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第1行： − 此词条暂由彩云小译翻译，翻译字数共3900，未经人工整理和审校，带来阅读不便，请见谅。+ − − {{short description|Mathematical model which describes the time dependence of a point in a geometrical space}} − − {{about|the general aspects of dynamical systems|technical details|Dynamical system (definition)|the study|Dynamical systems theory}} − − {{Redirect|Dynamical}} − − [[File:Lorenz attractor yb.svg|thumb|right|The [[Lorenz attractor]] arises in the study of the [[Lorenz system|Lorenz oscillator]], a dynamical system.]] − − The [[Lorenz attractor arises in the study of the Lorenz oscillator, a dynamical system.]] − − [洛伦兹吸引子出现在洛伦兹振子的研究中，一个动力系统。] − − − − In [[mathematics]], a '''dynamical system''' is a system in which a [[Function (mathematics)|function]] describes the [[time]] dependence of a [[Point (geometry)|point]] in a [[Manifold|geometrical space]]. Examples include the [[mathematical model]]s that describe the swinging of a clock [[pendulum]], [[fluid dynamics|the flow of water in a pipe]], and [[population dynamics|the number of fish each springtime in a lake]]. − − In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical 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 springtime in a lake. − − 在数学中，动力系统是一个函数描述几何空间中一个点的时间依赖性的系统。这些例子包括描述钟摆摆动的数学模型、管道中的水流以及每个春季湖中的鱼的数量。 − − − − At any given time, a dynamical system has a [[State (controls)|state]] given by a [[tuple]] of [[real numbers]] (a [[vector space|vector]]) that can be represented by a point in an appropriate [[state space]] (a geometrical [[manifold]]). The ''evolution rule'' of the dynamical system is a function that describes what future states follow from the current state. Often the function is [[Deterministic system (mathematics)|deterministic]], that is, for a given time interval only one future state follows from the current state.<ref>{{cite book |last=Strogatz |first=S. H. |year=2001 |title=Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry |location= |publisher=Perseus }}</ref><ref>{{cite book |first1=A. |last1=Katok |first2=B. |last2=Hasselblatt |title=Introduction to the Modern Theory of Dynamical Systems |location=Cambridge |publisher=Cambridge University Press |year=1995 |isbn=978-0-521-34187-5 |url-access=registration |url=https://archive.org/details/introductiontomo0000kato }}</ref> However, some systems are [[stochastic system|stochastic]], in that random events also affect the evolution of the state variables. − − At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables. − − 在任何给定的时间，一个动力系统都有一个由一组实数(一个向量)给出的状态，这些实数可以用一个适当的状态空间(一个几何流形)中的一个点来表示。动力系统的演化规则是一个描述当前状态下未来状态的函数。通常函数是确定的，也就是说，在给定的时间间隔内，当前状态只跟随一个未来状态。然而，有些系统是随机的，因为随机事件也影响状态变量的演化。 − − − − In [[physics]], a '''dynamical system''' is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives."<ref>{{cite web|title=Nature|url=http://www.nature.com/subjects/dynamical-systems|publisher=Springer Nature|accessdate= 17 February 2017}}</ref> In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. − − In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives." In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. − − 在物理学中，动力系统被描述为“状态随时间变化的粒子或粒子集合，因此遵循包含时间导数的微分方程。”为了对系统的未来行为做出预测，这些方程的解析解或者它们随着时间的推移通过计算机模拟来积分是可以实现的。 − − − − The study of dynamical systems is the focus of [[dynamical systems theory]], which has applications to a wide variety of fields such as mathematics, physics,<ref>{{cite journal|last1=Melby|first1=P. |display-authors=etal |title=Dynamics of Self-Adjusting Systems With Noise|journal= Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=15 |issue=3 |pages=033902 |date=2005|doi=10.1063/1.1953147|pmid=16252993 |bibcode=2005Chaos..15c3902M}}</ref><ref>{{cite journal|last1=Gintautas|first1=V. |display-authors=etal |title=Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics|journal=J. Stat. Phys. |volume=130|date=2008|doi=10.1007/s10955-007-9444-4|arxiv=0705.0311|bibcode=2008JSP...130..617G|s2cid=8677631 }}</ref> [[biology]],<ref>{{cite book |last1=Jackson |first1=T. |last2=Radunskaya |first2=A. |title=Applications of Dynamical Systems in Biology and Medicine |date=2015 |publisher=Springer }}</ref> [[chemistry]], [[engineering]],<ref>{{cite book |first=Erwin |last=Kreyszig |title=Advanced Engineering Mathematics |location=Hoboken |publisher=Wiley |year=2011 |isbn=978-0-470-64613-7 }}</ref> [[economics]],<ref>{{cite book |last=Gandolfo |first=Giancarlo |authorlink=Giancarlo Gandolfo |title=Economic Dynamics: Methods and Models |location=Berlin |publisher=Springer |edition=Fourth |year=2009 |origyear=1971 |isbn=978-3-642-13503-3 }}</ref> [[Cliodynamics|history]], and [[medicine]]. Dynamical systems are a fundamental part of [[chaos theory]], [[logistic map]] dynamics, [[bifurcation theory]], the [[self-assembly]] and [[self-organization]] processes, and the [[edge of chaos]] concept. − − The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept. − − 动力系统的研究是动态系统理论的重点，它应用于各种领域，如数学、物理、生物、化学、工程、经济、历史和医学。