We find a dichotomy between the system of difference equations $u_{n+1}=(a+c{v}_n)∕u_n$ and $v_{n+1}=(b+d{u}_{n+1})∕v_n$, $n=0,1,2,\dots ,$ and its perturbed system $u_{n+1}=(a+c{v}_n)∕u_n$ and $v_{n+1}=(b+d{u}_{n+1}+\eta v_n^2)∕(v_n+\eta v_n)$, $n=0,1,2,\dots ,$ where $a,b,c$ and $d$ are arbitrary positive real numbers, $\eta \ge 0$ and the initial values $u_0,v_0>0$, which originate from the Lyness difference equation with period-two coefficients. Namely, there are infinitely many initial conditions giving rise to periodic sequences with infinitely many different periods generated by the system of difference equations whereas all solutions of the perturbed system with $\eta >0$ are globally asymptotically stable.

我们发现差分方程组之间存在二分法 ${ü}_{ñ+\mathrm{1个}}=（\mathrm{一种}+C{v}_{ñ}）∕{ü}_{ñ}$ 和 $v_{ñ+\mathrm{1个}}=（b+d{ü}_{ñ+\mathrm{1个}}）∕v_{ñ}$， $ñ=0，\mathrm{1个}，2，\dots ，$ 及其扰动的系统 ${ü}_{ñ+\mathrm{1个}}=（\mathrm{一种}+C{v}_{ñ}）∕{ü}_{ñ}$ 和 $v_{ñ+\mathrm{1个}}=（b+d{ü}_{ñ+\mathrm{1个}}+\eta v_{ñ}^2）∕（v_{ñ}+\eta v_{ñ}）$， $ñ=0，\mathrm{1个}，2，\dots ，$ 哪里 $\mathrm{一种}，b，C$ 和 $d$ 是任意正实数， $\eta \ge 0$ 和初始值 ${ü}_0，v_0>0$，其源于具有两个周期系数的Lyness差分方程。就是说，存在无限多的初始条件，它们会产生由差分方程组生成的具有无限多个不同周期的周期序列，而扰动系统的所有解都具有$\eta >0$ 是全局渐近稳定的。