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Nonlinear Dynamics

, Volume 95, Issue 3, pp 1747–1765 | Cite as

First integrals and analytical solutions of some dynamical systems

  • B. U. Haq
  • I. NaeemEmail author
Original Paper

Abstract

This article investigates the first integrals and closed- form solutions of some nonlinear first-order dynamical systems from diverse areas of applied mathematics. We use the notion of artificial Hamiltonian, and we show that every first-order system of ordinary differential equations (ODEs) can be written in the form of an artificial Hamiltonian system [see Naz and Naeem (ZNA 73(4):323–330, 2018)]. One can also express the second-order ODE or system of second-order ODEs in the form of system of first-order artificial Hamiltonian system. Then the partial Hamiltonian approach is employed to compute the partial Hamiltonian operators and the corresponding first integrals. The first integrals are utilized to construct the closed-form solutions of two-stream model for tuberculosis and dengue fever, Duffing–van der Pol oscillator, nonlinear optical oscillators under parameter restrictions, nonlinear convection model and the two- dimensional galaxy model. We show that how one can apply the existing “partial Hamiltonian approach” for nonstandard Hamiltonian systems. This study provides a new way of solving the dynamical systems of first-order ODEs, second-order ODE and second-order systems of ODEs which are expressed into the artificial Hamiltonian system.

Keywords

Artificial Hamiltonian Phase space coordinates Gauge terms First integrals Exact solution 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

