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→Related fields： 第一遍翻译，到Projected dynamical systems之前
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。
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。
== Related fields ==
== 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。
=== Chaos theory ===
=== 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 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.
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。
=== Complex systems ===
=== 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 [[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.
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。
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。
=== Control theory ===
=== Control theory 控制理论===
:[[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 system]]s.
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，它部分地涉及到对动力系统行为的影响。
=== Ergodic theory ===
=== Ergodic theory 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]].
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。
=== Functional analysis ===
=== 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]].
:[[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.
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。
=== Graph dynamical systems ===
=== 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 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.
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(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。
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=== Projected dynamical systems ===
=== Projected dynamical systems ===
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。
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 ==
== Applications ==