Classification of van der Pol-type equations with respect to admitted approximate transformation groups transforming a small parameter is given. It is shown that approximate symmetries transforming the small parameter as well as the usual approximate symmetries can be used for approximate integration (e.g., by method of successive reduction of order) of ordinary differential equations with a small parameter.
V. A. Baikov, “Approximate symmetries of an equation of van der Pol type,” Differential Equations, vol. 30, no. 10, pp. 1684–1686 (1995), 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNetV. A. Baikov, R. K. Gazizov, and N. H. Ibragimov, “Approximate symmetries,” Mathematics of the USSR Sbornik, vol. 64, no. 2, pp. 427–441, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNetV. A. Baikov, R. K. Gazizov, and N. H. Ibragimov, “Approximate transformation groups and symmetry Lie algebras,” in CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, pp. 31–67, CRC Press, Florida, 1996, chapter 2.
View at: Google ScholarV. A. Baikov and N. H. Ibragimov, “Continuation of approximate transformation groups via multiple time scales method,” Nonlinear Dynamics, vol. 22, no. 1, pp. 3–13, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, 1989.
View at: Zentralblatt MATH | MathSciNetE. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1991.
View at: Zentralblatt MATH | MathSciNetN. H. Ibragimov, “Seven miniatures on group analysis,” Differential Equations, vol. 29, no. 10, pp. 1511–1520, 1993.
View at: Google Scholar | Zentralblatt MATHN. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, vol. 4 of Wiley Series in Mathematical Methods in Practice, John Wiley & Sons, Chichester, 1999.
View at: Zentralblatt MATH | MathSciNetA. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.
View at: Zentralblatt MATH | MathSciNetL. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982, English translation by W. F. Ames.
View at: Zentralblatt MATH | MathSciNet