动力系统理论 Dynamical Systems Theory

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Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

动力系统理论 Dynamical Systems Theory是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续，或者像cantor（康托尔）集一样在任意的时间集合上运行时，人们就能得到时间尺度上的动力方程。 算子 Operators是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。动力系统的有些情况也可以用混合算子 Mixed Operators来建模，如微分-差分方程。

 --趣木木（讨论）专有术语的基本格式不需要加括号  
--嘉树（讨论） 只找到了算子，没找到混合算子orz

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

这个理论处理动力系统的长期定性行为。如果能够得到解的话，还可以研究系统的运动方程。这些方程通常运用到机械或物理研究领域，如行星轨道和电子电路，以及出现在生物学，经济学和其他地方的系统。现代的研究大多集中在对混沌系统的研究上。

This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.

This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.

这个研究领域被称为动力系统，数学动力系统理论或动力系统的数学理论。

文件:Lorenz attractor yb.svg

The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.

洛伦兹吸引子 Lorenz Attractor是非线性动力系统的一个例子。对这个系统的研究产生了混沌理论。

Overview 综述

Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"

动力系统理论和混沌理论 Chaos Theory是用来处理动力系统的长期定性行为的理论。寻找动力系统方程的精确解通常是很难达到的。这两个理论的重点不在于找到精确解，而是回答如下的问题，如“系统长期来看是否会稳定下来，如果可以，那么可能的稳定状态是什么样的?”，或“系统长期的行为是否取决于其初始条件?”

An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.

An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.

描述给定动力系统的不动点或定态 Steady States是一个重要的目标。不动点或定态的变量值不会随时间的变化而变化。一些不动点是有吸引力的（attractive），即如果系统的初始值在它的附近，系统最终会收敛到这个不动点。

Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

人们还对动力系统的周期点 Periodic Points感兴趣，即系统在几个时间步之后会不断重复的状态。周期点也可以是有吸引力的。Sharkovskii定理描述了一维离散动力系统的周期点的个数。

Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos.^[1] The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.

Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.

即使是简单的非线性动力系统也常常表现出看似随机的行为，这种行为被称为混沌。混沌理论主要研究混沌的清晰定义和混沌的现象，它是动力系统理论的一个分支。

History 历史

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.

动态系统理论的概念起源于牛顿运动定律。与其他自然科学和工程学科一样，动力系统的演化规律也是通过一种预测系统在未来很短时间内的状态的关系隐含地给出的。

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.

在高速计算机器出现之前，解决动力系统问题需要复杂的数学技能，而且还只能解决一小类动力系统问题。

Some excellent presentations of mathematical dynamic system theory include Beltrami (1990), Luenberger (1979), Padulo & Arbib (1974), and Strogatz (1994).^[2]

Some excellent presentations of mathematical dynamic system theory include , , , and . To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008

优秀的数学动力系统理论包括，，，和。--嘉树（需要原文）。

 --趣木木（讨论）了解  会询问一下词源那边

Concepts 概念

Dynamical systems 动力系统

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.

动力系统是一个对任何描述了点的位置在周围环境随时间变化的“固定”规则的数学式。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。

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).

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）——的坐标系（coordinates ）--嘉树（这里翻译为坐标还是坐标系更好一点？）。

 --趣木木（讨论）按照之前的说法 由适当的状态空间中的一组点  感觉坐标更贴切  或者译为坐标组（一组包含很多的坐标点）

动力系统的演化是描述了在当前状态之后出现的未来状态的固定规则。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 动态主义

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.

动态主义(Dynamicism)，又称动态假设，或称认知科学、动态认知中的动态假设，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。

Nonlinear system 非线性系统

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，除了它的自变量包含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。

Related fields 相关领域

Arithmetic dynamics 算术动态系统

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,

p

-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 --嘉树（找不到原文）和/或代数点的数论性质。

 --趣木木（讨论）了解

Chaos theory 混沌理论

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。

Complex systems 复杂系统

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 --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。

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带来了广泛的新活力。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。

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 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。

 --趣木木（讨论）单独小词条可以考虑补充一下内容 
 --趣木木（讨论）添加补充内容：

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 遍历理论

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。

 --趣木木（讨论）百度百科简单补充：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 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。

Graph dynamical systems 图动力系统

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。

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.

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--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）。

 --趣木木（讨论） wiki上搜索该名词后的补充

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]\displaystyle{ \frac{dx(t)}{dt} = \Pi_K(x(t),-F(x(t))) }[/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值得注意。

Symbolic dynamics 符号动力学

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)。

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.^[3] 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）--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 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.

