# 动态系统理论

此词条暂由Henry翻译。

**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.

动态系统理论 动态系统理论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，是直接假定运动方程的推广，不受最小作用原理的 欧拉-拉格朗日方程Euler–Lagrange equations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，或者在任意的时间集合上运行时，得到时间尺度上的动态方程。有些情况也可以用混合算子来模拟，如微分差分方程。

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.

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，并研究系统的性质。如果可能的话，它能得出主要是机械或其他物理性质的系统的运动方程的解，如行星轨道和电子电路的行为，以及出现在生物学，经济学和其他地方的系统上。现代的许多研究都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.

这个研究领域也被称为“动力学系统”，“数学动态系统理论”或“动力学系统的数学理论”。

[] 洛仑兹吸引子Lorenz attractor是非线性动力系统的一个例子。对这个系统的研究助力了混沌理论的发展

## Overview

## Overview

概览

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?"

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?"

动态系统理论和混沌理论处理动力系统的长期定性行为。在这里，重点不是找到定义动力系统的方程的精确解(这通常是没有希望的) ，而是回答诸如“系统是否在长期会稳定下来，如果是，可能的稳定状态是什么? ”或者“系统的长期行为是否取决于它的初始条件? ”的问题。

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.

一个重要的目标是描述给定动力系统的固定点或稳定状态; 这些是不随时间变化的变量的值。这些不动点中的一些是“吸引”的，这意味着如果系统开始于这样的点的附近的状态，它会往不动点的方向收敛。

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.

类似地，人们对周期点感兴趣，即在几个时间步骤之后重复的系统状态。周期点也可以是吸引人的。Sharkovskii的定理是关于一维离散动力系统的周期点数的一个有趣陈述。

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”。处理混沌的清晰定义和对混沌进行研究的动力学系统分支，被称作混沌理论。

## History

## 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.

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Newtonian mechanics。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关系隐含地给出的，这种关系只给出了系统在未来很短时间内的状态。

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.

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.

在计算机出现之前，解决动力系统问题需要复杂的数学技术，而且只能在一小类动态系统中完成。

Some excellent presentations of mathematical dynamic system theory include **脚本错误：没有“Footnotes”这个模块。**, **脚本错误：没有“Footnotes”这个模块。**, **脚本错误：没有“Footnotes”这个模块。**, and **脚本错误：没有“Footnotes”这个模块。**.^{[2]}

Some excellent presentations of mathematical dynamic system theory include , , , and .

一些优秀的数学动态系统理论的演示包括 和 。

## Concepts

## Concepts

概念

### Dynamical systems

### 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.

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.

动力系统的概念是一个数学形式化的任意固定“规则”，它描述了一个点的位置在其环境空间的时间依赖性。这些例子包括描述钟摆摆动的数学模型、管道中的水流量以及每年春天湖中鱼的数量。

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，或者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微小变化。这些数字也是几何空间流形的坐标。动力系统的演变规则是一个固定的规则，描述了当前状态下的未来状态。该规则可以是确定性的(在给定的时间间隔内，只有一个未来状态在当前状态之后)或随机性的(状态的演变受到随机冲击)。

### Dynamicism

### Dynamicism

动力主义

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.

动态主义又被称为动态假设或认知科学或动态认知中的动态假设，是以哲学家蒂姆·范·格尔德的著作为代表的认知科学的一种新方法。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

### 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.

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，非线性系统是一个不是线性的系统--也就是说，一个不满足叠加原理的系统。从技术上讲，非线性是任何需要求解的变量都不能被写成独立分量的线性和的问题。非齐次系统除了自变量函数的存在外是线性的，根据严格的定义其他都是非线性的，但这类系统通常与线性系统一起研究，因为只要知道特定的解，它们就可以转化为线性系统。

## Related fields

## Related fields

相关领域

### Arithmetic dynamics

### 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.

算术动态系统是一个出现在20世纪90年代的领域，融合了数学的两个领域----动力系统和数论。经典的离散动力学是指研究复杂平面或实线的自映射的迭代。算术动态系统是研究在多项式或有理函数的重复应用下整数、有理数、并元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 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动力学系统的行为，即状态随时间演化的系统，这些系统可能表现出对初始条件高度敏感的动力学(通常称为蝴蝶效应)。由于这种敏感性，在初始条件下其表现为指数增长的扰动。混沌系统的行为看起来是随机的，即使这些系统是确定性的，这意味着它们未来的动力学完全由它们的初始条件定义，没有任何随机因素参与。这种行为被称为确定性混沌，或简单的混沌。

### Complex systems

### 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.

复杂系统是研究自然界、社会和科学中被认为是复杂的系统的共同性质的一个科学领域。它也被称为复杂系统理论、复杂性科学、复杂系统研究和复杂性科学。这些系统的关键问题在于它们的正式建模与模拟。从这个角度来看，在不同的研究背景中，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.

