Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

By using the KAMtheory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: xn+1 = (2axn)/(1 + x2n) – xn-1, n = 0, 1, …, where x-1, x0, ∈ (−∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions

Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12,  n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period

Global behavior of a three-dimensional linear fractional system of difference equations

AbstractWe investigate the global asymptotic behavior of solutions of the system of difference equations xn+1=a+xnb+yn,yn+1=c+ynd+zn,zn+1=e+znf+xn,n=0,1,…, where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x0, y0, and z0 are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters

Stability of the KTH order lyness\u27 equation with a period-k coefficient

We first investigate the Lyapunov stability of the period-three solution of Todd\u27s equation with a period-three coefficient: xn+1 = 1+x n + xn-1/pnxx-2, n = 0. 1, . . . where pn{ α, for n = 3l β for n = 3l +1 γ, for n = 3l + 2, l = 0, 1, . . . α, β, and γ positive. Then for k = 2,3, . . . we extend our stability result to the k-order equation, xn-1 = 1 + xn + . . . + xn...k+2/pnxn-k+1, n = 0. 1, . . . where pn is a periodic coefficient of period k with positive real values and x-k-1, . . ., x-1, x0 ∈ (0, ∞). © World Scientific Publishing Company