“动力系统理论 Dynamical Systems Theory”的版本间的差异

2020年7月7日 (二) 21:51的版本

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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.

动态系统理论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.

这个理论处理动力系统的长期定性行为，研究系统的性质，如果可能的话，解决方案的运动方程通常主要是机械或其他物理性质的系统，如行星轨道和行为的电子电路，以及系统出现在生物学，经济学和其他地方。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.

这个研究领域也被称为动力系统，数学动态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.

[] Lorenz 吸引子是非线性动力系统的一个例子。对这个系统的研究有助于产生混沌理论

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?"

动态系统理论和混沌理论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?”，“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，系统最终共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

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20世纪90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.

在快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 Beltrami (1990), Luenberger (1979), Padulo & Arbib (1974), and Strogatz (1994).^[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

Dynamical systems

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何固定的“规则” ，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ーー流形ーー的坐标。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，只有一个未来状态从当前状态跟随)或随机性的(状态的演变受到随机冲击)。

Dynamicism

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.

动态主义又称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

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，非线性是一个不是线性的系统ーー也就是说，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

Arithmetic dynamics

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20世纪90年代出现的一个领域，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 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 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 / 或复杂性科学。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.

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不足。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 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，部分涉及到对动力系统行为的影响。

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.

遍历理论是数学的一个分支，研究具有不变测度和相关问题的动力系统。它最初的发展是受到统计物理学问题的推动。

Functional analysis

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，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。它的使用一般被认为是数学家和物理学家维托沃尔泰拉和它的建立主要是归功于数学家斯蒂芬巴纳赫。

Graph dynamical systems

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 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数学理论，研究动力系统的行为，其解决方案限制在一个约束集。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流量给计划的微分方程。

Symbolic dynamics

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 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即使看似简单的系统如何显示令人困惑的非线性。

Topological dynamics

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来研究的。

Applications

In human development

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，人类发展的预测在同样的统计意义上成为可能，即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 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。呼吸的，循环的，神经的，骨骼肌的，知觉的，由大量相互作用的成分组成的。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

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”的过程在头脑中形成了一种新的秩序状态的结构(复杂性能的不断累积和崩溃)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 错误。

In second language development

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。

Notes

↑ 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

Dynamic Systems Encyclopedia of Cognitive Science entry.

Definition of dynamical system in MathWorld.

DSWeb Dynamical Systems Magazine

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

编者推荐

书籍推荐

Chaos: Making a New Science

The million-copy bestseller by National Book Award nominee and Pulitzer Prize finalist James Gleick—the author of Time Travel: A History—that reveals the science behind chaos theory。 Very readable, focuses on the scientists and mathematicians behind many of the key results in chaos. 本书由普利茨奖获得者James Gleick撰写。它是一本易读科普书籍，侧重介绍对混沌概念的发展做出重大贡献的科学家和数学家们。

Chaos and Fractals: An Elementary Introduction

This book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or physics. 这是一本关于混沌和分形的入门书籍。读者需要有一定的线性代数基础，但是不需要有微积分或物理学的基础。

牛津通识读本: Chaos

This book would not have been possible without my parents, of course, but I owe a greater debt than most to their faith, doubt, and hope, and to the love and patience of a, b, and c. Professionally my greatest debt is to Ed Spiegel, a father of chaos and my thesis Professor, mentor, and friend. 本书隶属于牛津通识读本系列。该书由Ed Spiegel的学生Leonard Smith撰写，语言生动简明。

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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−	== Overview ==	+	== Overview 综述 ==

	Dynamical systems theory and [[chaos theory]] deal with the long-term qualitative behavior of [[dynamical system]]s. 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 system]]s. 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?"