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-此词条暂由彩云小译翻译，翻译字数共1957，未经人工整理和审校，带来阅读不便，请见谅。
+#重定向 [[动力系统理论 Dynamical Systems Theory]]
-{{Use dmy dates|date=May 2019|cs1-dates=y}}
-'''Dynamical systems theory''' is an area of [[mathematics]] used to describe the behavior of the  [[complex systems|complex]] [[dynamical system]]s, usually by employing [[differential equations]] or [[difference equations]]. When differential equations are employed, the theory is called [[continuous time|''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 equation]]s of a [[Principle of least action|least action principle]]. When difference equations are employed, the theory is called [[discrete time|''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运行，或者是任意的时间集合，如 Cantor 集合，就得到了时间尺度上的动力学方程。有些情况也可以用混合算子来模拟，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 [[mechanics|mechanical]] or otherwise physical in nature, such as [[planetary orbit]]s and the behaviour of [[electronic circuit]]s, as well as systems that arise in [[biology]], [[economics]], and elsewhere. Much of modern research is focused on the study of [[chaotic system]]s.
-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.
-这个理论研究动力系统的长期定性行为，研究系统运动方程的性质，如果可能的话，研究系统的解决方案，这些系统通常主要是机械的或其他物理性质的，例如行星轨道和电子线路的行为，以及生物学、经济学和其他领域出现的系统。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。
-[[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]].]]
-The [[Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.]]
-[] Lorenz 吸引子是非线性动力系统的一个例子。研究这个系统有助于发展混沌理论
-== Overview ==
-Dynamical systems theory and [[chaos theory]] deal with the long-term qualitative behavior of [[dynamical system]]s. 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，如果是的话，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? ”
-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.
-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; 这些是变量的值，不会随着时间的推移而改变。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。
-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.
-类似地，人们对周期点感兴趣，即在几个时间步骤之后重复的系统状态。周期点也可以是吸引人的。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。
-Even simple [[nonlinear dynamical system]]s often exhibit seemingly random behavior that has been called ''chaos''.<ref>{{cite journal |last1=Grebogi |first1=C. |last2=Ott |first2=E. |last3=Yorke |first3=J. |year=1987 |title=Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics |journal=[[Science (journal)|Science]] |volume=238 |issue=4827 |pages=632–638 |jstor=1700479 |doi=10.1126/science.238.4827.632 |pmid=17816542 |bibcode=1987Sci...238..632G |s2cid=1586349 }}</ref> 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。
-== History ==
-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20世纪90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。
-Before the advent of [[computer|fast computing machines]], solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.
-Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.
-在快速计算机器出现之前，解决动力系统问题需要复杂的数学技术，而且只能在一小类动态系统中完成。
-Some excellent presentations of mathematical dynamic system theory include {{harvtxt|Beltrami|1990}}, {{harvtxt|Luenberger|1979}}, {{harvtxt|Padulo|Arbib|1974}}, and {{harvtxt|Strogatz|1994}}.<ref>Jerome R. Busemeyer (2008), [http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html "Dynamic Systems"]. To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008.  {{webarchive |url=https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html |date=June 13, 2008 }}</ref>
-Some excellent presentations of mathematical dynamic system theory include , , , and .
-一些优秀的数学动态系统理论的演示包括，，，和。
-== Concepts ==
-=== Dynamical systems ===
-{{main|Dynamical system (definition)}}
-The [[dynamical system]] concept is a mathematical [[formal system|formalization]] for any fixed "rule" that describes the [[time]] dependence of a point's position in its [[ambient space]].  Examples include the [[mathematical model]]s 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.
