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

Duffing-van der Pol-type oscillator systems

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.
Citation: Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete and 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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[15]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[25]

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

[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.

[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.

[28]

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

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

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

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[15]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[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.

[24]

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

[25]

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

[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.

[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.

[28]

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

[29]

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

[30]

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

[31]

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

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