动力系统理论 Dynamical Systems Theory

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动力系统理论 Dynamical Systems Theory是数学领域中的一部分．主要在描述复杂的动力系统，一般会用微分方程或差分方程来表示。当采用微分方程时，该理论被称为“连续动力系统”，若用差分方程来表示，则称为“离散动力系统”。若其时间只在一些特定区域连续，在其余区域离散，或时间是任意的时间集合（像康托尔集），需要用时标微积分来处理。有时也会需要用混合的算子来处理，像微分差分。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。当时间变量运行在一个某些区间离散、其他区间连续的集合、或者像cantor集一样任意的时间集合上时，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。某些情况下，也可以用混合算子 Mixed Operators来建模，如微分-差分方程。

该理论涉及动力学系统的长期定性行为，研究了通常以机械或物理性质为主的系统（例如行星轨道和行星）的运动方程式的性质以及其常用的的解决方案，电子电路的求解方式以及生物学，经济学等领域产生的系统。许多现代研究集中在混沌系统的研究上。

这个研究领域也被称为动力学系统，数学动力学系统理论或动力学系统数学理论。

Lorenz attractor是一个典型的非线性动态系统。研究这个系统有助于对混沌理论进行发展。

Overview 综述

动力系统理论和混沌理论 Chaos Theory都是用来处理动力系统的长期定性行为的理论。一般而言，很难对动力系统方程进行精确求解，但是对这两个理论的研究重点不在于找到精确解，而是为了解答类似于如下的问题，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?”等。

对给定动力系统的研究的一个重要方向就是求动力系统的不动点或稳态 Steady States。不动点或稳态的的值不会随时间的变化而变化，在不动点的附近，不动点对系统具有收敛性。也就是说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。

动力系统的周期点 Periodic Points也是一个具有前景的研究方向，周期点为系统在重复几个周期后之后的状态。周期点也是具有系统的收敛性，也可称做该点具有吸引力（attactive）的。Sharkovskii定理描述了一维离散动力系统的周期点的个数。

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chaos。动力学系统中涉及混沌的清晰定义和研究的分支称为混沌理论。

历史

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，动力系统的进化规则隐含地由一个关系给出，该关系给出了系统在未来很短时间内的状态。

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。

一些优秀的数学动力系统理论学家包括贝尔特拉米（Beltrami，1990年），龙伯格(Luenberger，1979年），帕杜罗&阿尔比布（Padulo＆Arbib，1974年）和斯托加茨（Strogatz，1994年）等在该领域做出了杰出的贡献。

概念

动力系统

动力系统概念是对描述了一个点的位置在其周围环境中随时间变化的任何“固定”规则的数学形式化。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。

动力系统的状态由实数的集合决定，或更一般地由适当的状态空间中的点集决定。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)——的坐标组。动力系统的演化规律是一种固定的规则，它描述了从当前状态得出的未来状态。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)。

动态主义

动态主义 Dynamicism，也称动态假设，或称认知科学的动态假设或动态认知，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。

非线性系统

在数学中，非线性系统是指不是线性的系统，即不满足叠加原理的系统。从技术上讲，非线性系统是无法解决的变量不能写成独立分量的线性和的任何问题。阿非均匀系统，其是直链距的函数的存在独立变量，是根据一个严格的定义非线性的，但是这样的系统通常被研究沿着线性系统，因为它们可以被转换成一个线性系统，只要一个特定的解决方案是已知的 在数学中，非线性系统 (Nonlinear System)是指系统不是线性的系统，即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，除了它的自变量函数以外，其他部分都是线性的。但非齐次系统通常可当做线性系统进行研究，因为只要知道特定解，它就可以转化为线性系统。

Related fields 相关领域

Arithmetic dynamics 算术动力学

算术动力学 (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的是复平面或实实数轴的自映射的迭代，算术动力学是在反复应用多项式或有理函数的情况下对整数，有理数，p进数(p-adic)和/或代数点的数论性质进行研究。

混沌理论

混沌理论(Chaos theory)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)）。由于这种敏感性，在初始条件下表现为扰动呈指数增长，因此混沌系统的行为似乎是随机的。即使这些系统是确定性的，也会发生这种情况，这意味着它们的未来动力完全由其初始条件定义，而没有涉及随机元素。这种行为称为确定性混乱，或简称为混乱。

