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第30行： − + − + − − {{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. + − 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，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]], 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. + − + − === Nonlinear system 非线性系统===+ − − {{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. − − − − − :[[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. − − − − − − − === Chaos theory 混沌理论=== − − :[[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 Butterfly Effect'''）。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，即它们未来的动力学变化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 复杂系统=== − − :[[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 Complex Systems'''是研究自然、社会和科学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的困难。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 [[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. − − 复杂系统的研究为许多科学领域带来了新的活力，在这些领域中，更为典型的还原论策略已经不足。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 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 Control Theory'''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。 − 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。设定值一般维持不变的控制称为调节，设定值快速变化，要求跟踪速度加速度等的控制称为伺服。它的其中一部分研究影响动力系统行为的各种因素。+ + − +
无编辑摘要
一些优秀的数学动力系统理论学家包括贝尔特拉米（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）<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>等在该领域做出了杰出的贡献。
一些优秀的数学动力系统理论学家包括贝尔特拉米（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）<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>等在该领域做出了杰出的贡献。
== Concepts 概念==
==概念==
=== Dynamical systems 动力系统===
===动力系统===
动力系统概念是对描述了一个点的位置在其周围环境中随时间变化的任何“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，都属于动力系统的概念范畴。
动力系统概念是对描述了一个点的位置在其周围环境中随时间变化的任何“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，都属于动力系统的概念范畴。
动力系统的状态由实数的集合决定，或更一般地由适当的状态空间中的点集决定。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)。
===动态主义 ===
=== Dynamicism 动态主义 ===
'''动态主义 Dynamicism'''，也称动态假设，或称认知科学的动态假设或动态认知，是以哲学家Tim van Gelder的著作为代表的认知科学的一种新取向。动态主义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。
'''动态主义 Dynamicism'''，也称动态假设，或称认知科学的动态假设或动态认知，是以哲学家Tim van Gelder的著作为代表的认知科学的一种新取向。动态主义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。
=== 非线性系统===
在数学中，非线性系统是指不是线性的系统，即不满足叠加原理的系统。从技术上讲，非线性系统是无法解决的变量不能写成独立分量的线性和的任何问题。阿非均匀系统，其是直链距的函数的存在独立变量，是根据一个严格的定义非线性的，但是这样的系统通常被研究沿着线性系统，因为它们可以被转换成一个线性系统，只要一个特定的解决方案是已知的
在数学中，'''非线性系统 (Nonlinear System)'''是指系统不是线性的系统，即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，除了它的自变量函数以外，其他部分都是线性的。但非齐次系统通常可当做线性系统进行研究，因为只要知道特定解，它就可以转化为线性系统。
在数学中，'''非线性系统 Nonlinear System'''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，除了它的自变量函数以外，其他部分都是线性的。但非齐次系统通常可当做线性系统进行研究，因为只要知道特定解，它就可以转化为线性系统。
== Related fields 相关领域==
== Related fields 相关领域==
=== Arithmetic dynamics 算术动力学===
=== Arithmetic dynamics 算术动力学===
'''算术动力学 (Arithmetic Dynamics)'''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的是复平面或实实数轴的自映射的迭代，算术动力学是在反复应用多项式或有理函数的情况下对整数，有理数，p进数(p-adic)和/或代数点的数论性质进行研究。
'''算术动力学 Arithmetic Dynamics'''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实数轴的自映射的迭代。算术动力学是在反复应用多项式或有理函数的情况下研究整数，有理数，p进数(p-adic)和/或代数点的数论性质。
===混沌理论===
混沌理论(Chaos theory)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 (Butterfly Effect)'''）。由于这种敏感性，在初始条件下表现为扰动呈指数增长，因此混沌系统的行为似乎是随机的。即使这些系统是确定性的，也会发生这种情况，这意味着它们的未来动力完全由其初始条件定义，而没有涉及随机元素。这种行为称为确定性混乱，或简称为混乱。
=== Control theory 控制理论===
=== 复杂系统===
'''复杂系统 (Complex Systems)'''是研究自然、社会和科学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的困难。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。
复杂系统的研究为许多科学领域带来了新的活力，在这些领域中，更为典型的简化主义策略已经不足以提供研究动力。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许多不同学科的问题的研究方法。
=== 控制理论===
'''[[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。
=== Ergodic theory 遍历理论===
=== 遍历理论===
:[[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 system]]s with an [[invariant measure]] and related problems. Its initial development was motivated by problems of [[statistical physics]].