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第7行： − 动态系统理论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，是直接假定运动方程的推广，不受最小作用原理的欧拉-拉格朗日方程的约束。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，或者在任意的时间集合上运行时，得到时间尺度上的动态方程。有些情况也可以用混合算子来模拟，如微分差分方程。+ 第15行：
第15行： − 这个理论处理动力系统的长期定性行为，研究系统的性质，如果可能的话，解决方案的运动方程通常主要是机械或其他物理性质的系统，如行星轨道和行为的电子电路，以及系统出现在生物学，经济学和其他地方。现代的研究大多集中在对混沌系统的研究上。+ 第23行：
第23行： − 这个研究领域也被称为动力系统，数学动态系统理论或动力系统的数学理论。+ 第39行：
第39行： − 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?”+ 第72行：
第72行： − ====动态系统理论的概念起源于牛顿运动定律。与其他自然科学和工程学科一样，动力系统的演化规律也是通过一种关系隐含地给出的，这种关系只给出了系统在未来很短时间内的状态。====+ 第88行：
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第102行： − 动力系统是一个对任何描述了点的位置在周围环境随时间变化的“固定”规则的数学形式化。举例来说，描述钟摆摆动、管道中的水流以及每年春天湖中鱼的数量的数学模型都是属于动力系统的概念范畴。+ 第110行：
第110行： − 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——的坐标。动力系统的演化是一个描述了在当前状态之后出现的未来状态的固定规则。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)。+ + 第132行：
第133行： − 在数学中，非线性系统是指系统不是线性的——也就是说，一个不满足叠加原理的系统。更通俗地说，非线性系统是需要求解的变量不能被写成它的独立分量的线性和的系统。非齐次系统根据定义严格来说是非线性的，但是它的自变量函数是线性的 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）+讨论如何翻译本句 。非齐次系统通常与线性系统一起研究，因为只要知道特解，它们就可以转化为线性系统。+ 第144行：
第145行： − 算术动态系统是20世纪90年代出现的一个领域，融合了动力系统和数论这两个数学领域。经典的离散动力学研究复平面或实直线的自映射的迭代。算术动态系统研究在多项式或有理函数的重复应用下整数、有理数、并元 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）原文找不到 和/或代数点的数论性质。+ 第154行：
第155行： − 混沌理论描述了某些状态随时间演化的动力系统的行为，这些系统可能表现出对初始条件高度敏感的特点（通常被称为蝴蝶效应）。由于这种受初始条件影响而指数增长的扰动，混沌系统的敏感性使它的行为看起来像是随机的。但是这种敏感性甚至会出现在确定的动力系统中，即它未来的动力学完全由它的初始条件定义，没有任何随机因素参与。这种现象被称为确定性混沌，或简单混沌。+ 第164行：
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第173行： − 还原论策略已经不足以研究许多科学领域的问题，而复杂系统的研究则为科学广泛带来了新的活力。因此，复杂系统通常被用作一个宽泛的研究方法术语，涵盖了许多不同的学科，包括神经科学、社会科学、气象学、化学、物理学、计算机科学、心理学、人工生命、进化计算、经济学、地震预测、分子生物学以及对活细胞的研究。+ 第182行：
第183行： − 控制理论是工程和数学的一个交叉学科，它部分地涉及到对动力系统行为的影响。+ 第202行：
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第232行： − 符号动力学是通过一个由抽象符号的无限序列组成的离散空间建立一个拓扑或光滑动力系统的实践。每一个抽象符号的无限序列序列对应于系统的一个状态，并由移位算子给出动力学(演化)。+ 第241行：
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第266行： − 在人类发展方面，动态系统理论已经被用来增强和简化 Erik Erikson 的社会心理发展8阶段理论，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的自组织性（self-organizing）和分形特性。利用数学模型，人类发展的自然进程被分为8个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、童年早期(4-7岁)、童年中期(7-11岁)、青春期(11-18岁)、成年早期(18-29岁)、成年中期(29-48岁)和老年成年期(48-78岁及以上)。+ 第285行：
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第292行： − 在运动生物力学中，动力系统理论新兴地成为运动表现建模的可行框架。从动力系统的角度来看，人类的运动系统是一个高度复杂的相互依赖的子系统网络（如呼吸、循环、神经、骨骼肌系统和知觉系统等），它们由大量相互作用的部分组成（包括血细胞、氧分子、肌肉组织、代谢酶、结缔组织和骨骼等)。动力系统理论中，运动模式通过物理系统和生物系统中的一般自组织过程出现。没有任何研究证实与这一框架的概念相关的任何主张。--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 没有研究证实？那就是说不可信吗？+ 第309行：
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第二遍翻译和校对
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.
