差分方程$z_{n+1}=\fracα{z_{n}}+ \fracβ{z_{n-1}}$的复动力学
Complex Dynamics of the Difference Equation $z_{n+1}=\fracα{z_{n}}+ \fracβ{z_{n-1}}$
摘要：
研究了具有复参数和任意复初始条件的二阶有理差分方程的动力学性质。还研究了两个相关的差分方程。这类方程的解在复平面上表现出许多丰富而复杂的渐近性态。对这三个差分方程的局部稳定性和周期性进行了分析。我们进一步展示了这个方程的解的几个有趣的特性，使用计算，这不会出现当我们考虑相同的方程具有正实参数和初始条件。许多有趣的观察使我们提出了几个开放的问题和最重要的关于混沌和高阶周期解和全球渐近收敛等差分方程的approaches。我们希望，这些复杂的差分方程的这些意见肯定会是新的附加到目前的研究艺术在理性差分方程在理解的行为，在复杂的域。
英文摘要：
The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex plane of such equations exhibit many rich and complicated asymptotic behavior. The analysis of the local stability of these three difference equations and periodicity have been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions. Many interesting observations led us to pose several open problems and conjectures of paramount importance regarding chaotic and higher order periodic solutions and global asymptotic convergence of such difference equations. It is our hope that these observations of these complex difference equations would certainly be new add-ons to the present art of research in rational difference equations in understanding the behavior in the complex domain.
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