Stability Chaos and Periodic Solution of Delayed Rational Recursive Sequence

Abstract

In this paper, we will investigate a non-linear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semi cycles and global asymptotic stability of all positive solutions of

s_n+1=As_n+Bs_n−l+(α+βs_n−k)/(A₀+B₀s_n−k), n=0,1,...

where the parameters A, B, α, β and A₀, B₀ and the initial conditions s_−r, s_−r+1, s_−r+2, ..., s₀ are arbitrary positive real numbers, r=max{l,k}. Finally, we study the global stability of this equation through numerically solved examples and confirm our theoretical discussion through it.