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Dynamical systems theory is an area of mathematics used to describe the behavior of the  complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

Dynamical systems theory is an area of mathematics used to describe the behavior of the  complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）感觉很棒，在翻译的过程中添加了自己的思考

    

'''动力系统理论（Dynamical Systems Theory）'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续，或者像cantor（康托尔）集一样在任意的时间集合上运行时，人们就能得到时间尺度上的动力方程。有些情况也可以用混合算子来建模，如微分-差分方程。

'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续，或者像cantor（康托尔）集一样在任意的时间集合上运行时，人们就能得到时间尺度上的动力方程。

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有术语的基本格式不需要加括号  '''动力系统理论 Dynamical Systems Theory'''进行一下修改

'''算子 Operators'''是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。动力系统的有些情况也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。

  --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）这里可以补充一下 混合算子的定义

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有术语的基本格式不需要加括号   

  --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 只找到了算子，没找到混合算子orz

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

 

This field of study is also called just ''dynamical systems'', ''mathematical dynamical systems theory'' or the ''mathematical theory of dynamical systems''.

This field of study is also called just ''dynamical systems'', ''mathematical dynamical systems theory'' or the ''mathematical theory of dynamical systems''.

The [[Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.]]

The [[Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.]]

'''洛伦兹吸引子(Lorenz Attractor)'''是非线性动力系统的一个例子。对这个系统的研究产生了混沌理论。

'''洛伦兹吸引子 Lorenz Attractor'''是非线性动力系统的一个例子。对这个系统的研究产生了混沌理论。