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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。

由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。

==编者推荐==

==编者推荐==

集智学园：[https://campus.swarma.org/course/697 非线性动力学与混沌]

[https://campus.swarma.org/ 集智学园]课程：[https://campus.swarma.org/course/697 非线性动力学与混沌]

非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。

非线性动力学和混沌理论是系统发展的，从一阶微分方程及其分岔开始，然后是相平面分析，极限环和它们的分岔，最终得到Lorenz方程，混沌，迭代映射，周期倍增，重整化，分形和奇怪吸引。