对复杂系统的研究给许多科学领域带来了新的活力，在这些领域，更典型的还原论策略已经有所欠缺。因此，复杂系统通常被用作一个宽泛的术语，涵盖了许多不同学科的问题研究方法，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本身本质的研究。

### 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 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，其中的一些部分是关于对动力系统的行为进行影响。

### Ergodic theory

### Ergodic theory

遍历理论Ergodic theory

- 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，研究具有不变测度和相关问题的动力系统。它最初的发展是由统计物理学问题推动的。

### Functional analysis

### 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.

泛函分析是数学，特别是分析的一个分支，研究向量空间和作用于向量空间的算子。它的历史根源在于研究函数空间，特别是函数的变换，例如 傅里叶变换Fourier transform，以及微分和积分方程的研究。函数式这个词的用法可以追溯到变分法，意味着一个函数的参数是一个函数。它的使用一般归功于数学家和物理学家维托沃尔泰拉而它的建立主要是归功于数学家斯蒂芬巴纳赫。

### Graph dynamical systems

### Graph dynamical systems

图形动力系统

- 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.

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.

图动态系统(GDS)的概念可以用来捕捉发生在图或网络上的各种过程。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)和全球动态的结果。

### Projected dynamical systems

### 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.

投影动力系统是一个研究动力系统行为的数学理论，其中的解限制在一个约束集。这门学科与最优化和平衡问题的静态世界以及常微分方程的动态世界都有联系和应用。一个投影动态系统是由计划的微分方程的流量给出。

### Symbolic dynamics

### 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.

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.

符号动力学是通过一个由无限抽象符号序列组成的离散空间建立一个拓扑或光滑动力系统的实践，每一个抽象符号序列对应于一个系统的状态，由移位算子给出动力学(演化)。

### System dynamics

### 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.

系统动力学是一种理解系统随时间变化行为的方法。它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。使用系统动力学不同于其他研究系统的方法是使用反馈循环、存量和流量。这些元素有助于描述看似简单的系统如何显示令人困惑的非线性。

### Topological dynamics

### 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.

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出发来研究的。

## Applications

## Applications

应用

### In human development

### In human development

在人类发展中

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，动态系统理论已经被用来增强和简化Erik Erikson的8个社会心理发展阶段，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Fibonacci sequence的自组织性和分形特性。利用数学模型，人类发展的自然进程被确定为八个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、幼儿期(4-7岁)、中期儿童期(7-11岁)、青春期(11-18岁)、青年期(18-29岁)、中期成年期(29-48岁)和老年期(48-78岁以上)。

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，年龄间隔之间的阶段转换代表了多层次的自我组织过程(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，以及在达到18岁这一关键点(成年早期)之后，男性的睾丸激素达到高峰，而女性的最佳生育期刚刚开始。同样，在30岁时，女性的最佳生育能力开始下降，在48岁这一阶段，从中年成年期过渡到老年成年期，开始绝经的平均年龄出现。

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，这些事件是衰老的物理生物吸引子。实际上，人类发展的预测在统计学意义上成为可能，在这个意义上，一年中不同时间的平均气温或降水量可以用来预测2010年的天气预报。人类发展的每一个预定阶段都遵循最佳的表观遗传生物模式。这种现象可以用生物DNA中斐波那契数的出现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

### In biomechanics

在生物力学上

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20世纪90年代运动生物力学，动态系统理论运动科学已经成为运动表现建模的可行框架。从动力系统的角度来看，人类的运动系统是一个高度复杂的相互依赖的子系统网络。呼吸系统，循环系统，神经系统，骨骼肌系统，感知系统，是由大量相互作用的成分组成的。（比如血细胞、氧分子、肌肉组织、代谢酶、结缔组织和骨骼)。在动态系统理论，运动模式通过物理系统和生物系统中自我组织的一般过程出现。没有任何研究证实与这一框架的概念应用相关的任何主张。

### In cognitive science

### In cognitive science

在认知科学领域

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.

