实参数多项式离散动力系统的稳定性
Stability of real parametric polynomial discrete dynamical systems
摘要：
我们扩展和改进现有的动力学的一般二次实多项式映射的系数取决于一个单一的参数$λ$的表征，并推广到三次实多项式映射，在一个一致的理论，进一步推广到真正的$m$次实多项式映射。实质上，我们给出了具有实不动点的实多项式映射的不动点稳定的条件。为了做到这一点，我们引入了规范多项式映射的概念，它与任何具有实不动点的同阶多项式映射拓扑共轭。典型多项式映射不动点的稳定性只依赖于一个特殊的函数，即乘积位置函数。这个乘积位置的值决定了所讨论的不动点的稳定性，当它分叉时，甚至当混沌出现时，因为它穿过我们所说的稳定带。这些稳定带的确切边界值尚未计算的区域类型大于一次多项式的次数高于三。
英文摘要：
We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter $λ$, and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real $m$-th degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of Canonical Polynomial Maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates, and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.
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