-动力系统的概念是一个数学形式化的任何固定的“规则” ，描述了一个点的位置在其环境空间的时间依赖性。这些例子包括描述钟摆摆动的数学模型、管道中的水流量以及每年春天湖中鱼的数量。
-A dynamical system has a ''state'' determined by a collection of [[real numbers]], or more generally by a [[Set (mathematics)|set]] of [[Point (geometry)|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 [[function (mathematics)|fixed rule]] that describes what future states follow from the current state.  The rule may be [[Deterministic system (mathematics)|deterministic]] (for a given time interval only one future state follows from the current state) or [[stochastic differential equation|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，或者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微小变化。这些数字也是几何空间ーー流形ーー的坐标。动力系统的演变规则是一个固定的规则，描述了当前状态下的未来状态。该规则可以是确定性的(在给定的时间间隔内，只有一个未来状态从当前状态跟随)或随机性的(状态的演变受到随机冲击)。
-=== Dynamicism ===
-[[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.
-动态主义又称为认知科学或动态认知中的动态假设或动态假设，是以哲学家蒂姆 · 范 · 格尔德的著作为代表的认知科学的一种新方法。认为微分方程比传统的计算机模型更适合于建立认知模型。
-=== Nonlinear system ===
-{{main|Nonlinear system}}
-In [[mathematics]], a [[nonlinear system]] is a system that is not [[linear system|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 [[Homogeneity (physics)|nonhomogeneous]] system, which is linear apart from the presence of a function of the [[independent variable]]s, 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，非线性是一个不是线性的系统ーー也就是说，一个不满足叠加原理的系统。从技术上讲，非线性是任何需要求解的变量不能被写成独立分量的线性和的问题。根据严格的定义，非齐次系统除了自变量函数的存在外是线性的，是非线性的，但是这类系统通常与线性系统一起研究，因为只要知道特定的解，它们就可以转化为线性系统。
-== Related fields ==
-=== Arithmetic dynamics ===
-:[[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 [[Iterated function|iteration]] of self-maps of the [[complex plane]] or [[real line]]. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, {{math|<var>p</var>}}-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，融合了数学的两个领域---- 动力系统和数论。经典的离散动力学是指研究复杂平面或实线的自映射的迭代。算术动态系统是研究在多项式或有理函数的重复应用下整数、有理数、-并元和/或代数点的数论性质。
-=== Chaos theory ===
-:[[Chaos theory]] describes the behavior of certain [[dynamical system (definition)|dynamical system]]s – 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 [[randomness|random]]. This happens even though these systems are [[deterministic system (philosophy)|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动力学系统的行为，即状态随时间演化的系统，这些系统可能表现出对初始条件高度敏感的动力学(通常称为蝴蝶效应)。由于这种敏感性，在初始条件下表现为一个指数增长的扰动，混沌系统的行为看起来是随机的。即使这些系统是确定性的，这意味着它们未来的动力学完全由它们的初始条件定义，没有任何随机因素参与。这种行为被称为确定性混沌，或简单的混沌。
-=== Complex systems ===
-:[[Complex systems]] is a scientific field that studies the common properties of [[system]]s considered [[Complexity|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 [[Scientific modelling|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界、社会和科学中被认为是复杂的系统的共同性质的一个科学领域。它也被称为复杂系统理论、复杂性科学、复杂系统研究和/或复杂性科学。这些系统的关键问题在于它们的正式建模与模拟难以实现。从这个角度来看，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。
-: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 [[cell (biology)|cell]]s 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.
-对复杂系统的研究给许多科学领域带来了新的活力，在这些领域，更典型的还原论策略已经不足。因此，复杂系统通常被用作一个宽泛的术语，涵盖了许多不同学科的问题研究方法，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本身本质的研究。
-=== Control theory ===
-:[[Control theory]] is an interdisciplinary branch of [[engineering]] and [[mathematics]], in part it deals with influencing the behavior of [[dynamical system]]s.
-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，部分涉及到对动力系统行为的影响。
-=== Ergodic theory ===
-:[[Ergodic theory]] is a branch of [[mathematics]] that studies [[dynamical system]]s with an [[invariant measure]] and related problems. Its initial development was motivated by problems of [[statistical physics]].
-Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.
