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Zbl 1131.34030
Bluman, George; Cheviakov, Alexei F.; Senthilvelan, M.
Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems. (English)
[J] J. Math. Anal. Appl. 339, No. 2, 1199-1209 (2008). ISSN 0022-247X

Summary: We consider a three-parameter class of Liénard type nonlinear dissipative systems of the form $\ddot x + (b+3kx)\dot x + k^2x^3 + bkx^2 +\lambda x = 0$. Since such dissipative systems admit an eight-parameter Lie group of point transformations, it follows that there exists a (complex) point transformation mapping such a system into the free particle system $\ddot x = 0$. Normally, such an explicit point transformation cannot be found. Here we find such an explicit point transformation through exploiting the group properties of the determining equations that lead to it. Consequently, we obtain the explicit general solution of such dissipative systems. Moreover, we completely characterize the asymptotic and/or finite time blow-up behaviour of such systems in terms of their three parameters and initial data.

MSC 2000:: *34C14 Symmetries, invariants
34A05 Methods of solution of ODE

Keywords: nonlinear oscillator; Lane-Emden equation; Liénard system; symmetry

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