On difference Riccati equation and continued fractions

2024/10/10 by Ivanov, Alexey V.
#37C40 #37C55 #37D25 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2410.07730

Abstract

We study a difference Riccati equation Φ(x) + ρ(x)/Φ(x-ω) = v(x) with 1-periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.

Related

• Nonautonomous Riccati difference equation with real k-periodic (k≥ 1) coefficients
• Continued fractions and Bessel functions
• Observations on continued fractions
• Periodic solutions of quaternionic Riccati equations with periodic coeffcients
• Continuants and convergence of certain continued fractions