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

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    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.


  • Keywords


    Chaos, Chaotic systems, conservative systems, Hamiltonian systems, Lyapunov exponents.

  • References


      [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 system”, 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 design”, 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 system”, Optik, Vol. 126, No. 20, (2015), pp. 2445–2452.

      [7] I. Klioutchnikov, M. Sigova and N. Beizerova, “Chaos theory in finance”, 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 control”, 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 encryption”, 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 architectures”, 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 method”, 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 sclerosis”, 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 method”, International Journal of PharmTech Research, Vol. 8, No. 6, (2015), pp. 156-166.

      [14] S. Vaidyanathan, “Adaptive control of the FitzHugh-Nagumo chaotic neuron model”, 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 capacitor”, Measurement, Vol. 129, (2018), pp. 227–235.

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

      [17] S. Vaidyanathan, “Synchronization of 3-cells cellular neural network (CNN) attractors via adaptive control method ”, 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 control”, 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 uncertainties”, Chinese Journal of Physics, Vol. 55, No. 3, (2017), pp. 594–605.

      [20] S. Vaidyanathan, “Adaptive synchronization of novel 3-D chemical chaotic reactor systems”, 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 Reactions”, 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 system”, Biosystems, Vol. 138, (2015), pp. 18–24.

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

      [24] S. Vaidyanathan, “Lotka-Volterra population biology models with negative feedback and their ecological monitoring”, 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 system”, 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 oscillator”, Optik, Vol. 165, (2018), pp. 275–287.

      [27] S. Vaidyanathan and S. Rasappan, “Hybrid synchronization of hyperchaotic Qi and L¨u systems by nonlinear control”, 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 Applications”, 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 nonlinearities”, International Journal of Modelling, Identification and Control, Vol. 23, No. 4, (2015), pp. 380–392.

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

      [31] 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.

      [32] 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.

      [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 attractors”, 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 nonlinearity”, 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 bifurcations”, 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 maps”, 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 series”, 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 encryption”, 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 system”, 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 robot”, 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 robot”, 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 design”, 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 design”, 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 design”, 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 circuits”, 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 simulation”, 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 implementation”, 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 design”, 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 implementation”, 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 implementation”, 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 simulation”, 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 implementation”, 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 points”, 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 ensemble”, 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 experiments”, 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 system”, 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 control”, 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 method”, 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 control”, Studies in Computational Intelligence, Vol. 636, (2016), pp.85–108.

      [61] S. Vaidyanathan, “A conservative hyperchaotic hyperjerk system based on memristive device”, 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 control”, 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 systems”, 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 H¥ filtering for discrete-time Markov jump LPV systems over unreliable communication channels”, 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 H¥ filtering for a class of discrete-time Markov jump Lur’e systems with redundant channels”, 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 channels”, 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 capability”, 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 scales”, Applied Mathematics and Computation, Vol. 313, (2017), pp. 24–36.

      [69] D.S. Tourigny, “Networks of planar Hamiltonian systems”, 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 systems”, 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 theorem”, Boundary Value Problems, Vol. 2018, No. 1, (2018), pp. 102.

      [72] F. Gesztesy and M. Zinchenko, “Renormalized oscillation theory for Hamiltonian systems”, 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 multistibility”, 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 dynamics”, 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 equilibria”, 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 series”, Physica D: Nonlinear Phenomena, Vol. 16, No. 3, (1985), pp. 285–317.


 

View

Download

Article ID: 16826
 
DOI: 10.14419/ijet.v7i4.16826




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.