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 human development 在人类发展中的应用

In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence.^[4] Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).^[4]

In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).

在人类发展方面，动力系统理论已经被用来增强和简化 Erik Erikson 的社会心理发展8阶段理论(Eight Stages of Psychosocial Development)，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的自组织性（self-organizing）和分形(Fractal)特性。利用数学模型，人类发展的自然进程被分为8个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、童年早期(4-7岁)、童年中期(7-11岁)、青春期(11-18岁)、成年早期(18-29岁)、成年中期(29-48岁)和老年成年期(48-78岁及以上)。

According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males^[5] and the period of optimal fertility begins in females.^[6] Similarly, at age 30 optimal fertility begins to decline in females,^[7] and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs.^[7]

According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males and the period of optimal fertility begins in females. Similarly, at age 30 optimal fertility begins to decline in females, and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs.

根据这个模型，年龄的阶段转换代表了多层次的自组织过程(例如，分子、基因、细胞、器官、器官系统、生物体、行为和环境)。例如，在从青春期向成年早期过渡的阶段中，在达到18岁这一关键年龄之后，男性的睾丸激素达到高峰，女性的最佳生育期开始。同样，在30岁时，女性的最佳生育能力开始下降；在从成年中期过渡到老年成年期时，48岁是绝经的平均年龄。

These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA^[8] and self-organizing properties of the Fibonacci numbers that converge on the golden ratio.

These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA and self-organizing properties of the Fibonacci numbers that converge on the golden ratio.

从斐波那契数学模型和动力系统理论的角度来看，上述事件是衰老的物理生物吸引子（physical bioattractors）。实际上，正如一年中不同时间的平均气温和降水量可以用来预测天气，预测人类的发展在统计意义上同样是可能的。人类发展的每个预定阶段（predetermined stages）都遵循最佳的表观遗传生物模式（epigenetic biological pattern）。这种现象可以用 DNA 中的斐波那契数和收敛于黄金分割比的斐波那契数的自组织特性来解释。

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.^[9] 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，动力系统理论新兴地成为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相关的任何主张。--嘉树（没有研究证实？那就是说不可信吗？所以这个文本的意思是什么。）

 ---趣木木（讨论）这个我问一下词源

In cognitive science 在认知科学中的应用

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(或者是)(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.^[10]

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具有新状态的结构，这个过程被称为“扇贝化（scalloping）”，也就是头脑的复杂性能的不断累积和崩溃的过程 --嘉树（扇贝化的翻译拿不准，这句话的描述比较抽象）。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.^[11]

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。

In second language development 在二语习得中的应用

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.^[12] 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.

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。

Notes 参考资料

↑ Grebogi, C.; Ott, E.; Yorke, J. (1987). "Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics". Science. 238 (4827): 632–638. Bibcode:1987Sci...238..632G. doi:10.1126/science.238.4827.632. JSTOR 1700479. PMID 17816542.
↑ Jerome R. Busemeyer (2008), "Dynamic Systems". To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期June 13, 2008，.
↑ MIT System Dynamics in Education Project (SDEP) -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期2008-05-09.
↑ ^4.0 ^4.1 Sacco, R.G. (2013). "Re-envisaging the eight developmental stages of Erik Erikson: The Fibonacci Life-Chart Method (FLCM)". Journal of Educational and Developmental Psychology. 3 (1): 140–146. doi:10.5539/jedp.v3n1p140.
↑ Kelsey, T. W. (2014). "A validated age-related normative model for male total testosterone shows increasing variance but no decline after age 40 years". PLOS One. 9 (10): e109346. Bibcode:2014PLoSO...9j9346K. doi:10.1371/journal.pone.0109346. PMC 4190174. PMID 25295520.
↑ Tulandi, T. (2004). Preservation of fertility. Taylor & Francis. pp. 1–20.
↑ ^7.0 ^7.1 Blanchflower, D. G. (2008). "Is well-being U-shaped over the life cycle?". Social Science & Medicine. 66 (8): 1733–1749. CiteSeerX 10.1.1.63.5221. doi:10.1016/j.socscimed.2008.01.030. PMID 18316146.
↑ Perez, J. C. (2010). (2010). "Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618". Interdisciplinary Sciences: Computational Life Sciences. 2 (3): 228–240. doi:10.1007/s12539-010-0022-0. PMID 20658335.
↑ Paul S Glazier, Keith Davids, Roger M Bartlett (2003). "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research". in: Sportscience 7. Accessed 2008-05-08.
↑ Lewis, Mark D. (2000-02-25). "The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development" (PDF). Child Development. 71 (1): 36–43. CiteSeerX 10.1.1.72.3668. doi:10.1111/1467-8624.00116. PMID 10836556. Retrieved 2008-04-04.
↑ Smith, Linda B.; Esther Thelen (2003-07-30). "Development as a dynamic system" (PDF). Trends in Cognitive Sciences. 7 (8): 343–8. CiteSeerX 10.1.1.294.2037. doi:10.1016/S1364-6613(03)00156-6. PMID 12907229. Retrieved 2008-04-04.
↑ "Chaos/Complexity Science and Second Language Acquisition". Applied Linguistics. 1997.