-遍历理论是数学的一个分支，研究具有不变测度和相关问题的动力系统。它最初的发展是受到统计物理学问题的推动。
-=== Functional analysis ===
-:[[Functional analysis]] is the branch of [[mathematics]], and specifically of [[mathematical analysis|analysis]], concerned with the study of [[vector space]]s and [[operator (mathematics)|operator]]s acting upon them. It has its historical roots in the study of [[functional space]]s, in particular transformations of [[function (mathematics)|functions]], such as the [[Fourier transform]], as well as in the study of [[differential equations|differential]] and [[integral equations]]. This usage of the word ''[[functional (mathematics)|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，特别是分析的一个分支，研究向量空间和作用于向量空间的算子。它的历史根源在于研究函数空间，特别是函数的变换，例如傅里叶变换，以及微分和积分方程的研究。函数式这个词的用法可以追溯到变分法，意味着一个函数的参数是一个函数。它的使用通常被认为是数学家和物理学家维托沃尔泰拉和它的建立主要被认为是数学家斯蒂芬巴纳赫。
-=== Graph dynamical systems ===
-:The concept of [[graph dynamical system]]s (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
-The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
-图动态系统(GDS)的概念可以用来捕捉图或网络上发生的大量过程。图动力系统的数学和计算分析的一个主要主题是关联它们的结构性质(例如:。网络连接)和全球动力学的结果。
-=== Projected dynamical systems ===
-:[[Projected dynamical systems]] it is a [[mathematics|mathematical]] theory investigating the behaviour of [[dynamical system]]s where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of [[Optimization (mathematics)|optimization]] and [[Equilibrium point|equilibrium]] problems and the dynamical world of [[ordinary differential equations]]. A projected dynamical system is given by the [[flow (mathematics)|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.
-投影动力系统它是一个数学理论研究的行为动力系统的解决方案是限制在一个约束集。这门学科与最优化和平衡问题的静态世界以及常微分方程的动态世界都有联系和应用。一个投影动态系统是由流量给计划的微分方程。
-=== Symbolic dynamics ===
-:[[Symbolic dynamics]] is the practice of modelling a topological or smooth [[dynamical system]] by a discrete space consisting of infinite [[sequence]]s 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无限抽象符号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，每一个抽象符号序列对应于一个系统的状态，由移位算子给出动力学(演化)。
-=== System dynamics ===
-:[[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.<ref name="sysdyn">[http://sysdyn.clexchange.org MIT System Dynamics in Education Project (SDEP)<!-- Bot generated title -->] {{webarchive|url=https://web.archive.org/web/20080509163801/http://sysdyn.clexchange.org/ |date=2008-05-09 }}</ref>  What makes using system dynamics different from other approaches to studying systems is the use of [[feedback]] loops and [[Stock and flow|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。使用系统动力学不同于其他研究系统的方法是使用反馈循环、存量和流量。这些元素有助于描述即使看似简单的系统如何显示令人困惑的非线性。
-=== Topological dynamics ===
-:[[Topological dynamics]] is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of [[general topology]].
-Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
-拓扑动力学是动态系统理论的一个分支，在这个分支中，动态系统的定性，渐近性质是从点集拓扑学的观点来研究的。
-== Applications ==
-=== In biomechanics ===
-In [[sports biomechanics]], dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance and efficiency. 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 and efficiency. 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20世纪90年代运动生物力学，运动科学领域出现了一个可行的运动表现和效率模型框架---- 动态系统理论。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。呼吸的，循环的，神经的，骨骼肌的，知觉的，由许多相互作用的成分组成的。血细胞、氧分子、肌肉组织、代谢酶、结缔组织和骨骼)。在动态系统理论，运动模式通过物理系统和生物系统中自我组织的一般过程出现。没有任何研究证实与该框架的概念应用相关的任何主张。
-=== In cognitive science ===
-Dynamical system theory has been applied in the field of [[neurodynamics|neuroscience]] and [[cognitive science|cognitive development]], especially in [[the neo-Piagetian theories of cognitive development]]. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and [[AI]]. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through [[state space]]. In other words, dynamicists argue that [[psychology]] should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.
-Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.
