Dichotomy between a generalized Lyness difference equation with period-two coefficients and its perturbation. (November 2020)

• Record Type: Journal Article
• Authors: Deng, Guifeng; Li, Xianyi; Lu, Qiuying; Qian, Lili
• Is Part Of: Applied mathematics letters. Volume 109(2020)
• Publication Date: 2020-11
• Subjects: Dichotomy -- Lyness difference equation with periodic two coefficients -- Periodicity -- Global asymptotic stability -- Lyapunov function; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2020.106522
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: We find a dichotomy between the system of difference equations u n + 1 = ( a + c v n ) ∕ u n and v n + 1 = ( b + d u n + 1 ) ∕ v n, n = 0, 1, 2, …, and its perturbed system u n + 1 = ( a + c v n ) ∕ u n and v n + 1 = ( b + d u n + 1 + η v n 2 ) ∕ ( v n + η v n ), n = 0, 1, 2, …, where a, b, c and d are arbitrary positive real numbers, η ≥ 0 and the initial values u 0, v 0 > 0, which originate from the Lyness difference equation with period-two coefficients. Namely, there are infinitely many initial conditions giving rise to periodic sequences with infinitely many different periods generated by the system of difference equations whereas all solutions of the perturbed system with η > 0 are globally asymptotically stable.