External links 外部链接

Dynamic Systems 动力系统 Encyclopedia of Cognitive Science entry. --嘉树（这是一个嵌套的链接，但好像只能鼠标点击并超链接到内嵌的那个）认知科学百科全书

Definition of dynamical system 动力系统的定义 in MathWorld. 在MathWorld.wolfram.com储存的定义

DSWeb Dynamical Systems Magazine 动力系统杂志

模板:Areas of mathematics

Category:Dynamical systems

类别: 动力系统

Category:Complex systems theory

范畴: 复杂系统理论

Category:Computational fields of study

类别: 研究的计算领域

This page was moved from wikipedia:en:Dynamical systems theory. Its edit history can be viewed at 动力系统/edithistory

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Chaos: Making a New Science

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Chaos and Fractals: An Elementary Introduction

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牛津通识读本: Chaos

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集智课程推荐

圣塔菲课程：Introduction to Dynamical Systems and Chaos

本课程中，主要介绍动力学系统和混沌系统，您将学到蝴蝶效应（butterfly effect）、奇异吸引子（attractors）等基本概念，以及如何应用于您感兴趣的领域。

动力系统分析

本课程主要讲授连续和离散动力系统的定态、极限环及其稳定性分析、动力学系统的结构稳定性和常见的分支类型以及分析方法，混沌概念等。

[1] Grebogi, C.; Ott, E.; Yorke, J. (1987). "Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics". Science. 238 (4827): 632–638. Bibcode:1987Sci...238..632G. doi:10.1126/science.238.4827.632. JSTOR 1700479. PMID 17816542.

[2] Jerome R. Busemeyer (2008), "Dynamic Systems". To Appear in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期June 13, 2008，.

[sysdyn-3] MIT System Dynamics in Education Project (SDEP) -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期2008-05-09.

[Sacco-4] 4.0 ^4.1 Sacco, R.G. (2013). "Re-envisaging the eight developmental stages of Erik Erikson: The Fibonacci Life-Chart Method (FLCM)". Journal of Educational and Developmental Psychology. 3 (1): 140–146. doi:10.5539/jedp.v3n1p140.

[5] Kelsey, T. W. (2014). "A validated age-related normative model for male total testosterone shows increasing variance but no decline after age 40 years". PLOS One. 9 (10): e109346. Bibcode:2014PLoSO...9j9346K. doi:10.1371/journal.pone.0109346. PMC 4190174. PMID 25295520.

[6] Tulandi, T. (2004). Preservation of fertility. Taylor & Francis. pp. 1–20.

[Social_Science_2008-7] 7.0 ^7.1 Blanchflower, D. G. (2008). "Is well-being U-shaped over the life cycle?". Social Science & Medicine. 66 (8): 1733–1749. CiteSeerX 10.1.1.63.5221. doi:10.1016/j.socscimed.2008.01.030. PMID 18316146.

[8] Perez, J. C. (2010). (2010). "Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618". Interdisciplinary Sciences: Computational Life Sciences. 2 (3): 228–240. doi:10.1007/s12539-010-0022-0. PMID 20658335.

[9] Paul S Glazier, Keith Davids, Roger M Bartlett (2003). "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research". in: Sportscience 7. Accessed 2008-05-08.

[10] Lewis, Mark D. (2000-02-25). "The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development" (PDF). Child Development. 71 (1): 36–43. CiteSeerX 10.1.1.72.3668. doi:10.1111/1467-8624.00116. PMID 10836556. Retrieved 2008-04-04.

[11] Smith, Linda B.; Esther Thelen (2003-07-30). "Development as a dynamic system" (PDF). Trends in Cognitive Sciences. 7 (8): 343–8. CiteSeerX 10.1.1.294.2037. doi:10.1016/S1364-6613(03)00156-6. PMID 12907229. Retrieved 2008-04-04.

[12] "Chaos/Complexity Science and Second Language Acquisition". Applied Linguistics. 1997.

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