Chinese Journal of Physics, Vol. 56, No. 3, (2018), pp. 793–806.

      [75] S. Zhang, Y. Zeng, Z. Li, M. Wang and L. Xiong, “A novel grid multiwing chaotic system with only non-hyperbolic equilibriaâ€, Pramana, Vol. 90, No. 5, (2018), pp. 63.

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

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    Vaidyanathan, S., Sambas, A., Zhang, S., Mohamed, M. A., & Mamat, M. (2018). A New Hamiltonian Chaotic System with Coexisting Chaotic Orbits and its Dynamical Analysis. International Journal of Engineering & Technology, 7(4), 2430-2436. https://doi.org/10.14419/ijet.v7i4.16826