First Integrals and Integrating Factors of Second-Order Autonomous Systems

[+] Author and Article Information
Tamás Kalmár-Nagy

Department of Fluid Mechanics,
Faculty of Mechanical Engineering,
Budapest University of
Technology and Economics,
Budapest 1111, Hungary
e-mail: jcnd@kalmarnagy.com

Balázs Sándor

Department of Hydraulic and
Water Resources Engineering,
Faculty of Civil Engineering,
Budapest University of
Technology and Economics,
Water Management Research Group—
Hungarian Academy of Sciences,
Budapest 1111, Hungary
e-mail: sandor.balazs@epito.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 8, 2017; final manuscript received May 21, 2018; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090901 (Jul 26, 2018) (7 pages) Paper No: CND-17-1409; doi: 10.1115/1.4040410 History: Received September 08, 2017; Revised May 21, 2018

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

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Gurevich, N. C. , 1961, The Stability of Motion, Pergamon Press, New York.
Rouche, N. , Habets, P. , and Laloy, M. , 1977, Stability Theory by Liapunov's Direct Method, Vol. 4, Springer, New York. [CrossRef]
González-López, A. , 1988, “ Symmetry and Integrability by Quadratures of Ordinary Differential Equations,” Phys. Lett. A, 133(4–5), pp. 190–194. [CrossRef]
Hydon, P. E. , 2000, Symmetry Methods for Differential Equations: A Beginner's Guide, Vol. 22, Cambridge University Press, Cambridge, UK. [CrossRef] [PubMed] [PubMed]
Arrigo, D. J. , 2015, Symmetry Analysis of Differential Equations: An Introduction, Wiley, Hoboken, NJ.
Sarlet, W. , and Bahar, L. Y. , 1980, “ A Direct Construction of First Integrals for Certain Non-Linear Dynamical Systems,” Int. J. Non-Linear Mech., 15(2), pp. 133–146. [CrossRef]
Olver, P. J. , 1995, Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, UK. [CrossRef]
Olver, P. J. , 2000, Applications of Lie Groups to Differential Equations, Vol. 107, Springer Science & Business Media, New York.
Bluman, G. , and Anco, S. , 2008, Symmetry and Integration Methods for Differential Equations, Vol. 154, Springer Science & Business Media, New York.
Muriel, C. , and Romero, J. L. , 2001, “ New Methods of Reduction for Ordinary Differential Equations,” IMA J. Appl. Math., 66(2), pp. 111–125. [CrossRef]
Muriel, C. , and Romero, J. L. , 2008, “ Integrating Factors and λ–Symmetries,” J. Nonlinear Math. Phys., 15(Suppl. 3), pp. 300–309. [CrossRef]
Muriel, C. , and Romero, J. L. , 2009, “ First Integrals, Integrating Factors and λ-Symmetries of Second-Order Differential Equations,” J. Phys. A: Math. Theor., 42(36), p. 365207. [CrossRef]
Duarte, L. G. S. , Duarte, S. E. S. , da Mota, L. A. C. P. , and Skea, J. E. F. , 2001, “ Solving Second-Order Ordinary Differential Equations by Extending the Prelle-Singer Method,” J. Phys. A: Math. General, 34(14), p. 3015. [CrossRef]
Cheb-Terrab, E. S. , and Roche, A. D. , 1999, “ Integrating Factors for Second-Order Odes,” J. Symbolic Comput., 27(5), pp. 501–519. [CrossRef]
Leach, P. G. L. , and Bouquet, S. É. , 2002, “ Symmetries and Integrating Factors,” J. Nonlinear Math. Phys., 9(Suppl. 2), pp. 73–91. [CrossRef]
Hale, J. K. , and Hüseyin, K. , 2012, Dynamics and Bifurcations, Springer-Verlag, New York.
Anco, S. C. , and Bluman, G. , 1998, “ Integrating Factors and First Integrals for Ordinary Differential Equations,” Eur. J. Appl. Math., 9(3), pp. 245–259. [CrossRef]
Ibragimov, N. H. , 2005, “ Invariant Lagrangians and a New Method of Integration of Nonlinear Equations,” J. Math. Anal. Appl., 304(1), pp. 212–235. [CrossRef]
Ibragimov, N. H. , 2006, “ Integrating Factors, Adjoint Equations and Lagrangians,” J. Math. Anal. Appl., 318(2), pp. 742–757. [CrossRef]
Cohen, A. , 1911, An Introduction to the Lie Theory of One-Parameter Groups: With Applications to the Solution of Differential Equations, Health & Company, Washington, DC.
Myra, J. P. , and Singer, M. F. , 1983, “ Elementary First Integrals of Differential Equations,” Trans. Am. Math. Soc., 279(1), pp. 215–229. [CrossRef]
Man, Y. K. , 1994, “ First Integrals of Autonomous Systems of Differential Equations and the Prelle-Singer Procedure,” J. Phys. A: Math. General, 27(10), p. L329. [CrossRef]
Chandrasekar, V. K. , Pandey, S. N. , Senthilvelan, M. , and Lakshmanan, M. , 2006, “ A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems,” J. Math. Phys., 47(2), p. 023508. [CrossRef]
Yaşar, E. , 2011, “ Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators,” Math. Probl. Eng., 2011, p. 916437. [CrossRef]
Cantwell, B. , 2002, Introduction to Symmetry Analysis, Vol. 29, Cambridge University Press, Cambridge, UK. [PubMed] [PubMed]
Baer, S. M. , and Thomas, E. , 1986, “ Singular Hopf Bifurcation to Relaxation Oscillations,” SIAM J. Appl. Math., 46(5), pp. 721–739. [CrossRef]
Erneux, T. , Baer, S. M. , and Mandel, P. , 1987, “ Subharmonic Bifurcation and Bistability of Periodic Solutions in a Periodically Modulated Laser,” Phys. Rev. A, 35(3), p. 1165. [CrossRef]
Beatty, J. , and Mickens, R. E. , 2005, “ A Qualitative Study of the Solutions to the Differential Equation,” J. Sound Vib., 283(1–2), pp. 475–477. [CrossRef]
Mickens, R. E. , 2006, “ Investigation of the Properties of the Period for the Nonlinear Oscillator,” J. Sound Vib., 292(3–5), pp. 1031–1035. [CrossRef]
Kalmár-Nagy, T. , and Erneux, T. , 2008, “ Approximating Small and Large Amplitude Periodic Orbits of the Oscillator,” J. Sound Vib., 313(3–5), pp. 806–811. [CrossRef]




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