AI 中文总结
研究通过用高斯（q）二项式系数替换经典二项式记忆核引入 q 变形分数阶映射，利用 Z 变换等推导特征方程确定稳定区域，分析记忆核渐近行为，扩展到非线性映射，数值模拟验证理论及参数间相互作用。
AI 中文摘要
我们通过用高斯（q）二项式系数替换离散分数动力学中的经典二项式记忆核，引入了一类 q 变形分数阶映射。所提出的框架在无记忆离散映射和经典分数阶映射之间进行插值，通过中间新月形几何形状统一了圆形和心形稳定区域。利用 Z 变换和 q 二项式定理，我们推导了特征方程并确定了复平面中的相关稳定区域。我们进一步分析了记忆核的渐近行为，表明经典分数核呈现幂律衰减，而 q 和（p，q）变形核呈现指数型局部化。该理论扩展到非线性逻辑型映射和更广泛的（p，q）变形框架，其中 p>q 的情况产生衰减的记忆核和稳定的动力学。数值模拟说明了理论结果以及变形参数、记忆效应和稳定几何之间的相互作用。
英文摘要
We introduce a class of \(q\)-deformed fractional order maps by replacing the classical binomial memory kernel in discrete fractional dynamics with Gaussian (\(q\)-) binomial coefficients. The proposed framework interpolates between memoryless discrete maps and classical fractional order maps, unifying circle- and cardioid-shaped stability regions through intermediate crescent geometries. Using the \(Z\)-transform and the \(q\)-binomial theorem, we derive characteristic equations and determine the associated stability regions in the complex plane. We further analyze the asymptotic behavior of the memory kernels, showing that the classical fractional kernel exhibits power-law decay, whereas the \(q\)- and \((p,q)\)-deformed kernels exhibit exponential-type localization. The theory is extended to nonlinear logistic-type maps and to a broader \((p,q)\)-deformed framework, where the regime \(p>q\) yields decaying memory kernels and stable dynamics. Numerical simulations illustrate the theoretical results and the interplay between the deformation parameters, memory effects, and stability geometry.
Comments29 pages, 54 figures
