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

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 \(f(y)=-y^{3}\).


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


Zbl 0592.39005
Full Text: DOI


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[3] DOI: 10.1016/0016-0032(86)90054-2 · Zbl 0592.39005
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