Stability analysis of a certain class of difference equations by using KAM theory

Abstract

Abstract 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.

Keywords

• Area-preserving map
• Difference equation
• KAM theory
• Periodic orbit