Advertisement

Acta Mechanica

, Volume 229, Issue 4, pp 1833–1848 | Cite as

Basic theory of fractional Mei symmetrical perturbation and its applications

  • Shao-Kai Luo
  • Ming-Jing Yang
  • Xiao-Tian Zhang
  • Yun Dai
Original Paper

Abstract

In this paper, we present a new method of fractional dynamics, i.e., the fractional Mei symmetrical perturbation method of a disturbed system, and explore the adiabatic invariant directly led by the perturbation. For a dynamical system which is disturbed by small forces of perturbation, the disturbed fractional generalized Hamiltonian equation is investigated and, under a more general kind of fractional infinitesimal transformation of a Lie group, the fractional Mei symmetrical definition and determining equation of a disturbed dynamical system are given; then, the determining equation of fractional Mei symmetrical perturbation is obtained. In particular, we present the fractional Mei symmetrical perturbation method of a disturbed dynamical system and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant directly led by the perturbation. As special cases, we obtain a new fractional Mei symmetrical conservation law of the undisturbed dynamical system and a Mei symmetrical perturbation theorem of the disturbed integer dynamical system. Also, as the fractional Mei symmetrical perturbation method’s applications, we find the adiabatic invariants of a disturbed fractional Duffing oscillator and a disturbed fractional Lotka biochemical oscillator. This work constructs a basic theoretical framework of fractional Mei symmetrical perturbation method and provides a general method of fractional dynamics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are very grateful to reviewers for their valuable comments which undoubtedly improved our manuscript.

