Global behaviours of rational difference equations of orders two and three with quadratic terms

Abstract

We determine the global behaviours of all solutions of the following rational difference equationsThese equations are related to each other via semiconjugate relations that also let us reduce them to first-order equations. Using this approach, we determine the forbidden sets of each equation explicitly and show that for initial values outside the forbidden sets, their solutions may converge to 0, or to a positive fixed point, or they may be periodic of period 2 or unbounded. In some cases, different types of solutions coexist depending on the initial values.

Global behaviour of a second-order nonlinear difference equation

“…This result was recently generalized by Sedaghat [13], who studied equation (1.3) with g . 0 and arbitrary initial conditions x 21 , x 0 [ R. The author proved that the trichotomy result holds for this case as well.…”

“…Notice that this generalizes the characterization of the forbidden set for equation (1.3) given in [13].…”

“…The key feature of the solutions to (1.1) is that the sequence {y n } defined by y n ¼ x n x n21 for all n $ 0 solves a rational difference equation of Möbius type y nþ1 ¼ y n a þ by n ; ð1:4Þ this is to say, the vectorization of equation (1.1) is semiconjugated to a Möbius map where the corresponding fibres are hyperbolas (see [12,13] for more details on semiconjugated maps). Further, equation (1.4) is reducible to a linear difference equation (see, e.g., [9,10] where the term h(n) depends only on the parameter a and the product a ¼ bx 21 x 0 for each n $ 0.…”

“…An important difference is that there are initial conditions for which a complete solution of (1.3) cannot be constructed. The set of such initial conditions is called the forbidden set, and it is also described in [13].…”

Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms

“…Equation (1.1) is the special case of a general second order quadratic fractional equation of the form x n+1 = Ax 2 n + Bx n x n−1 + Cx 2 n−1 + Dx n + Ex n−1 + F ax 2 n + bx n x n−1 + cx 2 n−1 + dx n + ex n−1 + f , n = 0, 1, · · · , (1.2) with nonnegative parameters and initial conditions such that A + B + C > 0, a + b + c + d + e + f > 0 and ax 2 n + bx n x n−1 + cx 2 n−1 + dx n + ex n−1 + f > 0 , n = 0, 1, · · · . Several global asymptotic results for some special cases of (1.2) were obtained in [4][5][6]15]. The systematic theory of the linear fractional difference equation…”