This study develops a comprehensive theoretical and computational framework for Random Nonlinear Iterated Function Systems (RNIFS), a generalization of classical IFS models that incorporates both nonlinearity and stochasticity. We establish mathematical guarantees for the existence and stability of invariant fractal attractors by leveraging contractivity conditions, Lyapunov-type criteria, and measure-theoretic arguments. Empirically, we design a set of high-resolution simulations across diverse nonlinear functions and probabilistic schemes to analyze the emergent attractors geometry and dimensionality. A box-counting method is used to estimate the fractal dimension, revealing attractors with rich internal structure and dimensions ranging from 1.4 to 1.89. Additionally, we present a case study comparing RNIFS to the classical Sierpi\'nski triangle, demonstrating the generalization's ability to preserve global shape while enhancing geometric complexity. These findings affirm the capacity of RNIFS to model intricate, self-similar structures beyond the reach of traditional deterministic systems, offering new directions for the study of random fractals in both theory and applications.

翻译：本研究针对随机非线性迭代函数系统（RNIFS）——一种融合了非线性与随机性的经典IFS模型推广形式——构建了完整的理论与计算框架。通过运用压缩性条件、李雅普诺夫型判据及测度论论证，我们为不变分形吸引子的存在性与稳定性建立了数学保证。在实证层面，我们设计了一系列涵盖不同非线性函数与概率方案的高分辨率模拟，以分析涌现吸引子的几何形态与维数特性。采用盒计数法估算分形维数，揭示了具有丰富内部结构且维数介于1.4至1.89之间的吸引子。此外，我们通过案例研究将RNIFS与经典谢尔宾斯基三角形进行对比，证明该推广模型能在保持整体形态的同时增强几何复杂性。这些发现证实了RNIFS能够模拟传统确定性系统无法实现的精细自相似结构，为随机分形的理论与应用研究提供了新方向。