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第39行：
第39行： − 动态系统理论和混沌理论处理动力系统的长期定性行为。在这里，重点不是找到定义动力系统的方程的精确解(这通常是没有希望的) ，而是回答诸如“系统是否会长期稳定下来，如果是，可能的稳定状态是什么? ”？或者“系统的长期行为是否取决于它的初始条件? ”+ + 第47行：
第48行： − 一个重要的目标是描述给定动力系统的固定点或稳定状态; 这些是不随时间变化的变量值。这些不动点中的一些是有吸引力的，这意味着如果系统开始于附近的状态，它会收敛到不动点。+ 第55行：
第56行： − 类似地，人们对周期点感兴趣，即系统在几个时间步骤之后重复的状态。周期点也可以是有吸引力的。关于一维离散动力系统的周期点数，Sharkovskii 定理是一个有趣的陈述。+ 第63行：
第64行： − 即使是简单的非线性动力系统也常常表现出看似随机的行为，这种行为被称为混沌。混沌理论是动力学系统的一个分支，主要研究混沌的清晰定义和混沌的研究。+ − −
→Overview： 阅读并修改完善句意句法
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
动态系统理论和混沌理论是用来处理动力系统的长期定性行为的理论。寻找动力系统方程的精确解通常是很难达到的。这两个理论的重点不在于找到精确解，而是回答如下的问题，如“系统长期来看是否会稳定下来，如果可以那么可能的稳定状态是什么样的?”，“系统长期的行为是否取决于其初始条件?”
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
描述给定动力系统的不动点或定态（steady states）是一个重要的目标。不动点或定态的变量值不会随时间的变化而变化。一些不动点是有吸引力的（attractive），即如果系统的初始值在它的附近，系统最终共会收敛到这个不动点。
Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.
Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.
人们还对动力系统的周期点感兴趣，即系统在几个时间步之后会不断重复的状态。周期点也可以是有吸引力的。Sharkovskii定理描述了一维离散动力系统的周期点的个数。
Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.
Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.
即使是简单的非线性动力系统也常常表现出看似随机的行为，这种行为被称为混沌。混沌理论主要研究混沌的清晰定义和混沌的现象，它是动力系统理论的一个分支。
== History ==
== History ==