Stability of non-autonomous difference equations: simple ideas leading to useful results

Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations

“…Hamaya uses Liapunov and semicycle methods in [6] to obtain sufficient conditions for the global attractivity of the origin for the following special case of (1) x n+1 = αx n + a tanh x n − k i=1 b i x n−i with 0 ≤ α < 1, a > 0 and b i ≥ 0. These results can also be obtained using only the contraction method in [13] and [17]; also see [12] for a discussion of alternative methods. The results in [17] are used in [18], Section 4.3D, to prove the global asymptotic stability of the origin for an autonomous special case of (1) with a i , b i ≥ 0 for all i and g n = g for all n, where g is a continuous, non-negative function.…”

“…The parameters a j , b j , j = k − 1, k which affect ρ but do not appear in (17) and (18) are not free. They satisfy (12) and (13) in Lemma 3 and for the complex conjugate pair of roots in Theorem 4 they take the forms…”

Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

“…Recently, growing interest has been paid to investigating the stability of linear difference systems with delay (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]). …”

Local stability implies global stability for the 2-dimensional Ricker map

“…For d = 1, i.e., for the 2-dimensional Ricker map R 2 , condition (1.4) is equivalent to 0 < α < 0.875. See also [16] and [15] in the topic.…”