复杂系统

复杂系统 (Complex Systems)是研究自然、社会和科学中复杂现象的共同性质的科学领域。它也被称为复杂系统理论、复杂性科学、复杂系统研究和关于复杂性的科学。这些系统的关键问题在于对系统的形式化建模与仿真的困难。因此，复杂系统是根据在不同的研究语境中的不同属性来定义的。

复杂系统的研究为许多科学领域带来了新的活力，在这些领域中，更为典型的简化主义策略已经不足以提供研究动力。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许多不同学科的问题的研究方法。

控制理论

控制理论（Control Theory）是工程和数学的一个交叉学科。控制理论是一个研究如何调整动态系统特性的理论，它也是工程和数学的一个交叉学科，逐渐的应用在许多社会科学中，例如心理学、社会学（社会学中的控制理论）、犯罪学及金融系统(Financial System)。控制理论一般的目的是借由控制器的动作让系统稳定，也就是系统维持在设定值，而且不会在设定值附近晃动。维持设定值保持小范围稳定甚至不变的控制行为称为控制调节，设定值快速变化，对于跟踪速度加速度等的控制要求较高的控制行为称为伺服。控制理论的研究的一部分研究对于动力系统行为的研究产生了深远的影响。

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.

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，研究有不变测度和相关问题的动力系统。它最初的发展受到了统计物理学的推动。 遍历理论是研究保测变换的渐近性态的数学分支。它起源于为统计力学提供基础的"遍历假设"研究，并与动力系统理论、概率论、信息论、泛函分析、数论等数学分支有着密切的联系。

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.

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将其结构特性（例如：网络连接性）与其所产生的全局动力学联系起来。

Projected dynamical systems 投影动力系统

Projected dynamical systems it is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.

Projected dynamical systems it is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation.

投影动力系统 Projected Dynamical Systems是研究解在一个约束集内的动力系统行为的数学理论。这门学科与静态世界中的最优化和平衡问题以及动态世界中的常微分方程都有联系和应用。一个投影动力系统是由投影微分方程的流形 flow给定的。 通过对投影微分方程的流分析，给出了一个投影动力系统的表达式：

[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.

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是通过离散空间对拓扑或平滑动力学系统进行建模的方法，该离散空间由无限的抽象符号序列组成，每个抽象符号对应于系统的一个状态，并且由移位运算符给出动力学（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. 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、存量(stocks)和流量(flows)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.

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 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. 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，动力系统理论在运动科学中崭露头角，成为一种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应用相关的任何主张。

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.

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是(或者就是)(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.

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有序的新状态结构，这个过程被称为“扇贝化 Scalloping”，也就是头脑的复杂表现的不断累积和崩溃的过程。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.

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

Main article: Dynamic approach to 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. 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语言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。

See also 参见

Related subjects

Related subjects

相关科目

• List of dynamical system topics 动力系统主题清单

• Baker's map

• Biological applications of bifurcation theory 分支理论的生物学应用

• Dynamical system (definition) 动力系统（定义）

• Embodied Embedded Cognition 具身认知

• Fibonacci numbers 斐波那契数

• Fractals 分形

• Gingerbreadman map

• Halo orbit Halo 轨道

• List of types of systems theory 动力系统类型清单

• Oscillation 振动性

• Postcognitivism

• Recurrent neural network 递归神经网络

• Combinatorics and dynamical systems 组合数学和动力系统

• Synergetics 协同学(Haken)

• Systemography

Related scientists

Related scientists

相关科学家

• People in systems and control

• Dmitri Anosov

• Vladimir Arnold

• Nikolay Bogolyubov

• Andrey Kolmogorov

• Nikolay Krylov

• Jürgen Moser

• Yakov G. Sinai

• Stephen Smale

• Hillel Furstenberg

• Grigory Margulis

• Elon Lindenstrauss

Further reading 拓展阅读

• 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 外部链接

• Dynamic Systems 动力系统 Encyclopedia of Cognitive Science entry. 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

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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