Nonautonomous Riccati difference equation with real $k$-periodic ($k\geq 1$) coefficients

TL;DR

This paper analyzes the behavior of non-autonomous Riccati difference equations with periodic coefficients, focusing on the structure of solutions and forbidden sets, providing detailed insights into their dynamics.

Contribution

It offers a detailed analysis of the solutions and forbidden sets for Riccati difference equations with periodic coefficients, extending understanding of their dynamic behavior.

Findings

Characterization of forbidden sets for periodic Riccati equations

Analysis of solution behavior based on initial conditions

Insights into the impact of periodic coefficients on solution dynamics

Abstract

We study the non-autonomous Riccati difference equation \[x_{n+1}=\frac{a_nx_n+b_n}{c_nx_n+d_n}, \ n=0,1,2,\cdots\] where $(a_n)_{n\geq0}, \ (b_n)_{n\geq0}, \ (c_n)_{n\geq0}, \ \text{and} \ (d_n)_{n\geq0}$ are $k$-periodic sequences, $k\geq 1$, with initial value $x_0 \in \mathbb{R}$. Precisely we give a detailed analysis of the forbidden set and the character of the solutions.