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September  2011, 10(5): 1377-1391. doi: 10.3934/cpaa.2011.10.1377

Duffing--van der Pol--type oscillator system and its first integrals

Zhaosheng Feng 1, , Guangyue Gao 2, and  Jing Cui 2,

1. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539
2. 

Department of Mathematics, University of Texas{Pan American, Edinburg, Texas 78539, United States, United States

Received  March 2009 Revised  February 2010 Published  April 2011

In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing--van der Pol--type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. We apply the Lie symmetry method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are discussed accordingly.
Keywords: parametric solution., Lie symmetry method, Lie point symmetry, prolonged infinitesimal operator, diffeomorphism, First integral, van der Pol oscillator, Duffing oscillator.
Mathematics Subject Classification: Primary: 76M60; Secondary: 37K10, 34L3.
Citation: Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377
References:
[1]

M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform," SIAM, Philadelphia, 1981.  Google Scholar

[2]

J. A. Almendral and M. A. F. Sanjuán, Integrability and symmetries for the Helmholtz oscillator with friction, J. Phys. A (Math. Gen.), 36 (2003), 695-710.  Google Scholar

[3]

A. Canada, P. Drabek and A. Fonda, "Handbook of Differential Equations: Ordinary Differential Equations," Volumes 2-3, Elsevier, 2005.  Google Scholar

[4]

V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. Lond. Ser. A, 461 (2005), 2451-2476.  Google Scholar

[5]

L. G. S. Duarte, S. E. S. Duarte, A. C. P. da Mota and J. E. F. Skea, Solving the second-order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A (Math. Gen.), 34 (2001), 3015-3024.  Google Scholar

[6]

G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz," F. Vieweg u. Sohn, Braunschweig, 1918. Google Scholar

[7]

Z. Feng, On traveling wave solutions of the Burgers-Korteweg-de Vries equation, Nonlinearity, 20 (2007), 343-356.  Google Scholar

[8]

Z. Feng, The first-integral method to the Burgers-Korteweg-de Vries equation, J. Phys. A (Math. Gen.), 35 (2002), 343-350.  Google Scholar

[9]

Z. Feng, G. Chen and S. B. Hsu, A qualitative study of the damped Duffing equation and applications, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1097-1112.  Google Scholar

[10]

Z. Feng and Q. G. Meng, Exact solution for a two-dimensional KdV-Burgers-type equation with nonlinear terms of any order, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 285-291.  Google Scholar

[11]

Z. Feng and Q. G. Meng, First integrals for the damped Helmholtz oscillator, Int. J. Comput. Math. 87 (2010), 2798-2810.  Google Scholar

[12]

Z. Feng, S. Zheng and D. Y. Gao, Traveling wave solutions to a reaction-diffusion equation, Z. angew. Math. Phys., 60 (2009), 756-773.  Google Scholar

[13]

G. Gao and Z. Feng, First integrals for the Duffng-van der Pol-type oscillator, E. J. Diff. Equs., 2010 (2010), 1-12.  Google Scholar

[14]

M. Gitterman, "The Noisy Oscillator: the First Hundred Years, from Einstein until Now," World Scientific Publishing, Singapore, 2005.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.  Google Scholar

[16]

P. Holmes and D. Rand, Phase portraits and bifurcations of the non-linear oscillator: $\ddotx +(\alpha +\gamma x^2) \dotx + \beta x + \delta x^3=0$, Int. J. Non-Linear Mech., 15 (1980), 449-458.  Google Scholar

[17]

P. E. Hydon, "Symmetry Methods for Differential Equations," Cambridge University Press, New York, 2000.  Google Scholar

[18]

E. I. Ince, "Ordinary Differential Equations," Dover, New York, 1956. Google Scholar

[19]

D. W. Jordan and P. Smith, "Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers," Oxford University Press, New York, 2007.  Google Scholar

[20]

M. Lakshmanan and S. Rajasekar, "Nonlinear Dynamics: Integrability, Chaos and Patterns," Springer Verlag, New York, 2003.  Google Scholar

[21]

P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer Verlag, New York, 1993.  Google Scholar

[22]

M. Prelle and M. Singer, Elementary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-229.  Google Scholar

[23]

A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, London: CRC Press, 2003.  Google Scholar

[24]

A. D. Polyanin, V. F. Zaitsev and A. Moussiaux, "Handbook of First Order Partial Differential Equations," Taylor & Francis, London, 2002.  Google Scholar

[25]

S. N. Rasband, Marginal stability boundaries for some driven, damped, non-linear oscillators, Int. J. Non-Linear Mech., 22 (1987), 477-495.  Google Scholar

[26]

M. Senthil Velan and M. Lakshmanan, Lie symmetries and infinite-dimensional Lie algebras of certain nonlinear dissipative systems, J. Phys. A (Math. Gen.), 28 (1995), 1929-1942.  Google Scholar

[27]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701-710, 754-762. Google Scholar

