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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.

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<font color='red'><s>包含<s></font>函数以外其他部分都是线性的。但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 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<font color='red'><s>包含</s></font>函数以外其他部分都是线性的。<font color='red'>但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。</font><font color='blue'>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。</font>

== Related fields 相关领域==

== Related fields 相关领域==

=== Arithmetic dynamics 算术动态系统===

=== Arithmetic dynamics <font color='red'>算术动态系统</font><font color='blue'>算术动力学</font>===

:[[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 [[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.

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|存疑]]） 和/或代数点的数论性质。

'''<font color='red'>算术动态系统</font><font color='blue'>算术动力学</font> 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<font color='blue'>实直线</font><font color='blue'>实数轴</font>的自映射的迭代。<font color='red'>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|存疑]]） 和/或代数点的数论性质。</font><font color='blue'>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。</font>

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<font color='blue'>它们</font>未来的动力<s>学</s>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<font color='blue'>的困难</font>。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 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.

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。

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。

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。

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。

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。<font color='red'>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。</font><font color='blue'>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。</font>

 

=== Projected dynamical systems 投影动力系统===

=== Projected dynamical systems 投影动力系统===

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'''流形 flow'''给定的。

'''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<font color='blue'>和<s>，并且都有相互联系的</s></font>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：

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>

'''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。

'''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、<font color='red'>股票（stocks）和流形（flows）</font><font color='blue'>存量(stocks)和流量(flows)--[[用户:木子二月鸟|木子二月鸟]]（[[用户讨论:木子二月鸟|原wiki里这两个词指向:https://en.wikipedia.org/wiki/Stock_and_flow ，估计应该是存量和流量的意思，也可以群里大家讨论一下]]）</font>--[[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。