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

• Main Authors: Senada Kalabušić, Emin Bešo, Naida Mujić, Esmir Pilav
• Format: Article
• Language: English
• Published: SpringerOpen 2019-05-01
• Series: Advances in Difference Equations
• Subjects: Area-preserving map; Difference equation; KAM theory; Periodic orbit
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2148-7

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.