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

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

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.
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, (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.

[3]

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

[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.

[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.

[6]

G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz,", F. Vieweg u. Sohn, (1918).

[7]

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

[8]

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

[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.

[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.

[11]

Z. Feng and Q. G. Meng, First integrals for the damped Helmholtz oscillator,, Int. J. Comput. Math. \textbf{87} (2010), 87 (2010), 2798.

[12]

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

[13]

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

[14]

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

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983).

[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.

[17]

P. E. Hydon, "Symmetry Methods for Differential Equations,", Cambridge University Press, (2000).

[18]

E. I. Ince, "Ordinary Differential Equations,", Dover, (1956).

[19]

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

[20]

M. Lakshmanan and S. Rajasekar, "Nonlinear Dynamics: Integrability, Chaos and Patterns,", Springer Verlag, (2003).

[21]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Springer Verlag, (1993).

[22]

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

[23]

A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations,", 2nd edition, (2003).

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

show all references

References:
[1]

M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform,", SIAM, (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.

[3]

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

[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.

[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.

[6]

G. Duffing, "Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz,", F. Vieweg u. Sohn, (1918).

[7]

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

[8]

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

[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.

[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.

[11]

Z. Feng and Q. G. Meng, First integrals for the damped Helmholtz oscillator,, Int. J. Comput. Math. \textbf{87} (2010), 87 (2010), 2798.

[12]

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

[13]

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

[14]

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

[15]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983).

[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.

[17]

P. E. Hydon, "Symmetry Methods for Differential Equations,", Cambridge University Press, (2000).

[18]

E. I. Ince, "Ordinary Differential Equations,", Dover, (1956).

[19]

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

[20]

M. Lakshmanan and S. Rajasekar, "Nonlinear Dynamics: Integrability, Chaos and Patterns,", Springer Verlag, (2003).

[21]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Springer Verlag, (1993).

[22]

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

[23]

A. D. Polyanin and V. F. Zaitsev, "Handbook of Exact Solutions for Ordinary Differential Equations,", 2nd edition, (2003).

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[29]

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

[1]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[2]

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

[3]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453

[4]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

[5]

Dmitry Treschev. Oscillator and thermostat. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1693-1712. doi: 10.3934/dcds.2010.28.1693

[6]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977

[7]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925

[8]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121

[9]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451

[10]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[11]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2006, 5 (3) : 515-528. doi: 10.3934/cpaa.2006.5.515

[12]

Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2007, 6 (1) : 69-82. doi: 10.3934/cpaa.2007.6.69

[13]

Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083

[14]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[15]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[16]

János Karsai, John R. Graef. Attractivity properties of oscillator equations with superlinear damping. Conference Publications, 2005, 2005 (Special) : 497-504. doi: 10.3934/proc.2005.2005.497

[17]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[18]

Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1775-1799. doi: 10.3934/dcds.2012.32.1775

[19]

Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943

[20]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]