An order \(\varepsilon^2\) perturbation procedure for second-order nonlinear difference equations. (English) Zbl 0963.39005

Mickens, Ronald E.

J. Difference Equ. Appl. 6, No. 3, 337-349 (2000).

The author presents an order \(\varepsilon^{2}\) perturbation procedure for calculating uniformly valid approximations to periodic solutions of the second-order difference equation \[ \frac{y_{k+1}-2y_{k}+y_{k-1}}{4\sin^{2}(h/2)}+y_{k}= \varepsilon f(y_{k}), \] where \(0<\varepsilon\ll 1\), \(f(y)\) is a polynomial of \(y\) and \(h\in (0,1)\) a parameter. The procedure is a direct extension of the well-known Lindstedt-Poincaré method in the asymptotic theory of ordinary differential equations. The order \(\varepsilon\) calculation was given in an earlier work by the author [J. Franklin Inst. 321, 39-47 (1986; Zbl 0592.39005)]; the method is illustrated by considering the case where \(f(y)=-y^{3}\).

Reviewer: Oleg Anashkin (Simferopol)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis

Keywords:

periodic solutions; perturbation methods; second-order difference equation; Lindstedt-Poincaré method; asymptotic approximation

Citations:

Zbl 0592.39005

References:

[1]	Levy, H. and Lessman, F. 1961. ”Finite Difference Equations”. Edited by: New York: Macmillan. · Zbl 0092.07702
[2]	Mickens, R.E. 1981. ”Introduction to Nonliear Oscillations”. Edited by: Vol. 2, New York: Cambridge University Press.
[3]	DOI: 10.1016/0016-0032(86)90054-2 · Zbl 0592.39005 · doi:10.1016/0016-0032(86)90054-2
[4]	Mickens R.E., Multi-discrete time method 324 pp 263– (1987) · Zbl 0629.39002
[5]	Mickens R.E., World Scientific (1994)
[6]	Nayfeh A.H., Wiley-Interscience (1973)