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Nonlinear Dynamics

, Volume 95, Issue 3, pp 2063–2077 | Cite as

Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian conservative chaotic systems

  • Guoyuan QiEmail author
Original Paper
  • 188 Downloads

Abstract

Four sub-Euler equations for four sub-rigid bodies are generalized by extending the dimension of the state space from 3D to 4D. Six integrated 4D Euler equations are proposed by combining any two of the four sub-Euler equations with two common axes. These 4D equations are essential in providing the symplectic structures for the dynamics of rigid body and fluid mechanics and generalized Hamiltonian systems. The conservation of both the Hamiltonian and Casimir energies is proved for the six 4D Euler equations. Conservative chaos is more advantageous than dissipative chaos regarding ergodicity, the distribution of probability, and fractional dimensions in the application of chaos-based secure communications and generation of pseudo-random numbers. Six 4D Hamiltonian chaotic systems are proposed through breaking of the conservation of Casimir energies and preserving of the Hamiltonian energies, one of which is analyzed in detail. This system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 80 K), a large state amplitude and energy, and power spectral density with a wide bandwidth. The system passed the NIST tests performed on it. Therefore, strong pseudo-randomness of this Hamiltonian conservative chaotic system is confirmed. The Casimir power method is verified as an alternative analytical measuring index of orbital mode to the Lyapunov exponent. The force interaction and exchange in Casimir energy are the causes of chaos production. The mechanism underlying the transition from regular orbits to irregular orbits to stronger irregular orbits is studied using the Casimir power and the variability of physical parameters of the chaotic system. The supremum is also found using the property of Hamiltonian conservation.

Keywords

4D Euler equations Hamiltonian conservative chaotic system Rigid body Casimir power NIST tests Chaos bound 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (61873186) and the Tianjin Natural Science Foundation (17JCZDJC38300). We thank Richard Haase, Ph.D., from Liwen Bianji, Edanz Group China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript.

References

  1. 1.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, 2nd edn. Springer, Berlin (2002). Chapter 1zbMATHGoogle Scholar
  2. 2.
    Qi, G.: Energy cycle of brushless DC motor chaotic system. Appl. Math. Model. 51, 686–697 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gluhovsky, A.: Energy-conserving and Hamiltonian low-order models in geophysical fluid dynamics. Nonlinear Process. Geophys. 13, 125–133 (2006)CrossRefGoogle Scholar
  4. 4.
    Shamolin, M.V.: Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field. J. Math. Sci. 165, 743–754 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Taylor, J.R.: Classical Mechanics. University Science Books, Sausalito (2005)zbMATHGoogle Scholar
  6. 6.
    Fomenko, A.T.: Integrability and Nonintegrability in Geometry and Mechanics, Chapter 4. Kluwer Academic Publishers, Dordrecht (1988)CrossRefGoogle Scholar
  7. 7.
    Bogoyavlensky, O.I.: Integrable Euler equations on SO(4) and their physical applications. Commun. Math. Phys. 93, 417–436 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Perelomov, A.M.: Motion of four-dimensional rigid body around a fixed point: an elementary approach I. J. Phys. A Math. Gen. 38, L801–L807 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, H., Tadmor, E., Wei, D.: Global regularity of the 4D restricted Euler equations. Physica D 239, 1225–1231 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sprott, J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65, 537–543 (1997)CrossRefGoogle Scholar
  11. 11.
    Thomas, R.: Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, “labyrinth chaos”. Int. J. Bifurc. Chaos 9, 1889–1905 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cang, S., Wu, A., Wang, Z., Chen, Z.: Four-dimensional autonomous dynamical systems with conservative flows: two-case study. Nonlinear Dyn. 89, 2495–2508 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hoover, W.G.: Remark on “some simple chaotic flows”. Phys. Rev. E 51(1), 759–760 (1995)CrossRefGoogle Scholar
  14. 14.
    Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50(2), 647–50 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sprott, J.C.: Elegant Chaos–Algebraically Simple Chaotic Flows. World Scientific Publishing, Singapore (2010). Chapter 4CrossRefzbMATHGoogle Scholar
  16. 16.
    Vaidyanathan, S., Volos, C.: Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Arch. Control Sci. 25(3), 333–353 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mahmoud, G.M., Ahmed, M.E.: Analysis of chaotic and hyperchaotic conservative complex nonlinear systems. Miskolc Math. Notes 18(1), 315–326 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astrophys. J. 69, 73–79 (1964)MathSciNetGoogle Scholar
  19. 19.
    Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics-Integrability, Chaos, and Patterns. Springer, Berlin (2012). Chapter 7zbMATHGoogle Scholar
  20. 20.
    Eckhardt, B., Hose, G., Pollak, E.: Quantum mechanics of a classically chaotic system: observations on scars, periodic orbits, and vibrational adiabaticity. Phys. Rev. E 39, 3776–3793 (1989)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qi, G., Wyk, M.Avan, van Wyk, B.J., Chen, G.: On a new hyperchaotic system. Phys. Lett. A 372, 124–136 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Qi, G., Zhang, J.: Energy cycle and bound of Qi chaotic system. Chaos Solitons Fractals 99, 7–15 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Qi, G., Liang, X.: Mechanism and energy cycling of Qi four-wing chaotic system. Int. J. Bifurc. Chaos 27(12), 1750180-1-15 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
  25. 25.
    Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, Y., Qi, G.: Mechanical analysis and bound of plasma chaotic system. Chaos Soliton. Fractals 108, 187–195 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Qi, G., Hu, J.: Force analysis and energy operation of chaotic system of permanent-magnet synchronous motor. Int. J. Bifurc. Chaos 27, 1750216-1-18 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Qi, G., Liang, X.: Mechanical analysis of Qi four-wing chaotic system. Nonlinear Dyn. 86(2), 1095–1106 (2016)CrossRefGoogle Scholar
  29. 29.
    Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefzbMATHGoogle Scholar
  30. 30.
    Rukhin, A., et al.: A Statistical test suite for random and pseudorandom number generators for cryptographic applications. National Institute of Standards and Technology, Technology Administration U.S. Department of Commerce, Special Publication 800-22 (2001)Google Scholar
  31. 31.
    Liang, X., Qi, G.: Mechanical analysis and energy cycle of Chen chaotic system. Braz. J. Phys. 47, 1–7 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and AutomationTianjin Polytechnic UniversityTianjinChina

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