标题： 平稳测度和具有光滑加性噪声的随机动力系统族的李雅普诺夫景观的高效计算 显示英文标题

标题： Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise

摘要： 我们提出了一种高效且经过验证的方法，用于近似具有光滑加性噪声的随机动力系统的平稳测度。 该方法通过基于傅里叶逼近的有限维约化，利用了相关转移算子的强正则化特性。 显式的误差界使该方法适用于计算机辅助证明和严格数值研究；特别是其效率 {\em 使参数空间的系统探索成为可能}。 该方法提供了对平稳测度的访问，并支持对系统关键统计特性的分析。 作为应用，我们研究了噪声诱导现象，重点关注在具有高斯加性噪声的随机单峰映射族中从正到负的李雅普诺夫指数的转变（通常称为噪声诱导秩序）。 通过分析李雅普诺夫指数作为系统参数的函数，我们在参数空间中识别了沿超曲面的转变。 我们考虑的参数包括高斯噪声的标准差（强度）和单峰映射的形状。 显示英文摘要

摘要： We present an efficient and validated method for approximating the stationary measures of random dynamical systems with smooth additive noise. The approach leverages the strong regularizing properties of the associated transfer operator through a finite-dimensional reduction based on Fourier approximation. Explicit error bounds make the method suitable for use in computer-assisted proofs and rigorous numerical investigations; in particular, its efficiency {\em enables systematic explorations of parameter space}. The method provides access to the stationary measure and supports the analysis of key statistical properties of the system. As an application, we study noise-induced phenomena, focusing on the transition from positive to negative Lyapunov exponent (commonly known as Noise Induced Order) in families of random unimodal maps with Gaussian additive noise. By analyzing the Lyapunov exponent as a function of the system parameters, we identify transitions along a hypersurface in parameter space. The parameters we consider include the standard deviation (intensity) of the Gaussian noise and the shape of the unimodal map.