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Duffing-van der Pol-type oscillator systems
1. | Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539 |
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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