We prove the existence of Noise Induced Order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belosouv-Zabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the stationary measure in the $L^{1}$ norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure of the system varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a H\"older way when the noise amplitude increases.

中文翻译：

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噪声诱导阶次的存在，计算机辅助证明

我们证明了 Matsumoto-Tsuda 模型中噪声诱导阶次的存在，它最初是在 1983 年通过数值模拟发现的。这是著名的 Belosouv-Zabotinsky 反应模型，这是一种混沌化学反应，由具有加性噪声的一维随机动力系统组成。模拟结果表明，噪声幅度的增加会导致李雅普诺夫指数从正减少到负；我们给出了这种转变存在的数学证明。我们使用的方法依赖于一些计算机辅助估计，它提供了 $L^{1}$ 范数中平稳度量的经过认证的近似值。这是通过显式函数分析估计与有效算法一起工作来实现的。该方法具有足够的通用性，可以适用于具有加性噪声的单位间隔上的任何分段可微动力系统。我们还证明，如果系统受到扰动，系统的平稳测度以 Lipschitz 方式变化，并且当噪声幅度增加时，系统的 Lyapunov 指数以 H 更旧的方式变化。