Global behaviour of the Riccati difference equation of order two

Abstract

The second order rational difference equationassociated with a linear third order difference equation in the same way that the first order Riccati equation (c ¼ 0) is associated with a linear second order equation. This association and other features are used to study the global behaviour of solutions. If a; b $ 0 and a þ b; c . 0 then the above equation has a unique positive fixed point that is stable and attracts all orbits with initial points outside a set M of Lebesgue measure zero in the plane. However, within M there is an invariant subset containing periodic orbits of all possible periods.

A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers

“…Besides these studies, for related studies on solving difference equations and systems of difference equations and investigating the asymptotic behavior of their solutions, see [20,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Semiconjugate Factorization and Reduction of Order in Difference Equations

“…The factor equation in Corollary 16 is known as the Riccati difference equation of order k; see [4].…”

Factorizations and Reductions of Order in Quadratic and other Non-recursive Higher Order Difference Equations

“…This states that the magnitude of a quantity x n at time n is a fraction of the difference between its values in the two immediately preceding times; however, we cannot determine the sign of x n from (3). As a possible physical interpretation of (3) imagine a node in a circuit that in every second n fires a pulse x n that may go either to the right (if x n > 0) or to the left (if x n < 0) but the amplitude |x n | of the pulse obeys Eq.…”

“…As a possible physical interpretation of (3) imagine a node in a circuit that in every second n fires a pulse x n that may go either to the right (if x n > 0) or to the left (if x n < 0) but the amplitude |x n | of the pulse obeys Eq. (3). With regard to the variety of solutions for (3) we note that the direction of each pulse is entirely unpredictable, regardless of the directions of previous pulses emitted by the node; hence a large number of solutions are possible for (3).…”