References

  1. 1.
    Noether, E.: Invariante Variationsprobleme. Nachr. K\(\ddot{o}\)nig. Gesell. Wissen., G\(\ddot{o}\)ttingen, Math.-Phys. Kl., Heft 2, 235–257 (1918). (English translation in transport theory and statistical physics 1(3), 186–207 (1971))Google Scholar
  2. 2.
    Kara, A.H., Mahomed, F.M., Naeem, I., Wafo Soh, C.: Partial Noether operators and first integrals via partial Lagrangians. Math. Methods Appl. Sci. 30(16), 2079–2089 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Naz, R., Naeem, I., Mahomed, F.M.: A partial Lagrangian approach to mathematical models of epidemiology. Math. Probl. Eng. (2015).  https://doi.org/10.1155/2015/602915
  4. 4.
    Laplace, P.S.: Celestial Mechanics (English translation), New York (1966)Google Scholar
  5. 5.
    Dorodnitsyn, V., Kozlov, R.: Invariance and first integrals of continuous and discrete Hamiltonian equations. J. Eng. Math. 66, 253–270 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bahar, L.Y., Kwatny, H.G.: Extension of Noether’s theorem to constrained non-conservative dynamical systems. Int. J. Nonlinear Mech. 22(2), 125–138 (1987)CrossRefGoogle Scholar
  7. 7.
    Bahar, L.Y., Kwatny, H.G.: Dynamic response of some dissipative systems by means of functions of matrices. J. Sound Vib. 137(3), 433–442 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bahar, L.Y., Kwatny, H.G.: Conservation laws for dissipative systems possessing classical normal modes. J. Sound Vib. 102(4), 551–562 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bahar, L.Y.: Response, stability and conservation laws for the sleeping top problem. J. Sound Vib. 158(1), 25–34 (1992)CrossRefGoogle Scholar
  10. 10.
    Naz, R., Mahomed, F.M., Chaudhry, A.: A partial Hamiltonian approach to current value Hamiltonian systems. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3600–3610 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Naz, R.: The applications of the partial Hamiltonian approach to mechanics and other areas. Int. J. Nonlinear Mech. 86, 1–6 (2016)CrossRefGoogle Scholar
  12. 12.
    Naz, R., Naeem, I.: The artificial Hamiltonian, first integrals and closed-form solutions of dynamical systems for epidemics. ZNA 73(4), 323–330 (2018)Google Scholar
  13. 13.
    Naz, R., Mahomed, F.M.: Characterization of partial Hamiltonian operators and related first integrals. Discrete Cont. Dyn. Syst. Ser. S 11(4), 723–734 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Naz, R., Freire, I.L., Naeem, I.: Comparison of different approaches to construct first integrals for ordinary differential equations. Abstr. Appl. Anal. (2014).  https://doi.org/10.1155/2014/978636
  15. 15.
    Naeem, I., Mahomed, F.M.: Approximate partial Noether operators and first integrals for coupled nonlinear oscillators. Nonlinear Dyn. 57(1–2), 303–311 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Naeem, I., Mahomed, F.M.: Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn. 45(3–4), 367–383 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13, 129–152 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kovacic, I.: Conservation laws of two coupled nonlinear oscillators. Int. J. Nonlinear Mech. 41, 751–760 (2006)CrossRefGoogle Scholar
  19. 19.
    Gao, G., Feng, Z.: First integrals for the Duffing–van der Pol type oscillator. Electron. J. Differ. Equ. 19, 123–133 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Duarte, L.G.S., Duarte, S.E.S., Da Mota, A.C.P., Skea, J.E.F.: Solving the second-order ordinary differential equations by extending the Prelle-Singer method. J. Phys. A (Math. Gen.) 34, 3015–3024 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mahomed, K.S., Moitsheki, R.J.: First integrals of generalized Ermakov systems via the Hamiltonian formulation. Int. J. Mod. Phys. B (2016).  https://doi.org/10.1142/S0217979216400191
  22. 22.
    Milonni, P.W., Eberly, J.H.: Laser Physics. Wiley, New Jersey (2010)CrossRefGoogle Scholar
  23. 23.
    Contopoulos, G.: On the existence of a third integral of motion. Astron. J. 68, 1–14 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Waluya, S.B., Van Horssen, W.T.: On approximations of first integrals for strongly nonlinear oscillators. Nonlinear Dyn. 32, 109–141 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Waluya, S.B., Van Horssen, W.T.: Asymptotic approximations of first integrals for a nonlinear oscillator. Nonlinear Anal. 51(8), 1327–1346 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Waluya, S.B., Van Horssen, W.T.: On approximations of first integrals for a system of weakly nonlinear, coupled harmonic oscillators. Nonlinear Dyn. 30, 243–266 (2002)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Van Horssen, W.T.: On integrating factors for ordinary differential equations. Nieuw Archief voor Wiskunde 15, 15–26 (1997)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Van Horssen, W.T.: A perturbation method based on integrating factors. SIAM J. Appl. Math. 59(4), 1427–1443 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Van Horssen, W.T.: A perturbation method based on integrating vectors and multiple scales. SIAM J. Appl. Math. 59(4), 1444–1467 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Van Horssen, W.T.: On integrating vectors and multiple scales for singularly perturbed ordinary differential equations. Nonlinear Anal. TMA 46, 19–43 (2001)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Unal, G.: Approximate generalized symmetries, normal forms and approximate first integrals. Phys. Lett. A 269, 13–30 (2000)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries. Math. USSR Sb. 64, 427–441 (1989)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Differential equations with a small parameter: exact and approximate symmetries. In: Ibragimov, N.H. (ed.) CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3. CRC Press, Boca Raton (1996)Google Scholar
  34. 34.
    Johnpillai, A.G., Kara, A.H.: Variational formulation of approximate symmetries and conservation laws. Int. J. Theor. Phys. 40(8), 1501–1509 (2001)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Johnpillai, A.G., Kara, A.H., Mahomed, F.M.: Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter. J. Comput. Appl. Math. 223(1), 508–518 (2009)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wang, P.: Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system. Nonlinear Dyn. 68, 53–62 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Unal, G., Gorali, G.: Approximate first integrals of a galaxy model. Nonlinear Dyn. 28, 195–211 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Driessche, P.V.D., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Khan, N.A., Jamil, M., Ali, S.A., Khan, N.A.: Solutions of the force-free Duffing–van der Pol oscillator equation. Int. J. Differ. Equ. (2011).  https://doi.org/10.1155/2011/852919
  40. 40.
    Hassan, A.U., Hodaei, H., Miri, M.A., Khajavikhan, M., Christodoulides, D.N.: Integrable nonlinear parity-time symmetric optical oscillator. Phys. Rev. E 93, 042219 (2016)CrossRefGoogle Scholar
  41. 41.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, Weinheim (1995)CrossRefGoogle Scholar
  42. 42.
    Naz, R., Naeem, I.: Generalization of approximate partial Noether approach in phase space. Nonlin Dyn. 88(1), 735–748 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kovacic, I.: Conservation laws of two coupled non-linear oscillators. Int. J. Nonlinear Mech. 41, 751–760 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsLahore University of Management SciencesLahore CantonmentPakistan

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