[28]

B. van der Pol and J. van der Mark, Frequency demultiplication, Nature, 120 (1927), 363-364. Google Scholar

[29]

V. F. Zaitsev and A. D. Polyanin, "Handbook of Ordinary Differential Equations," Fizmatlit, Moscow, 2001 (in Russian).  Google Scholar

show all references

References:
[1]

M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform," SIAM, Philadelphia, 1981.  Google Scholar

[2]

J. A. Almendral and M. A. F. Sanjuán, Integrability and symmetries for the Helmholtz oscillator with friction, J. Phys. A (Math. Gen.), 36 (2003), 695-710.  Google Scholar

[3]

A. Canada, P. Drabek and A. Fonda, "Handbook of Differential Equations: Ordinary Differential Equations," Volumes 2-3, Elsevier, 2005.  Google Scholar

[4]

V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations, Proc. R. Soc. Lond. Ser. A, 461 (2005), 2451-2476.  Google Scholar

[5]

L. G. S. Duarte, S. E. S. Duarte, A. C. P. da Mota and J. E. F. Skea, Solving the second-order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A (Math. Gen.), 34 (2001), 3015-3024.  Google Scholar

[6]

G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz," F. Vieweg u. Sohn, Braunschweig, 1918. Google Scholar

[7]

Z. Feng, On traveling wave solutions of the Burgers-Korteweg-de Vries equation, Nonlinearity, 20 (2007), 343-356.  Google Scholar

[8]

Z. Feng, The first-integral method to the Burgers-Korteweg-de Vries equation, J. Phys. A (Math. Gen.), 35 (2002), 343-350.  Google Scholar

[9]

Z. Feng, G. Chen and S. B. Hsu, A qualitative study of the damped Duffing equation and applications, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1097-1112.  Google Scholar

[10]

Z. Feng and Q. G. Meng, Exact solution for a two-dimensional KdV-Burgers-type equation with nonlinear terms of any order, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 285-291.  Google Scholar

[11]

Z. Feng and Q. G. Meng, First integrals for the damped Helmholtz oscillator, Int. J. Comput. Math. 87 (2010), 2798-2810.  Google Scholar

[12]

Z. Feng, S. Zheng and D. Y. Gao, Traveling wave solutions to a reaction-diffusion equation, Z. angew. Math. Phys., 60 (2009), 756-773.  Google Scholar

[13]

G. Gao and Z. Feng, First integrals for the Duffng-van der Pol-type oscillator, E. J. Diff. Equs., 2010 (2010), 1-12.  Google Scholar

[14]

M. Gitterman, "The Noisy Oscillator: the First Hundred Years, from Einstein until Now," World Scientific Publishing, Singapore, 2005.  Google Scholar

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.  Google Scholar

[16]

P. Holmes and D. Rand, Phase portraits and bifurcations of the non-linear oscillator: $\ddotx +(\alpha +\gamma x^2) \dotx + \beta x + \delta x^3=0$, Int. J. Non-Linear Mech., 15 (1980), 449-458.  Google Scholar

[17]

P. E. Hydon, "Symmetry Methods for Differential Equations," Cambridge University Press, New York, 2000.  Google Scholar

[18]

E. I. Ince, "Ordinary Differential Equations," Dover, New York, 1956. Google Scholar

[19]

D. W. Jordan and P. Smith, "Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers," Oxford University Press, New York, 2007.  Google Scholar

[20]

M. Lakshmanan and S. Rajasekar, "Nonlinear Dynamics: Integrability, Chaos and Patterns," Springer Verlag, New York, 2003.  Google Scholar

[21]

P. J. Olver, "Applications of Lie Groups to Differential Equations," Springer Verlag, New York, 1993.  Google Scholar

[22]

M. Prelle and M. Singer, Elementary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-229.  Google Scholar

[23]

A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, London: CRC Press, 2003.  Google Scholar

[24]

A. D. Polyanin, V. F. Zaitsev and A. Moussiaux, "Handbook of First Order Partial Differential Equations," Taylor & Francis, London, 2002.  Google Scholar

[25]

S. N. Rasband, Marginal stability boundaries for some driven, damped, non-linear oscillators, Int. J. Non-Linear Mech., 22 (1987), 477-495.  Google Scholar

[26]

M. Senthil Velan and M. Lakshmanan, Lie symmetries and infinite-dimensional Lie algebras of certain nonlinear dissipative systems, J. Phys. A (Math. Gen.), 28 (1995), 1929-1942.  Google Scholar

[27]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Review, 1 (1920), 701-710, 754-762. Google Scholar

[28]

B. van der Pol and J. van der Mark, Frequency demultiplication, Nature, 120 (1927), 363-364. Google Scholar

[29]

V. F. Zaitsev and A. D. Polyanin, "Handbook of Ordinary Differential Equations," Fizmatlit, Moscow, 2001 (in Russian).  Google Scholar

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