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Stability of linear differential equations with a distributed delay
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 |
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). Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[28] |
B. van der Pol and J. van der Mark, Frequency demultiplication,, Nature, 120 (1927), 363. Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[28] |
B. van der Pol and J. van der Mark, Frequency demultiplication,, Nature, 120 (1927), 363. Google Scholar |
[29] |
V. F. Zaitsev and A. D. Polyanin, "Handbook of Ordinary Differential Equations,", Fizmatlit, (2001).
|
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