-动力系统理论已经应用于神经科学和认知发展领域，特别是在新皮亚杰学派。人们相信，物理学理论比句法学理论和人工智能理论更能代表认知发展。它还认为微分方程式是建立人类行为模型的最合适的工具。这些方程被解释为通过状态空间代表一个主体的认知轨迹。换句话说，动力学家认为心理学应该(或者是)描述(通过微分方程)在一定的环境和内部压力下的认知和行为。混沌理论的语言也经常被采用。
-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.<ref>{{cite journal|title=The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development|journal=Child Development|date=2000-02-25|first=Mark D.|last=Lewis|volume=71|issue=1|pages=36–43|id= |url=http://home.oise.utoronto.ca/~mlewis/Manuscripts/Promise.pdf|accessdate=2008-04-04|doi=10.1111/1467-8624.00116|pmid=10836556 |citeseerx=10.1.1.72.3668}}</ref>
-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.
-在这个过程中，学习者的思维达到了一种不平衡的状态，旧的模式被打破了。这是认知发展的阶段性转变。自我组织(连贯形式的自发创造)作为活动水平相互联系而产生。新形成的宏观和微观结构相互支持，加速了这一过程。这些联系通过一个被称为“扇贝化”的过程(复杂表现的反复累积和崩溃) ，在头脑中形成了一种新的秩序状态的结构这种新的，新奇的状态是渐进的，离散的，特殊的和不可预知的。
-Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the [[A-not-B error]].<ref>{{cite journal|title=Development as a dynamic system|journal=Trends in Cognitive Sciences|date=2003-07-30|first=Linda B.|last=Smith|author2=Esther Thelen|volume=7|issue=8|pages=343–8|id= |url=http://www.indiana.edu/~cogdev/labwork/dynamicsystem.pdf|accessdate=2008-04-04|doi=10.1016/S1364-6613(03)00156-6|pmid=12907229|citeseerx=10.1.1.294.2037|s2cid=5712760}}</ref>
-Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error.
-动态系统理论最近被用来解释儿童发展中一个长期没有答案的问题，即 A-not-B 错误。
-===In second language development===
-{{Main|Dynamic approach to second language development}}
-The application of Dynamic Systems Theory to study [[second language acquisition]] is attributed to [[Diane Larsen-Freeman]] who published an article in 1997 in which she claimed that [[second language acquisition]] should be viewed as a developmental process which includes [[language attrition]] as well as language acquisition.<ref>{{cite web|url=https://academic.oup.com/applij/article-abstract/18/2/141/134192|title=Chaos/Complexity Science and Second Language Acquisition|date=1997|publisher=Applied Linguistics}}</ref> In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive.
-The application of Dynamic Systems Theory to study second language acquisition is attributed to Diane Larsen-Freeman who published an article in 1997 in which she claimed that second language acquisition should be viewed as a developmental process which includes language attrition as well as language acquisition. In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive.
