The Bifurcation of Cycle Length and Global Asymptotic Stability in a Rational Difference Equation with Higher Order

ABSTRACT

A new bifurcation case for the cycle length is found in this paper for rational difference equations, which is shown out from the following fifth order rational difference equation

x_n+1 = (x_n-1x_n-2 + x_n-1x_n-4 + x_n-2x_n-4 + 1+ a) / ( x_n-1x_n-2x_n-4 + x_n-2 + x_n-4 + a ) , n= 0,1,2,...,

where a ∈[0,∞) and the initial values x^-4 , x^-3, x^-2 , x^-1, x⁰ ∈ (0,∞) . Mainly, the perturbation of the initial values may lead to the essential variation of the cycle length rule for the nontrivial solutions of the equation. That is, with the change of the initial values, the successive lengths of positive and negative semicycles for nontrivial solutions of this equation is found to periodically occur with multiple different prime periods, respectively, 4 ,12 . Furthermore, in any one fixed period, the successive occurring order of positive and negative semicycles is completely inverse, i.e., for the period 4 , the order is either 3⁺ ,1^- or 3^- ,1⁺ in a period, and for the period 12 , the order is either 5⁺ ,2^- ,1⁺ ,1^- ,1⁺,2^- or 5^- ,2⁺ ,1⁺ ,1^-,1⁺ ,2⁺ in a period. This rule is different from the known one we have obtained for various rational difference equations. By the use of the rule its positive equilibrium point is verified to be globally asymptotically stable.

Keywords:

Rational difference equation, bifurcation of cycle length, perturbation, global asymptotic stability, semicycle, periodicity.