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第69行： − − :[[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 Ergodic Theory'''是数学的一个分支，研究有不变测度和相关问题的动力系统。它最初的发展受到了统计物理学的推动。 − 遍历理论是研究保测变换的渐近性态的数学分支。它起源于为统计力学提供基础的"遍历假设"研究，并与动力系统理论、概率论、信息论、泛函分析、数论等数学分支有着密切的联系。 − − − − === Functional analysis 泛函分析=== − :[[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]].+ − + − '''泛函分析 Functional analysis'''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。 + − === Graph dynamical systems 图动力系统===+ − :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.+ − + − '''图动力系统 Graph dynamical systems (GDS)'''可以用来描绘图或网络上发生的各种过程。图动力系统的数学和计算分析的一个主要主题是将其结构特性（例如：网络连接性）与其所产生的全局动力学联系起来。 − − === Projected dynamical systems 投影动力系统=== − − :[[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 Projected Dynamical Systems'''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和应用。一个投影动力系统是由投影微分方程的'''流形 flow'''给定的。 − 通过对投影微分方程的流分析，给出了一个投影动力系统的表达式： + − 其中K为约束集。这种形式的微分方程因具有不连续的向量场而值得注意。+ − − − − − :[[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 Symbolic Dynamics'''是通过离散空间对拓扑或平滑动力学系统进行建模的方法，该离散空间由无限的抽象符号序列组成，每个抽象符号对应于系统的一个状态，并且由移位运算符给出动力学（演化）。 − − − === System dynamics 系统动力学=== − − :[[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 System Dynamics'''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、存量(stocks)和流量(flows)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。 − − === Topological dynamics 拓扑动力学=== − − :[[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 Topological Dynamics'''是动力系统理论的一个分支。在拓朴动力学中，动力系统的定性性质和渐近性质是从一般拓扑学的观点来研究的。 − − == Applications 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. 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，动力系统理论在运动科学中崭露头角，成为一种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应用相关的任何主张。 − − === In cognitive science 在认知科学中的应用=== − − 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. − − 动力系统理论已经被应用于神经科学和认知发展领域，特别是在认知发展的'''新皮亚杰学派 neo-Piagetian'''中。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。人们还相信微分方程是人类行为建模最合适的工具。人们认为微分方程可以解释为通过状态空间代表一个主体的认知轨迹的算式。换句话说，动力学家认为心理学应该是(或者就是)(通过微分方程)描述在一定的环境和内部压力下的主体的认知和行为的学科。混沌理论在相关领域也经常被采用。 − − − − 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有序的新状态结构，这个过程被称为“扇贝化 Scalloping”，也就是头脑的复杂表现的不断累积和崩溃的过程。这种新的状态是渐进的、离散的、异质的的和不可预知的。+ + + − 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}}</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.+ − 动力系统理论最近还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。+ + + − + − {{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.+ − +
无编辑摘要
'''[[控制理论]]（Control Theory）'''是工程和数学的一个交叉学科。控制理论是一个研究如何调整动态系统特性的理论，它也是工程和数学的一个交叉学科，逐渐的应用在许多社会科学中，例如心理学、社会学（社会学中的控制理论）、犯罪学及'''金融系统(Financial System)'''。控制理论一般的目的是借由控制器的动作让系统稳定，也就是系统维持在设定值，而且不会在设定值附近晃动。维持设定值保持小范围稳定甚至不变的控制行为称为控制调节，设定值快速变化，对于跟踪速度加速度等的控制要求较高的控制行为称为伺服。控制理论的研究的一部分研究对于动力系统行为的研究产生了深远的影响。