动态系统理论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，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。
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。如果能够得到解的话，还可以研究系统的运动方程。这些方程通常主要是机械的或物理的，如行星轨道和电子电路，以及出现在生物学，经济学和其他地方的系统。现代的研究大多集中在对混沌系统的研究上。
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.
这个研究领域被称为动力系统，数学动力系统理论或动力系统的数学理论。
[[Image:Lorenz attractor yb.svg|thumb|240px|right|The [[Lorenz attractor]] is an example of a [[non-linear]] dynamical system. Studying this system helped give rise to [[chaos theory]].]]
[[Image:Lorenz attractor yb.svg|thumb|240px|right|The [[Lorenz attractor]] is an example of a [[non-linear]] dynamical system. Studying this system helped give rise to [[chaos theory]].]]
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?”，或“系统长期的行为是否取决于其初始条件?”
The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.
The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.
动态系统理论的概念起源于牛顿运动定律。与其他自然科学和工程学科一样，动力系统的演化规律也是通过一种预测系统在未来很短时间内的状态的关系隐含地给出的。
Some excellent presentations of mathematical dynamic system theory include , , , and .
Some excellent presentations of mathematical dynamic system theory include , , , and .
优秀的数学动力系统理论包括，，，和。
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|讨论]]）需要原文。
== 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.
动力系统是一个对任何描述了点的位置在周围环境随时间变化的“固定”规则的数学式。举例来说，描述钟摆摆动、管道中的水流以及每年春天湖中鱼的数量的数学模型，都属于动力系统的概念范畴。
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）——的坐标系（coordinates ）--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）这里翻译为坐标还是坐标系更好一点？
。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。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)。
In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
在数学中，非线性系统是指系统不是线性的——也就是说，一个不满足叠加原理的系统。更通俗地说，非线性系统是待求解变量不能被写成其独立分量的线性和的系统。非齐次系统根据定义严格来说是非线性的，除了它的自变量包含函数以外其他部分都是线性的。但非齐次系统通常与线性系统一起研究，因为只要知道特解，它们就可以转化为线性系统。
== Related fields 相关领域==
== Related fields 相关领域==
Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a polynomial or rational function.
Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a polynomial or rational function.
算术动态系统是20世纪90年代出现的一个领域，融合了动力系统和数论这两个数学领域。经典的离散动力学研究复平面或实直线的自映射的迭代。算术动态系统研究内容是在多项式或有理函数中的整数、有理数、并元 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）原文找不到 和/或代数点的数论性质。
Chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
Chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
混沌理论描述了某些状态随时间演化的动力系统的行为，这些系统可能表现出对初始条件高度敏感的特点（通常被称为蝴蝶效应）。由于扰动受初始条件影响而指数增长，因此混沌系统具有敏感性，敏感性使它的行为看起来是随机的。但是这种敏感性也会出现在确定的动力系统中，即未来的动力学完全由它的初始条件定义，没有任何随机因素参与的系统中。这种现象被称为确定性混沌，或简单混沌。
Complex systems is a scientific field that studies the common properties of systems considered complex in nature, society, and science. It is also called complex systems theory, complexity science, study of complex systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal modeling and simulation. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.
Complex systems is a scientific field that studies the common properties of systems considered complex in nature, society, and science. It is also called complex systems theory, complexity science, study of complex systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal modeling and simulation. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.
复杂系统是研究自然、社会和科学 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）+ 自然、社会和科学为什么会并列呢？ 中复杂现象的共同性质的科学领域。它也被称为复杂系统理论、复杂性科学、复杂系统研究和关于复杂性的科学。这些系统的关键问题在于对系统的形式化建模与仿真。因此，在不同的研究语境中，复杂系统是根据其不同的属性来定义的。
复杂系统是研究自然、社会和科学 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）+ 自然、社会和科学为什么会并列呢？ 中复杂现象的共同性质的科学领域。它也被称为复杂系统理论、复杂性科学、复杂系统研究和关于复杂性的科学。这些系统的关键问题在于对系统的形式化建模与仿真。因此，复杂系统是根据在不同的研究语境中的不同属性来定义的。
The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences, social sciences, meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves.
The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences, social sciences, meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves.
