Dichotomy between a generalized Lyness difference equation with period-two coefficients and its perturbation

• Published in: Applied mathematics letters Vol. 109; p. 106522
• Main Authors: Deng, Guifeng, Li, Xianyi, Lu, Qiuying, Qian, Lili
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.11.2020
• Subjects: Dichotomy; Global asymptotic stability; Lyapunov function; Lyness difference equation with periodic two coefficients; Periodicity; Dichotomy; Lyness difference equation with periodic two coefficients; Global asymptotic stability; Periodicity; Lyapunov function
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2020.106522

We find a dichotomy between the system of difference equations un+1=(a+cvn)∕un and vn+1=(b+dun+1)∕vn, n=0,1,2,…, and its perturbed system un+1=(a+cvn)∕un and vn+1=(b+dun+1+ηvn2)∕(vn+ηvn), n=0,1,2,…, where a,b,c and d are arbitrary positive real numbers, η≥0 and the initial values u0,v0>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.