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December  2014, 7(6): 1231-1257. doi: 10.3934/dcdss.2014.7.1231

Duffing-van der Pol-type oscillator systems

Zhaosheng Feng 1,

1. 

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

Received  January 2013 Revised  September 2013 Published  June 2014

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. After applying the method of differentiable dynamics to analyze the bifurcation set and bifurcations of equilibrium points, we use the Lie symmetry reduction 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 included accordingly.
Keywords: autonomous system, Lie symmetry, infinitesimal generator, bifurcation, Van der Pol oscillator, first integral, Duffing equation, diffeomorphism..
Mathematics Subject Classification: Primary: 34C14, 37G10; Secondary: 34C0.
Citation: Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1231-1257. doi: 10.3934/dcdss.2014.7.1231
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. doi: 10.1088/0305-4470/36/3/308.  Google Scholar

[3]

H. W. Broer, B. Krauskopf and G. Vegter, Global Analysis of Dynamical Systems, Institue of Physics Publishing, London, 2001. doi: 10.1887/0750308036.  Google Scholar

[4]

A. Canada, P. Drabek and A. Fonda, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier/North-Holland, Amsterdam, 2006.  Google Scholar

[5]

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. doi: 10.1098/rspa.2005.1465.  Google Scholar

[6]

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. doi: 10.1088/0305-4470/34/14/308.  Google Scholar

[7]

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

[8]

Z. Feng, On traveling wave solutions of the Burgers-Korteweg-de Vries equation, Nonlinearity, 20 (2007), 343-356. doi: 10.1088/0951-7715/20/2/006.  Google Scholar

[9]

Z. Feng, The first-integral method to the Burgers-Korteweg-de Vries equation, J. Phys. A (Math. Gen.), 35 (2002), 343-349. doi: 10.1088/0305-4470/35/2/312.  Google Scholar

[10]

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. doi: 10.3934/dcdsb.2006.6.1097.  Google Scholar

[11]

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

[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. doi: 10.1007/s00033-008-8092-0.  Google Scholar

[13]

G. Gao and Z. Feng, First integrals for the Duffng-van der Pol-type oscillator, E. J. Diff. Equs., 19 (2010), 123-133.  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, 1990.  Google Scholar

[16]

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.  Google Scholar

[17]

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

[18]

P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511623967.  Google Scholar

[19]

E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.  Google Scholar

[20]

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

[21]

M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns, Springer Verlag, New York, 2003. doi: 10.1007/978-3-642-55688-3.  Google Scholar

[22]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[23]

M. Prelle and M. Singer, Elementary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.  Google Scholar

[24]

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

[25]

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

[26]

S. N. Rasband, Marginal stability boundaries for some driven, damped, non-linear oscillators, Int. J. Non-Linear Mech., 22 (1987), 477-495. doi: 10.1016/0020-7462(87)90038-2.  Google Scholar

[27]

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. doi: 10.1088/0305-4470/28/7/015.  Google Scholar

[28]

F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis, I, Communications of the Mathematics Institute, Rijksuniversiteit Utrecht, 1974, 1-59.  Google Scholar

[29]

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

[30]

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

[31]

V. F. Zaitsev and A. D. Polyanin, Handbook of Ordinary Differential Equations (in Russian), Fizmatlit, Moscow, 2001. 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. doi: 10.1088/0305-4470/36/3/308.  Google Scholar

[3]

H. W. Broer, B. Krauskopf and G. Vegter, Global Analysis of Dynamical Systems, Institue of Physics Publishing, London, 2001. doi: 10.1887/0750308036.  Google Scholar

[4]

A. Canada, P. Drabek and A. Fonda, Handbook of Differential Equations: Ordinary Differential Equations, Elsevier/North-Holland, Amsterdam, 2006.  Google Scholar

[5]

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. doi: 10.1098/rspa.2005.1465.  Google Scholar

[6]

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. doi: 10.1088/0305-4470/34/14/308.  Google Scholar

[7]

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

[8]

Z. Feng, On traveling wave solutions of the Burgers-Korteweg-de Vries equation, Nonlinearity, 20 (2007), 343-356. doi: 10.1088/0951-7715/20/2/006.  Google Scholar

[9]

Z. Feng, The first-integral method to the Burgers-Korteweg-de Vries equation, J. Phys. A (Math. Gen.), 35 (2002), 343-349. doi: 10.1088/0305-4470/35/2/312.  Google Scholar

[10]

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. doi: 10.3934/dcdsb.2006.6.1097.  Google Scholar

[11]

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

[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. doi: 10.1007/s00033-008-8092-0.  Google Scholar

[13]

G. Gao and Z. Feng, First integrals for the Duffng-van der Pol-type oscillator, E. J. Diff. Equs., 19 (2010), 123-133.  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, 1990.  Google Scholar

[16]

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974.  Google Scholar

[17]

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

[18]

P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, New York, 2000. doi: 10.1017/CBO9780511623967.  Google Scholar

[19]

E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.  Google Scholar

[20]

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

[21]

M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns, Springer Verlag, New York, 2003. doi: 10.1007/978-3-642-55688-3.  Google Scholar

[22]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[23]

M. Prelle and M. Singer, Elementary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.  Google Scholar

[24]

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

[25]

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

[26]

S. N. Rasband, Marginal stability boundaries for some driven, damped, non-linear oscillators, Int. J. Non-Linear Mech., 22 (1987), 477-495. doi: 10.1016/0020-7462(87)90038-2.  Google Scholar

[27]

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. doi: 10.1088/0305-4470/28/7/015.  Google Scholar

[28]

F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis, I, Communications of the Mathematics Institute, Rijksuniversiteit Utrecht, 1974, 1-59.  Google Scholar

[29]

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

[30]

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

[31]

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

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