还原论策略已经不足以研究许多科学领域的问题，而复杂系统的研究则为科学带来了广泛的新活力。复杂系统通常被用作一个应用广泛的研究方法术语，并涵盖许多不同的学科，包括神经科学、社会科学、气象学、化学、物理学、计算机科学、心理学、人工生命、进化计算、经济学、地震预测、分子生物学以及对活细胞的研究等。
Control theory is an interdisciplinary branch of engineering and mathematics, in part it deals with influencing the behavior of dynamical systems.
Control theory is an interdisciplinary branch of engineering and mathematics, in part it deals with influencing the behavior of dynamical systems.
控制理论是工程和数学的一个交叉学科，它的其中一部分研究影响动力系统行为的各种因素。
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.
泛函分析是数学分析的一个分支，研究向量空间和作用于向量空间的算子。它源于对函数空间的研究，特别是对函数变换的研究，例如傅里叶变换，微积分方程的研究等。泛函分析的名称“Functional Analysis”中，“functional”这个词的用法可以追溯到变分法，也就是说一个函数的参数是一个函数。这个词的使用一般被认为归功于数学家和物理学家Vito Volterra，它的建立主要归功于数学家Stefan Banach。
泛函分析是数学分析的一个分支，研究向量空间和作用于向量空间的算子。它源于对函数空间的研究，特别是对函数变换的研究，例如傅里叶变换，微积分方程的研究等。泛函分析的名称“Functional Analysis”中，“functional”这个词的用法可以追溯到变分法，也就是说函数的参数是一个函数。这个词的使用一般被认为归功于数学家和物理学家Vito Volterra，和数学家Stefan Banach。
Projected dynamical systems it is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.
Projected dynamical systems it is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.
投影动力系统是研究解在一个约束集内的动力系统行为的数学理论。这门学科与静态世界中的最优化和平衡问题以及动态世界中的常微分方程都有联系，并且都有相互联系的应用。一个投影动力系统是由投影微分方程的流给定的--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这句啥意思不懂。
投影动力系统是研究解在一个约束集内的动力系统行为的数学理论。这门学科与静态世界中的最优化和平衡问题以及动态世界中的常微分方程都有联系，并且都有相互联系的应用。一个投影动力系统是由投影微分方程的流给定的--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|这句话的数学原理对我来说过于深奥，因此不确定翻译的内容对不对]]）。
Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.
Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.
符号动力学是通过一个由抽象符号的无限序列组成的离散空间建立一个拓扑或光滑动力系统的方法。每一个抽象符号的无限序列序列对应于系统的一个状态，并由移位算子给出动力学(演化)。
System dynamics is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system. What makes using system dynamics different from other approaches to studying systems is the use of feedback loops and stocks and flows. These elements help describe how even seemingly simple systems display baffling nonlinearity.
System dynamics is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system. What makes using system dynamics different from other approaches to studying systems is the use of feedback loops and stocks and flows. These elements help describe how even seemingly simple systems display baffling nonlinearity.
系统动力学是一种理解系统随时间变化行为的方法。它是用来处理影响整个系统行为和状态的内部反馈回路和时间延迟的方法。系统动力学不同于其他系统研究方法的地方在于它使用了反馈环、股票（stocks）和流（flows）--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 翻译stocks 和 folows的元素。这些元素有助于描述看似简单的系统如何显示复杂的非线性行为。
系统动力学是一种理解系统随时间变化行为的方法。它是用来处理影响整个系统行为和状态的内部反馈回路和时间延迟的方法。系统动力学不同于其他系统研究方法的地方在于它使用了反馈环、股票（stocks）和流（flows）--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|如何翻译stocks 和 folows]]）的元素。这些元素有助于描述看似简单的系统如何显示复杂的非线性行为。
=== In human development 人类发展中的应用===
=== In human development 在人类发展中的应用===
In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).
In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).
在人类发展方面，动力系统理论已经被用来增强和简化 Erik Erikson 的社会心理发展8阶段理论，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的自组织性（self-organizing）和分形特性。利用数学模型，人类发展的自然进程被分为8个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、童年早期(4-7岁)、童年中期(7-11岁)、青春期(11-18岁)、成年早期(18-29岁)、成年中期(29-48岁)和老年成年期(48-78岁及以上)。
=== In biomechanics 在生物力学中的应用===
=== In biomechanics 在运动生物力学中的应用===
In [[sports biomechanics]], dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.<ref>Paul S Glazier, Keith Davids, Roger M Bartlett (2003). [http://www.sportsci.org/jour/03/psg.htm "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research"]. in: Sportscience 7. Accessed 2008-05-08.</ref> There is no research validation of any of the claims associated to the conceptual application of this framework.
