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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(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|找不到原文]]） 和/或代数点的数论性质。
'''算术动态系统 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|找不到原文]]） 和/或代数点的数论性质。
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）了解
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）了解
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(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。
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(Complex Systems)'''是研究自然、社会和科学 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|科学为什么会和自然、社会并列呢？]]）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。
'''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。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。
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(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 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（英语：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。
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（英语：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。
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.
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 Ergodic Theory'''是数学的一个分支，研究有不变测度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 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(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。
'''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。
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.
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关联它们的结构性质(例如:网络连接)和结构性质造成的网络整体的动态结果。
'''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关联它们的结构性质(例如:网络连接)和结构性质造成的网络整体的动态结果。
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(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，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'''流行(flow)'''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]]）。
'''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，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'''流行(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：
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：
:<math>
:<math>
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（Symbolic Dynamics）'''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)。
'''符号动力学 Symbolic Dynamics'''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(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、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。
'''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、股票（stocks）和流形（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。
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。
'''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==
== Applications 应用==