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'''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。<font color="red">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。</font> <font color="blue"> 当时间变量运行在一个某些区间离散、其他区间连续的集合、或者像cantor集一样任意的时间集合上时，人们就能得到时间尺度上的动力方程。</font>

'''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。当时间变量运行在一个某些区间离散、其他区间连续的集合、或者像cantor集一样任意的时间集合上时，人们就能得到时间尺度上的动力方程。

<font color="red">'''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。</font> <font color="blue">这句话的英文原文在哪里？</font> <font color="red">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。</font> <font color="blue"> 某些情况下，也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。</font>

<font color="red">'''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。</font> <font color="blue">这句话的英文原文在哪里？</font>  

--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这个是补充内容，摘自百度百科 https://baike.baidu.com/item/%E7%AE%97%E5%AD%90

 

某些情况下，也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。

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.

这个理论处理动力系统的长期定性行为，<font color="blue">并且研究系统的动力方程的规律，努力求得可能的解。这些系统通常是一些自然领域里的机械系统或其他物理系统，例如行星轨道和电子电路，也包括一些生物学、经济学和其他学科里的系统。</font>现代的研究大多集中在对混沌系统的研究上。

这个理论处理动力系统的长期定性行为，<font color="blue">并且研究系统的动力方程的规律，努力求得可能的解。这些系统通常是一些自然领域里的机械系统或其他物理系统，例如行星轨道和电子电路，也包括一些生物学、经济学和其他学科里的系统。</font>现代的研究大多集中在对混沌系统的研究上。

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'''<font color="red">定态</font> <font color="blue">稳态</font>Steady States'''是一个重要的目标。不动点或<font color="red">定态</font> <font color="blue">稳态</font>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，系统最终会收敛到这个不动点。

描述给定动力系统的不动点或'''稳态 Steady States'''是一个重要的目标。不动点或稳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 Periodic Points'''感兴趣，<font color='red'>即系统在几个时间步之后会不断重复的状态。</font><font color='blue'>即系统在重复几个时间步之后的状态。</font>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。

人们还对动力系统的'''周期点 Periodic Points'''感兴趣，即系统在重复几个时间步之后的状态。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。

Some excellent presentations of mathematical dynamic system theory include , , , and . To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008

Some excellent presentations of mathematical dynamic system theory include , , , and . To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008

优秀的数学动力系统理论包括，，，和。--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|存疑]]）。

Some excellent presentations of mathematical dynamic system theory include Beltrami (1990), Luenberger (1979), Padulo & Arbib (1974), and Strogatz (1994).[2]  

一些优秀的数学动力系统理论包括贝尔特拉米（Beltrami，1990年），龙伯格(Luenberger，1979年），帕杜罗&阿尔比布（Padulo＆Arbib，1974年）和斯托加茨（Strogatz，1994年）。</font>

一些优秀的数学动力系统理论包括贝尔特拉米（Beltrami，1990年），龙伯格(Luenberger，1979年），帕杜罗&阿尔比布（Padulo＆Arbib，1974年）和斯托加茨（Strogatz，1994年）。

== Concepts 概念==

== Concepts 概念==

The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient space.  Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient space.  Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space.  Small changes in the state of the system correspond to small changes in the numbers.  The numbers are also the coordinates of a geometrical space—a manifold.  The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state.  The rule may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space.  Small changes in the state of the system correspond to small changes in the numbers.  The numbers are also the coordinates of a geometrical space—a manifold.  The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state.  The rule may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

动力系统的状态是由一组实数决定的，更广泛地说，是由适当的状态空间中的一组点决定的。系统状态的微小变化对应于数字的变化。这些数字也是几何空间——'''流形 Manifold'''——的坐标组。<font color='red'>动力系统的演化是描述了在当前状态之后出现的未来状态的固定规则。</font><font color='blue'>动力系统的演化规律是一种固定的规则，它描述了从当前状态得出的未来状态。</font>这个规则可以是确定性的(在给定的时间间隔内，有且仅有一个未来状态在当前状态之后出现)，或随机性的(状态的演化受到随机因素的影响)。

动力系统的状态是由一组实数决定的，更广泛地说，是由适当的状态空间中的一组点决定的。系统状态的微小变化对应于数字的变化。这些数字也是几何空间——'''流形 Manifold'''——的坐标组。动力系统的演化规律是一种固定的规则，它描述了从当前状态得出的未来状态。这个规则可以是确定性的(在给定的时间间隔内，有且仅有一个未来状态在当前状态之后出现)，或随机性的(状态的演化受到随机因素的影响)。

=== Dynamicism 动态主义===

=== Dynamicism 动态主义 0713 here===

[[Dynamicism]], also termed the ''dynamic hypothesis'' or the ''dynamic hypothesis in cognitive science'' or ''dynamic cognition'', is a new approach in [[cognitive science]] exemplified by the work of philosopher [[Tim van Gelder]]. It argues that [[differential equations]] are more suited to modelling [[cognition]] than more traditional [[computer]] models.

[[Dynamicism]], also termed the ''dynamic hypothesis'' or the ''dynamic hypothesis in cognitive science'' or ''dynamic cognition'', is a new approach in [[cognitive science]] exemplified by the work of philosopher [[Tim van Gelder]]. It argues that [[differential equations]] are more suited to modelling [[cognition]] than more traditional [[computer]] models.