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第3行： − − '''Dynamical systems theory''' is an area of [[mathematics]] used to describe the behavior of the [[complex systems|complex]] [[dynamical system]]s, usually by employing [[differential equations]] or [[difference equations]]. When differential equations are employed, the theory is called [[continuous time|''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 equation]]s of a [[Principle of least action|least action principle]]. When difference equations are employed, the theory is called [[discrete time|''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. − + − '''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。 − − − 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 [[mechanics|mechanical]] or otherwise physical in nature, such as [[planetary orbit]]s and the behaviour of [[electronic circuit]]s, as well as systems that arise in [[biology]], [[economics]], and elsewhere. Much of modern research is focused on the study of [[chaotic system]]s. − − 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，并且研究系统的动力方程的规律，努力求得可能的解。这些系统通常是一些自然领域里的机械系统或其他物理系统，例如行星轨道和电子电路，也包括一些生物学、经济学和其他学科里的系统。现代的研究大多集中在对混沌系统的研究上。 − − − This field of study is also called just ''dynamical systems'', ''mathematical dynamical systems theory'' or the ''mathematical theory of dynamical systems''. − − This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems. − − 这个研究领域也被称为动力系统，数学动力系统理论或动力系统的数学理论。 − − [[Image:Lorenz attractor yb.svg|thumb|240px|right|The [[Lorenz attractor]] is an example of a [[non-linear]] dynamical system. Studying this system helped give rise to [[chaos theory]].]] − − The [[Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.]] − − '''洛伦兹吸引子 Lorenz Attractor'''是非线性动力系统的一个例子。对这个系统的研究产生了混沌理论。 + + + − 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 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?" − − + + − 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'''稳态 Steady States'''是一个重要的目标。不动点或稳态的变量值不会随时间的变化而变化。一些不动点是有吸引力的（attractive），即如果系统的初始值在它的附近，系统最终会收敛到这个不动点。+ − − − − Similarly, one is interested in ''periodic points'', states of the system that repeat after several timesteps. Periodic points can also be attractive. [[Sharkovskii's theorem]] is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. − − Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. − − 人们还对动力系统的'''周期点 Periodic Points'''感兴趣，即系统在重复几个时间步之后的状态。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 system]]s often exhibit seemingly random behavior that has been called ''chaos''.<ref>{{cite journal |last=Grebogi |first=C. |last2=Ott |first2=E. |last3=Yorke |first3=J. |year=1987 |title=Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics |journal=[[Science (journal)|Science]] |volume=238 |issue=4827 |pages=632–638 |jstor=1700479 |doi=10.1126/science.238.4827.632 |pmid=17816542 |bibcode=1987Sci...238..632G }}</ref> The branch of dynamical systems that deals with the clean definition and investigation of chaos is called ''[[chaos theory]]''. − − Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory. − − 即使是简单的非线性动力系统也常常表现出看似随机的行为，这种行为被称为混沌。混沌理论主要研究混沌的清晰定义和混沌的现象，它是动力系统理论的一个分支。 − − == History 历史== − − The concept of dynamical systems theory has its origins in [[Newtonian mechanics]]. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. − − The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. − − 动态系统理论的概念起源于牛顿运动定律。与其他自然科学和工程学科一样，动力系统的演化规律也是通过一种预测系统在未来很短时间内的状态的关系隐含地给出的。 − − − − Before the advent of [[computer|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. − + − − Some excellent presentations of mathematical dynamic system theory include {{harvtxt|Beltrami|1990}}, {{harvtxt|Luenberger|1979}}, {{harvtxt|Padulo|Arbib|1974}}, and {{harvtxt|Strogatz|1994}}.<ref>Jerome R. Busemeyer (2008), [http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html "Dynamic Systems"]. To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008. {{webarchive |url=https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html |date=June 13, 2008 }}</ref> − − Some excellent presentations of mathematical dynamic system theory include , , , and . To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008 − − Some excellent presentations of mathematical dynamic system theory include Beltrami (1990), Luenberger (1979), Padulo & Arbib (1974), and Strogatz (1994).[2] − − − 一些优秀的数学动力系统理论包括贝尔特拉米（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）。
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'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量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。
'''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。
[[File:360px-Lorenz_attractor_yb.svg.png|thumb|240px|right|[[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 ==
== 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。
动力系统理论和'''混沌理论 Chaos Theory'''是用来处理动力系统的长期定性行为的理论。寻找动力系统方程的精确解通常是很难达到的。这两个理论的重点不在于找到精确解，而是回答如下的问题，如“系统长期来看是否会稳定下来，如果可以，那么可能的稳定状态是什么样的?”，或“系统长期的行为是否取决于其初始条件?”
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'''<ref>{{cite journal |last=Grebogi |first=C. |last2=Ott |first2=E. |last3=Yorke |first3=J. |year=1987 |title=Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics |journal=[[Science (journal)|Science]] |volume=238 |issue=4827 |pages=632–638 |jstor=1700479 |doi=10.1126/science.238.4827.632 |pmid=17816542 |bibcode=1987Sci...238..632G }}</ref>。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。
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）<ref>Jerome R. Busemeyer (2008), [http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html "Dynamic Systems"]. To Appear in: ''Encyclopedia of cognitive science'', Macmillan. Retrieved 8 May 2008. {{webarchive |url=https://web.archive.org/web/20080613053119/http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html |date=June 13, 2008 }}</ref>等在该领域做出了杰出的贡献。
== Concepts 概念==
== Concepts 概念==