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.

动力系统理论已经应用于神经科学和认知发展领域，特别是在新皮亚杰学派。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。它还认为微分方程式是建模人类行为的最合适的工具。这些方程解释为代表一个主体的认知轨迹通过状态空间。换句话说，动力学家认为心理学应该(或者是)描述(通过微分方程)在一定的环境和内部压力下的认知和行为。混沌理论的语言也经常被采用。

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.

在这个过程中，学习者的思维达到了一种不平衡的状态，旧的模式被打破了。这是认知发展的阶段性转变。自我组织(连贯形式的自发创造)作为活动水平相互联系而产生。新形成的宏观和微观结构相互支持，加速了这一过程。这些联系通过一个被称为“扇贝化”的过程(复杂表现的反复累积和崩溃) ，在头脑中形成了一种新的秩序状态的结构。这种新的，新奇的状态是渐进的，离散的，特殊的和不可预知的。

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.

动态系统理论最近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谬误。

### In second language development

### In second language development

在第二语言的发展中

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.

动态系统理论在第二语言习得研究中的应用归功于戴安娜·拉森-弗里曼，她在1997年发表的一篇文章中声称，二语习得应该被看作是一个包括语言流失和语言习得在内的发展过程。她在文章中主张，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、适应性的动态系统。

## See also

## See also

参见

- Related subjects

Related subjects

相关科目

动态系统主题清单

贝克地图

分岔理论的生物学应用

动态系统（定义）

具身认知

斐波那契数列

分形

姜饼人地图

哈罗轨迹

系统理论类型清单

振荡

后认知主义

循环神经理论

组合学和动态理论

协同论

系统学

- Related scientists

Related scientists

相关科学家

系统论及控制论著名人士

德米特里·阿诺索夫

弗拉基米尔·阿诺尔德

尼古拉·博戈柳博夫

奥黛丽·柯尔莫哥洛夫

尼古拉·米特罗法诺维奇·克雷洛夫

尤尔根·莫瑟

雅科夫·G·西奈

斯蒂芬·斯梅尔

希尔勒·弗斯滕伯格

格里戈里·马古利斯

埃隆·林登斯特劳斯

## Notes

## Notes

注释

- ↑ 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.

## Further reading

## Further reading

进一步阅读

- Abraham, Frederick D.; Abraham, Ralph; Shaw, Christopher D. (1990).
*A Visual Introduction to Dynamical Systems Theory for Psychology*. Aerial Press. ISBN 978-0-942344-09-7. OCLC 24345312. https://books.google.com/books?id=oMbaAAAAMAAJ.

- Beltrami, Edward J. (1998).
*Mathematics for Dynamic Modeling*(2nd ed.). Academic Press. ISBN 978-0-12-085566-7. OCLC 36713294. https://books.google.com/books?id=MmoiJPsuFW8C.

- Hájek, Otomar (1968).
*Dynamical systems in the plane*. Academic Press. OCLC 343328. https://books.google.com/books?id=czhPAQAAIAAJ.

- Luenberger, David G. (1979).
*Introduction to dynamic systems: theory, models, and applications*. Wiley. ISBN 978-0-471-02594-8. OCLC 4195122. https://books.google.com/books?id=mvlQAAAAMAAJ.

- Michel, Anthony; Kaining Wang; Bo Hu (2001).
*Qualitative Theory of Dynamical Systems*. Taylor & Francis. ISBN 978-0-8247-0526-8. OCLC 45873628. https://books.google.com/books?id=ZPGBnZGV2_cC.

- Padulo, Louis; Arbib, Michael A. (1974).
*System theory: a unified state-space approach to continuous and discrete systems*. Saunders. ISBN 9780721670355. OCLC 947600. https://books.google.com/books?id=6UPvAAAAMAAJ.

- Strogatz, Steven H. (1994).
*Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering*. Addison Wesley. ISBN 978-0-7382-0453-6. OCLC 49839504. https://archive.org/details/nonlineardynamic00stro.

## External links

## External links

外部链接

- Dynamic Systems Encyclopedia of Cognitive Science entry.

- Definition of dynamical system in MathWorld.

- DSWeb Dynamical Systems Magazine

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

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