'''[[控制理论]]（Control Theory）'''是工程和数学的一个交叉学科。控制理论是一个研究如何调整动态系统特性的理论，它也是工程和数学的一个交叉学科，逐渐的应用在许多社会科学中，例如心理学、社会学（社会学中的控制理论）、犯罪学及'''金融系统(Financial System)'''。控制理论一般的目的是借由控制器的动作让系统稳定，也就是系统维持在设定值，而且不会在设定值附近晃动。维持设定值保持小范围稳定甚至不变的控制行为称为控制调节，设定值快速变化，对于跟踪速度加速度等的控制要求较高的控制行为称为伺服。控制理论的研究的一部分研究对于动力系统行为的研究产生了深远的影响。
=== 遍历理论===
=== 遍历理论===
'''[[遍历理论]](Ergodic Theory)'''是数学的一个分支，它起源于为统计力学提供基础的"遍历假设"研究，并与动力系统理论、概率论、信息论、泛函分析、数论等数学分支有着密切的联系。
===泛函分析===
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]](Functional analysis)'''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，其创立很大程度上归功于数学家Stefan Banach。。
===图动力系统===
'''[[图动力系统]](Graph dynamical systems (GDS))'''可以用来描绘图或网络上发生的各种过程。图动力系统的数学和计算分析的一个主要主题是将其结构特性（例如：网络连接性）与其所产生的全局动力学联系起来。
===投影动力系统===
'''[[投影动力系统]](Projected Dynamical Systems)'''一种数学理论，用于研究将解决方案限制为约束集的动力系统的行为。这门学科与静态理论中的最优化和平衡问题以及动态理论中的常微分方程都有联系和应用。一个投影动力系统是由投影微分方程的[[流形]]给定的。
通过对投影微分方程的[[流形]]分析，给出了一个投影动力系统的表达式：
:<math>
:<math>
\frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t)))
\frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t)))
</math>
</math>
其中K为约束集。这种形式的微分方程因具有不连续的向量场而受到许多研究人员的注意。
===[[符号动力学]]===
=== Symbolic dynamics 符号动力学===
'''[[符号动力学]](Symbolic Dynamics)'''是通过离散空间对拓扑或平滑动力学系统进行建模的方法，该离散空间由无限的抽象符号序列组成，每个抽象符号对应于系统的一个状态，并且动态（演化）由移位运算符给出。
===[[系统动力学]]===
'''[[系统动力学]](System Dynamics)'''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<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>。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、存量(stocks)和流量(flows)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。
===[[拓扑动力学]]===
'''[[拓扑动力学]](Topological Dynamics)'''是动力系统理论的一个分支。在拓朴动力学中，动力系统的定性性质和渐近性质是从一般拓扑学的观点来研究的。
== 应用==
===在运动生物力学中的应用===
在运动生物力学中，动力系统理论在运动科学中展露头角，成为一种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应用相关的任何主张。
===在认知科学中的应用===
动力系统理论已经被应用于神经科学和认知发展领域，特别是在认知发展的'''新皮亚杰学派（neo-Piagetian）'''中。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。人们还相信微分方程是人类行为建模最合适的工具。人们认为微分方程可以解释为通过状态空间代表一个主体的认知轨迹的算式。换句话说，动力学家认为心理学应该是(或者就是)(通过微分方程)描述在一定的环境和内部压力下的主体的认知和行为的学科。混沌理论在相关领域也经常被采用。
===In second language development 在二语习得中的应用===
在学习的过程中，旧的模式被打破，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))随'''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有序的新状态结构，这个过程被称为“扇贝化(Scalloping)”，也就是头脑的复杂表现的不断累积和崩溃的过程。这种新的状态是渐进的、离散的、异质的的和不可预知的<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>。
动态系统理论最近已被用来解释儿童发展中一个长期未解决的问题，称为A-not-B错误。[6]
动力系统理论最近还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<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}}</ref>。
===在第二语言发展中的应用===
动力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认为，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。她在文章中认为，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、反馈敏感的和适应性的动力系统。
动力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认为，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<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>。她在文章中认为，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、反馈敏感的和适应性的动力系统。
== See also 参见==
== See also 参见==