Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

IFSL > Volume 10 > On the Renormalization Group Techniques for the...
< Back to Volume

On the Renormalization Group Techniques for the Cubic-Quintic Duffing Equation

Full Text PDF


We apply the renormalization group techniques for solving the nonlinear cubic-quintic Duffing equation in the presence of an external periodic, non-autonomous force with an additional damping term. We also make a comparative study with the multiple-time scale approach and show that the correction to the frequency is the same.


International Frontier Science Letters (Volume 10)
S. Das "On the Renormalization Group Techniques for the Cubic-Quintic Duffing Equation", International Frontier Science Letters, Vol. 10, pp. 1-7, 2016
Online since:
Dec 2016

[1] L.Y. Chen, N. Goldenfeld, Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys Rev. 54 (1996) 376-394.

[2] L.Y. Chen, N. Goldenfeld, Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett. 73(10) (1994) 1311.

[3] D. Banerjee, J.K. Bhattacharjee, Analyzing jump phenomena and stability in nonlinear oscillators using renormalization group arguments, Am. J. Phys. 78(2) (2010) 142-149.

[4] R.E.L. DeVille et al., Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Physica D: Nonlinear Phenomena. 237(8) (2008) 1029- 1052.

[5] N. Goldenfeld, D. Pines , Westview Press, Lectures on phase transitions and the Renormalization Group, (1992).

[6] D.J. Amit, V.M. Mayor, Field Theory; the Renormalization Group and Critical Phenomena: Graphs to Computers, World Scientific Press, (2005).

[7] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford, Clarendon, (1989).

[8] T. Tao et al., Renormalization Group Method for Soliton Evolution in a Perturbed KdV Equation, Chin. Phys. Lett. 26(6) (2009) 060501.

[9] V. Chua, Cubic-Quintic Duffing Oscillators. (unpublished).

[10] S.K. Lai et al., Applied Mathematical Modelling, Simulation and Computation for Engineering and Environmental systems. 33 (2009) 852.

[11] J.I. Ramos, On Linstedt-Poincaré techniques for the quintic Duffing equation, Applied Mathematics and Computation. 193(2) (2007) 303-310.

[12] C.W. Lim et al., Nonlinear free vibration of an elastically-restrained beam with a point mass via the Newton-harmonic balancing approach, International Journal of Nonlinear Sciences and Numerical simulation. 10(5) (2009) 661-674.

[13] A. Beléndez et al., Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions, Journal of Applied Mathematics. 2012 (2012) 286290.

[14] M. Lakshmanan, S. Rajasekar, Nonlinear dynamics : Integrability, Chaos and Patterns, Advanced Texts in Physics, Springer-Verlag, Berlin, (2003).

[15] B. Bagchi et al., Nonlinear dynamics of a position-dependent mass-driven Duffing-type oscillator, J. Phys. A: Math. Theor. 46(3) (2012) 032001.

[16] V. Chithika Ruby, M. Senthilvelan, M. Lakshmanan, Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator, J. Phys. A: Math. Theor. 45(38) (2012) 382002.

[17] A. Lindstedt, Abh. K. Akad. Wiss. St. Petersburg. 31(4) (1882).

[18] N. Minorsky, Nonlinear oscillations, Melbourne : Krieger, (1974).

Show More Hide