-动态系统理论在二语习得研究中的应用归功于 Diane Larsen-Freeman，她在1997年发表的一篇文章中声称，二语习得应该被看作是一个包括语言流失和语言习得在内的发展过程。她在文章中主张，语言应该被看作是一个动态的、复杂的、非线性的、混沌的、不可预知的、对初始条件敏感的、开放的、自组织的、反馈敏感的、适应性的动态系统。
-== See also ==
-;Related subjects
-Related subjects
-相关科目
-{{div col|colwidth=20em}}
-* [[List of dynamical system topics]]
-* [[Baker's map]]
-* [[Biological applications of bifurcation theory]]
-* [[Dynamical system (definition)]]
-* [[Embodied Embedded Cognition]]
-* [[Fibonacci numbers]]
-* [[Fractals]]
-* [[Gingerbreadman map]]
-* [[Halo orbit]]
-* [[List of types of systems theory]]
-* [[Oscillation]]
-* [[Postcognitivism]]
-* [[Recurrent neural network]]
-* [[Combinatorics and dynamical systems]]
-* [[Synergetics (Haken)|Synergetics]]
-* [[Systemography]]
-{{div col end}}
-;Related scientists
-Related scientists
-相关科学家
-{{div col|colwidth=20em}}
-* [[People in systems and control]]
-* [[Dmitri Anosov]]
-* [[Vladimir Arnold]]
-* [[Nikolay Bogolyubov]]
-* [[Andrey Kolmogorov]]
-* [[Nikolay Mitrofanovich Krylov|Nikolay Krylov]]
-* [[Jürgen Moser]]
-* [[Yakov G. Sinai]]
-* [[Stephen Smale]]
-* [[Hillel Furstenberg]]
-* [[Grigory Margulis]]
-* [[Elon Lindenstrauss]]
-{{div col end}}
-== Notes ==
-{{reflist}}
-== Further reading ==
-*{{cite book |first1=Frederick D. |last1=Abraham |first2=Ralph |last2=Abraham |authorlink2=Ralph Abraham (mathematician)|first3=Christopher D. |last3=Shaw |title=A Visual Introduction to Dynamical Systems Theory for Psychology |url=https://books.google.com/books?id=oMbaAAAAMAAJ |year=1990 |publisher=Aerial Press |isbn=978-0-942344-09-7 |oclc=24345312}}
-*{{cite book |first=Edward J. |last=Beltrami |title=Mathematics for Dynamic Modeling |url=https://books.google.com/books?id=MmoiJPsuFW8C |year=1998 |publisher=Academic Press |isbn=978-0-12-085566-7 |edition=2nd |oclc=36713294}}
-*{{cite book |first=Otomar |last=Hájek |title=Dynamical systems in the plane |url=https://books.google.com/books?id=czhPAQAAIAAJ |year=1968 |publisher=Academic Press |oclc=343328 |authorlink=Otomar Hájek |ref=harv}}
-*{{cite book |first=David G. |last=Luenberger |title=Introduction to dynamic systems: theory, models, and applications |url=https://books.google.com/books?id=mvlQAAAAMAAJ |year=1979 |publisher=Wiley |isbn=978-0-471-02594-8 |oclc=4195122 |authorlink=David Luenberger |ref=harv}}
-*{{cite book |first1=Anthony |last1=Michel |author2=Kaining Wang |author3=Bo Hu |title=Qualitative Theory of Dynamical Systems |url=https://books.google.com/books?id=ZPGBnZGV2_cC |year=2001 |publisher=Taylor & Francis |isbn=978-0-8247-0526-8 |oclc=45873628}}
-*{{cite book |first1=Louis |last1=Padulo |first2=Michael A. |last2=Arbib |authorlink2=Michael A. Arbib |title=System theory: a unified state-space approach to continuous and discrete systems |url=https://books.google.com/books?id=6UPvAAAAMAAJ |year=1974 |publisher=Saunders |isbn=9780721670355 |oclc=947600 |ref=harv}}
-* {{cite book |first=Steven H. |last=Strogatz |title=Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering |year=1994 |publisher=Addison Wesley |isbn=978-0-7382-0453-6 |oclc=49839504 |authorlink=Steven Strogatz |ref=harv |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }}
-== External links ==
-*[https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html Dynamic Systems] Encyclopedia of Cognitive Science entry.
-*[http://mathworld.wolfram.com/DynamicalSystem.html Definition of dynamical system] in MathWorld.
-*[http://www.dynamicalsystems.org/ DSWeb] Dynamical Systems Magazine
-{{Areas of mathematics | state=collapsed}}
-[[Category:Dynamical systems]]
-Category:Dynamical systems
-类别: 动力系统
-[[Category:Complex systems theory]]
-Category:Complex systems theory
-范畴: 复杂系统理论
-[[Category:Computational fields of study]]
-Category:Computational fields of study
-类别: 研究的计算领域
-<noinclude>
-<small>This page was moved from [[wikipedia:en:Dynamical systems theory]]. Its edit history can be viewed at [[动力系统理论/edithistory]]</small></noinclude>
-[[Category:待整理页面]]

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