References

  1. 1.
    Mei, F.X.: Form invariance of Lagrange system. J. Beijing Inst. Technol. 2, 120–124 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Noether, E.: Invariant variational problems. Math. Phys. Klasse 2, 235–257 (1918)zbMATHGoogle Scholar
  3. 3.
    Lutzky, M.: Dynamical symmetries and conserved quantities. J. Phys. A 12, 973–981 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jiang, W.A., Luo, S.K.: Mei symmetry leading to Mei conserved quantity of generalized Hamilton systems. Acta Phys. Sin. 60, 060201 (2011)zbMATHGoogle Scholar
  5. 5.
    Luo, S.K., Guo, Y.X., Mei, F.X.: Form invariance and Hojman conserved quantity for nonholonomic mechanical system. Acta Phys. Sin. 53, 2413–2418 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Wu, H.B., Mei, F.X.: Form invariance and Lie symmetry of the generalized Hamiltonian system. Acta Mech. Solida Sin. 17, 370–373 (2004)Google Scholar
  7. 7.
    Jia, L.Q., Wang, X.X., Zhang, M.L., Han, Y.L.: Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system. Nonlinear Dyn. 69, 1807–1812 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, P., Xue, Y.: Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod. Nonlinear Dyn. 83, 1815–1822 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, P., Fang, J.H., Ding, N.: Two types of new conserved quantities and Mei symmetry of mechanical systems in phase space. Commun. Theor. Phys. 48, 993–995 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, M.J., Fang, J.H., Lu, K.: Perturbation to Mei symmetry and generalized Mei adiabatic invariants for Birkhoffian systems. Int. J. Theor. Phys. 49, 427–437 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X.W.: Global Analysis for Birkhoff Systems. Henan University Press, Kaifeng (2002)Google Scholar
  12. 12.
    Cai, J.L.: Conformal invariance and conserved quantity for the nonholonomic system of Chetaev’s type. Int. J. Theor. Phys. 49, 201–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huang, W.L., Cai, J.L.: Inverse problems of Mei symmetry for nonholonomic systems with variable mass. J. Mech. 31, 1–9 (2015)CrossRefGoogle Scholar
  14. 14.
    Cai, J.L.: Conformal invariance of Mei symmetry for the non-holonomic systems of non-Chetaev’s type. Nonlinear Dyn. 69, 487–493 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang, Y., Mei, F.X.: Form invariance for systems of generalized classical mechanics. Chin. Phys. 12, 1058–1061 (2003)CrossRefGoogle Scholar
  16. 16.
    Luo, S.K.: Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian canonical equation in a singular system. Acta Phys. Sin. 53, 5–10 (2004)zbMATHGoogle Scholar
  17. 17.
    Luo, S.K., Dai, Y., Zhang, X.T., Yang, M.J.: Fractional conformal invariance method for finding conserved quantities of dynamical systems. Int. J. Non-Linear Mech. 97, 107–114 (2017)Google Scholar
  18. 18.
    Luo, S.K.: Form invariance and Lie symmetries of rotational relativistic Birkhoff system. Chin. Phys. Lett. 19, 449–451 (2002)CrossRefGoogle Scholar
  19. 19.
    Luo, S.K., Zhang, Y.F.: Advances in the Study of Dynamics of Constrained System. Science Press, Beijing (2008)Google Scholar
  20. 20.
    Burgers, J.M.: Die adiabatischen invarianten bedingt periodischenr systems. Ann. Phys. 52, 195–202 (1917)CrossRefGoogle Scholar
  21. 21.
    Kruskal, M.: Asymptotic theory of Hamiltonian and other system with all solutions nearly periodic. J. Math. Phys. 3, 806–828 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Djukic, D.S.: Adiabatic invariants for dynamical systems with one degree of freedom. Int. J. Nonlinear Mech. 16, 489–498 (1981)CrossRefzbMATHGoogle Scholar
  23. 23.
    Bulanov, S.V., Shasharina, S.G.: Behaviour of adiabatic invariant near the separatrix in a stellarator. Nucl. Fus. 32, 1531–1543 (1992)CrossRefGoogle Scholar
  24. 24.
    Notte, J., Fajans, J., Chu, R., Wurtele, J.S.: Experimental breaking of an adiabatic invariant. Phys. Rev. Lett. 70, 3900–3903 (1993)CrossRefGoogle Scholar
  25. 25.
    Ding, N., Fang, J.H.: Mei adiabatic invariants induced by perturbation of Mei symmetry for nonholonomic controllable mechanical systems. Commun. Theor. Phys. 54, 785–791 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Song, C.J., Zhang, Y.: Perturbation to Mei symmetry and adiabatic invariants for disturbed El-Nabulsi’s fractional Birkhoff system. Commun. Theor. Phys. 64, 171–176 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1982)zbMATHGoogle Scholar
  28. 28.
    Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Agrawal, O.P., Muslih, S., Baleanu, D.: Generalized variational calculus in terms of multi-parameters fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, 4756–4767 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Baleanu, D., Agrawal, O.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56, 1087–1092 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Golmankhaneh, A.K., Yengejeh, A.M., Baleanu, D.: On the fractional Hamilton and Lagrange mechanics. Int. J. Theor. Phys. 51, 2909–2916 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Baleanu, D., Muslih, S.I., Rabei, E.M.: On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative. Nonlinear Dyn. 53, 67–74 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Baleanu, D., Trujillo, J.: A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 15, 1111–1115 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czechoslov. J. Phys. 52, 1247–1253 (2002)CrossRefzbMATHGoogle Scholar
  35. 35.
    Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, 1816–1820 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14, 523–537 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, Y.: Fractional differential equations of motion in terms of combined Riemann–Liouville derivatives. Chin. Phys. B 21, 084502 (2012)CrossRefGoogle Scholar
  38. 38.
    Tarasov, V.E.: Fractional Dynamics. Higher Education Press, Beijing (2010)CrossRefzbMATHGoogle Scholar
  39. 39.
    Luo, S.K., Xu, Y.L.: Fractional Birkhoffian mechanics. Acta Mech. 226, 829–844 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    He, J.M., Xu, Y.L., Luo, S.K.: Stability for manifolds of equilibrium state of fractional Birkhoffian systems. Acta Mech. 226, 2135–2146 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Luo, S.K., He, J.M., Xu, Y.L.: Fractional Birkhoffian method for equilibrium stability of dynamical systems. Int. J. Nonlinear Mech. 78, 105–111 (2016)CrossRefGoogle Scholar
  42. 42.
    Xu, Y.L., Luo, S.K.: Fractional Nambu dynamics. Acta Mech. 226, 3781–3793 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Luo, S.K., Zhang, X.T., He, J.M.: A general method of fractional dynamics, i.e., fractional Jacobi last multiplier method, and its applications. Acta. Mech 228, 157–174 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhang, Y., Zhai, X.H.: Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 81, 469–480 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Yan, B., Zhang, Y.: Noether’s theorem for fractional Birkhoffian systems of variable order. Acta Mech. 227, 2439–2449 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Jia, Q.L., Wu, H.B., Mei, F.X.: Noether symmetries and conserved quantities for fractional forced Birkhoffian systems. J. Math. Anal. Appl. 442, 782–795 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Song, C.J., Zhang, Y.: Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems. Int. J. Nonlinear Mech. 90, 32–38 (2017)CrossRefGoogle Scholar
  48. 48.
    Luo, S.K., Li, L.: Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives. Nonlinear Dyn. 73, 639–647 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Luo, S.K., Zhang, X.T., He, J.M., Xu, Y.L.: On the families of fractional dynamical models. Acta Mech. 228, 3741–3754 (2017)Google Scholar
  50. 50.
    Li, L., Luo, S.K.: Fractional generalized Hamiltonian mechanics. Acta Mech. 224, 1757–1771 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Luo, S.K., Li, L.: Fractional generalized Hamiltonian equations and its integral invariants. Nonlinear Dyn. 73, 339–346 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Luo, S.K., Li, L., Xu, Y.L.: Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems. Acta Mech. 225, 2653–2666 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Zhang, X.T., He, J.M., Luo, S.K.: A new type of fractional Lie symmetrical method and its applications. Int. J. Theor. Phys. 56, 971–990 (2017)CrossRefzbMATHGoogle Scholar
  54. 54.
    Luo, S.K., Dai, Y., Zhang, X.T., He, J.M.: A new method of fractional dynamics, i.e., fractional Mei symmetrical method for finding conserved quantity, and its applications to physics. Int. J. Theor. Phys. 55, 4298–4309 (2016)CrossRefzbMATHGoogle Scholar
  55. 55.
    Xu, Y.L., Luo, S.K.: Stability for manifolds of equilibrium state of fractional generalized Hamiltonian systems. Nonlinear Dyn. 76, 657–672 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Luo, S.K., He, J.M., Xu, Y.L.: A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications. Appl. Math. Comput. 269, 77–86 (2015)MathSciNetGoogle Scholar
  57. 57.
    Luo, S.K., He, J.M., Xu, Y.L., Zhang, X.T.: Fractional generalized Hamilton method for equilibrium stability of dynamical systems. Appl. Math. Lett. 60, 14–20 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Luo, S.K., Xu, Y.L.: Fractional Lorentz–Dirac model and its dynamical behaviors. Int. J. Theor. Phys. 54, 572–581 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Luo, S.K., He, J.M., Xu, Y.L., Zhang, X.T.: Fractional relativistic Yamaleev oscillator model and its dynamical behaviors. Found. Phys. 46, 776–786 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Continuous transformation groups of fractional differential equations. Vestn. USATU 9, 125–135 (2007)Google Scholar
  61. 61.
    Gazizov, R.K., Kasatkin, A.A., Lukashchuk, S.Y.: Symmetry properties of fractional diffusion equations. Phys. Scr. 136, 014016 (2009)CrossRefGoogle Scholar
  62. 62.
    Duffing, G.: Erzwunge Schweingungen bei Veranderlicher Eigenfrequenz. F. Viewigu Sohn, Braunschweig (1918)zbMATHGoogle Scholar
  63. 63.
    Sato, S.: Universal scaling property in bifurcation structure of Duffing’s and generalized Duffing’s equation. Phys. Rev. A. 28, 1654–1658 (1981)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Ueda, Y.: Random phenomena resulting from non-linearity in system described by Duffing’s equation. Int. J. Nonlinear Mech. 73, 481–491 (1985)CrossRefGoogle Scholar
  65. 65.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (1995)CrossRefzbMATHGoogle Scholar
  66. 66.
    Chen, Y.F., Zheng, J.H., Wu, X.Y., Wang, J.: On high-accuracy approximate solution of undamped Duffing equation. Mech. Sci. Technol. Aerosp. Eng. 27, 1591–1594 (2008)Google Scholar
  67. 67.
    Albert Luo, C.J., Huang, J.Z.: Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator. Nonlinear Dyn. 72, 417–438 (2013)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Dutt, R.: Application of Hamilton–Jacobi theory to the Lotka–Volterra oscillator. Bull. Math. Biol. 38, 459–465 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Institute of Mathematical Mechanics and Mathematical PhysicsZhejiang Sci-Tech UniversityHangzhouPeople’s Republic of China

Personalised recommendations