In [[sports biomechanics]], dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.<ref>Paul S Glazier, Keith Davids, Roger M Bartlett (2003). [http://www.sportsci.org/jour/03/psg.htm "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research"]. in: Sportscience 7. Accessed 2008-05-08.</ref> There is no research validation of any of the claims associated to the conceptual application of this framework.
In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems. There is no research validation of any of the claims associated to the conceptual application of this framework.
In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems. There is no research validation of any of the claims associated to the conceptual application of this framework.
在运动生物力学中，动力系统理论新兴地成为对运动表现建模的可行框架。从动力系统的角度来看，人类的运动系统是由高度复杂和相互依赖的子系统网络（如呼吸、循环、神经、骨骼肌系统和知觉系统等）组成的，它们由大量相互作用的部分组成（包括血细胞、氧分子、肌肉组织、代谢酶、结缔组织和骨骼等)。动力系统理论中，运动模式通过物理系统和生物系统中的一般自组织过程出现。没有任何研究证实与这一框架的概念相关的任何主张。--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|没有研究证实？那就是说不可信吗？所以这个文本的意思是什么。]]）
In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization (the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called scalloping (the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable.
In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization (the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called scalloping (the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable.
在学习的过程中，旧的模式被打破了，学习者的思维达到了一种不平衡的状态。这是认知发展的阶段性转变。自组织(连贯的自发创造(the spontaneous creation of coherent forms))在活动水平（activity levels）相互联系时产生。新形成的宏观和微观结构相互支持，加速了这一过程。这些联系在头脑中形成了一种具有新状态的结构，这个过程被称为“扇贝化（scalloping）”，也就是头脑的复杂性能的不断累积和崩溃的过程 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）扇贝化的翻译拿不准，而且这句话的描述比较抽象 。这种新的状态是渐进的、离散的、异质的的和不可预知的。
在学习的过程中，旧的模式被打破了，学习者的思维达到了一种不平衡的状态。这是认知发展的阶段性转变。自组织(连贯的自发创造(the spontaneous creation of coherent forms))在活动水平（activity levels）相互联系时产生。新形成的宏观和微观结构相互支持，加速了这一过程。这些联系在头脑中形成了一种具有新状态的结构，这个过程被称为“扇贝化（scalloping）”，也就是头脑的复杂性能的不断累积和崩溃的过程 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|扇贝化的翻译拿不准，这句话的描述比较抽象]]）。这种新的状态是渐进的、离散的、异质的的和不可预知的。
* [[Biological applications of bifurcation theory]] 分支理论的生物学应用
* [[Biological applications of bifurcation theory]] 分支理论的生物学应用
* [[Dynamical system (definition)]]动力系统（定义）
* [[Dynamical system (definition)]] 动力系统（定义）
* [[Embodied Embedded Cognition]] 具身认知
* [[Embodied Embedded Cognition]] 具身认知
* [[List of types of systems theory]] 动力系统类型清单
* [[List of types of systems theory]] 动力系统类型清单
* [[Oscillation]]振动性
* [[Oscillation]] 振动性
* [[Postcognitivism]]
* [[Postcognitivism]]
* [[Recurrent neural network]]递归神经网络
* [[Recurrent neural network]] 递归神经网络
* [[Combinatorics and dynamical systems]] 组合数学和动力系统
* [[Combinatorics and dynamical systems]] 组合数学和动力系统
== External links 外部链接==
== External links 外部链接==
*[https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html Dynamic Systems 动力系统] Encyclopedia of Cognitive Science entry. --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这是一个嵌套的链接，但好像只能超链接到内嵌的那个-- 认知科学百科全书
*[https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html Dynamic Systems 动力系统] Encyclopedia of Cognitive Science entry. --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|这是一个嵌套的链接，但好像只能鼠标点击并超链接到内嵌的那个]]）认知科学百科全书
*[http://mathworld.wolfram.com/DynamicalSystem.html Definition of dynamical system 动力系统的定义] in MathWorld. 在MathWorld.wolfram.com储存的定义
*[http://mathworld.wolfram.com/DynamicalSystem.html Definition of dynamical system 动力系统的定义] in MathWorld. 在MathWorld.wolfram.com储存的定义