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29. 5. 2019.
Stability analysis of a certain class of difference equations by using KAM theory
By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form xn+1=f(xn)xn−1,n=0,1,…$x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots $ , f:(0,+∞)→(0,+∞)$f:(0,+\infty )\to (0,+\infty )$, f is sufficiently smooth and the initial conditions are x−1,x0∈(0,+∞)$x_{-1}, x _{0}\in (0,+\infty )$. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α1$\alpha _{1}$. Then we apply the results to several difference equations.