噪声诱导阶存在性的计算机辅助证明
Existence of Noise Induced Order, a Computer Aided Proof
摘要：
本文证明了1983年首次通过数值模拟发现的Matsumoto-Tsuda模型中噪声诱导阶的存在性。这是一个著名的Belosouv-Zabotinsky反应的模型，这是一个混沌化学反应，由一个带有加性噪声的一维随机动力系统组成。仿真结果表明，噪声幅值的增加会导致系统的李雅普诺夫指数由正变负，并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提供了一个认证的近似的平稳措施在$L^{1}$规范。这是实现了明确的功能分析估计一起工作的一个有效的算法。该方法具有一般性，适用于单位区间上任何带加性噪声的分段可微动力系统。我们还证明了当系统受到扰动时，系统的平稳测度以Lipschitz方式变化，当噪声幅值增大时，系统的李雅普诺夫指数以Hölder方式变化.
英文摘要：
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 Hölder way when the noise amplitude increases.
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