动力系统是混沌理论、逻辑映射动力学、分岔理论、自组装和自我组织过程以及混沌概念边缘的基本组成部分。 − − − − ==Overview== − − The concept of a dynamical system has its origins in [[Newtonian mechanics]]. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a [[differential equation]], [[Recurrence relation|difference equation]] or other [[Time scale calculus|time scale]].) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as ''solving the system'' or ''integrating the system''. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a ''[[trajectory]]'' or ''[[orbit (dynamics)|orbit]]''. − − The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a trajectory or orbit. − − 动力系统的概念起源于20世纪牛顿运动定律。正如其他自然科学和工程学科一样，动力系统的演化规律是一种隐含的关系，它只在很短的时间内给出系统的状态。(这种关系要么是微分方程，要么是差分方程，要么是其他时间尺度。)为了确定所有未来时间的状态，需要多次迭代该关系——每次迭代一小步。迭代过程称为系统求解或系统集成。如果这个系统能够被解决，给定一个初始点，它就有可能确定它未来的所有位置，一组被称为轨道或轨道的点。 − − − − Before the advent of [[computers]], finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. − − Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. − − 在计算机出现之前，找到一个轨道需要复杂的数学技术，而且只能为一小类动力系统完成。在电子计算机上实现的数值方法简化了确定动力系统轨道的任务。 − − − − For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: − − For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: − − 对于简单的动力系统，知道轨迹通常是足够的，但是大多数动力系统太复杂，以至于无法用单个轨迹来理解。出现这些困难的原因是: − − * The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as [[Lyapunov stability]] or [[structural stability]]. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their [[Equivalence relation|equivalence]] changes with the different notions of stability. − − * The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. [[Linear dynamical system]]s and [[Poincaré–Bendixson theorem|systems that have two numbers describing a state]] are examples of dynamical systems where the possible classes of orbits are understood. − − * The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have [[bifurcation theory|bifurcation points]] where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the [[Turbulence|transition to turbulence of a fluid]]. − − * The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for [[ergodic theory|ergodic systems]] and a more detailed understanding has been worked out for [[Anosov diffeomorphism|hyperbolic systems]]. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of [[statistical mechanics]] and of [[chaos theory|chaos]]. − − − − ==History== − − − − Many people regard French mathematician [[Henri Poincaré]] as the founder of dynamical systems.<ref>Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"." ''Physics Reports'' 193.3 (1990): 137-163.</ref> Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the [[Poincaré recurrence theorem]], which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. − − Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. − − 许多人认为法国数学家亨利 · 庞加莱是动力学系统的创始人。庞加莱出版了两本经典专著《天体力学》(1892-1899)和《天体力学》(1905-1910)。其中，他成功地将他们的研究成果应用于三体运动问题，并详细研究了解的行为(频率、稳定性、渐近性等)。这些论文包括《庞加莱始态复现定理，其中指出，某些系统将在一个足够长但有限的时间后，回到一个非常接近初始状态。 − − − − [[Aleksandr Lyapunov]] developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. − − Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. − − 亚历山大·李亚普诺夫发展了许多重要的近似方法。他在1899年发展的方法使得定义常微分方程组的稳定性成为可能。他创立了动力系统稳定性的现代理论。 − − − − In 1913, [[George David Birkhoff]] proved Poincaré's "[[Poincaré–Birkhoff theorem|Last Geometric Theorem]]", a special case of the [[three-body problem]], a result that made him world-famous. In 1927, he published his ''[http://www.ams.org/online_bks/coll9/ Dynamical Systems]''. Birkhoff's most durable result has been his 1931 discovery of what is now called the [[ergodic theorem]]. Combining insights from [[physics]] on the [[ergodic hypothesis]] with [[measure theory]], this theorem solved, at least in principle, a fundamental problem of [[statistical mechanics]]. The ergodic theorem has also had repercussions for dynamics. − − In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his [http://www.ams.org/online_bks/coll9/ Dynamical Systems]. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics. − − 1913年，乔治·戴维·伯克霍夫证明了庞加莱的“最后几何定理” ，这是三体的一个特例，这个结果使他闻名世界。1927年，他出版了《 [http://www.ams.org/online_bks/coll9/ 动力系统]》。伯克霍夫最持久的成果是他在1931年发现的遍历定理。这个定理结合了遍历假设物理学和测量理论的见解，至少在原则上解决了统计力学的一个基本问题。遍历定理也对动力学产生了影响。 − − − − [[Stephen Smale]] made significant advances as well. His first contribution was the [[Horseshoe map|Smale horseshoe]] that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. − − Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. − − 斯蒂芬 · 斯梅尔也取得了重大进展。他的第一个贡献是斯梅尔马蹄铁，开启了动力系统的重要研究。他还概述了许多其他人开展的一个研究项目。 − − − − [[Oleksandr Mykolaiovych Sharkovsky]] developed [[Sharkovsky's theorem]] on the periods of [[discrete dynamical system]]s in 1964. One of the implications of the theorem is that if a discrete dynamical system on the [[real line]] has a [[periodic point]] of period 3, then it must have periodic points of every other period. − − Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. − − 在1964年，Oleksandr mykolaivych Sharkovsky 发展了 Sharkovsky 关于离散动力系统周期的定理。这个定理的一个含义是，如果实线上的一个离散动力系统有一个周期为3的周期点，那么它必须有每隔一个周期的周期点。 − − − − In the late 20th century, Palestinian mechanical engineer [[Ali H. Nayfeh]] applied [[nonlinear dynamics]] in [[mechanics|mechanical]] and [[engineering]] systems.<ref name="Rega">{{cite book |last1=Rega |first1=Giuseppe |chapter=Tribute to Ali H. Nayfeh (1933-2017) |title=IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems |date=2019 |publisher=[[Springer Science+Business Media|Springer]] |isbn=9783030236922 |url=https://books.google.com/books?id=pAilDwAAQBAJ&pg=PA1 |pages=1–2}}</ref> His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of [[machines]] and [[structures]] that are common in daily life, such as [[ships]], [[crane (machine)|cranes]], [[bridges]], [[buildings]], [[skyscrapers]], [[jet engines]], [[rocket engines]], [[aircraft]] and [[spacecraft]].<ref name="fi">{{cite web |title=Ali Hasan Nayfeh |url=https://www.fi.edu/laureates/ali-hasan-nayfeh |website=[[Franklin Institute Awards]] |publisher=[[The Franklin Institute]] |accessdate=25 August 2019 |date=4 February 2014}}</ref> − − In the late 20th century, Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft. − − 在20世纪后期，巴勒斯坦机械工程师 Ali h. Nayfeh 将非线性动力学应用于机械和工程系统。他在应用非线性动力学方面的开创性工作对日常生活中常见的机器和结构的建造和维护产生了影响，如船舶、起重机、桥梁、建筑物、摩天大楼、喷气发动机、火箭发动机、飞机和航天器。 − − ==Basic definitions== − − {{Main|Dynamical system (definition)}} − − A dynamical system is a [[manifold]] ''M'' called the phase (or state) space endowed with a family of smooth evolution functions Φ<sup>''t''</sup> that for any element ''t'' ∈ ''T'', the time, map a point of the [[phase space]] back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set ''T''. When ''T'' is taken to be the reals, the dynamical system is called a ''[[Flow (mathematics)|flow]]''; and if ''T'' is restricted to the non-negative reals, then the dynamical system is a ''semi-flow''. When ''T'' is taken to be the integers, it is a ''cascade'' or a ''map''; and the restriction to the non-negative integers is a ''semi-cascade''. − − A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions Φ<sup>t</sup> that for any element t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade. − − 动力系统是一个称为相空间的流形，它具有一族光滑演化函数 φ < sup > t </sup > ，对于任意元素 t ∈ t，时间将相空间的一点映射回相空间。光滑度的概念随着应用和流形类型的改变而改变。集合 t 有几种选择。当 t 被看作是实数时，动力系统被称为流; 如果 t 被限制为非负实数时，动力系统就是半实数流。当 t 被视为整数时，它是级联或映射; 对非负整数的限制是半级联。 − − − − Note: There is a further technical condition that Φ<sup>''t''</sup> is an action of ''T'' on ''M''. That includes the facts that Φ<sup>''0''</sup> is the identity function and that Φ<sup>''s+t''</sup> is the composition of Φ<sup>''s''</sup> and Φ<sup>''t''</sup>. This is a [[semigroup action]], which doesn't require the existence of negative values for ''t'', and doesn't require the functions Φ<sup>''t''</sup> to be invertible. − − Note: There is a further technical condition that Φ<sup>t</sup> is an action of T on M. That includes the facts that Φ<sup>0</sup> is the identity function and that Φ<sup>s+t</sup> is the composition of Φ<sup>s</sup> and Φ<sup>t</sup>. This is a semigroup action, which doesn't require the existence of negative values for t, and doesn't require the functions Φ<sup>t</sup> to be invertible. − − 注: 还有一个技术条件，即 φ < sup > t </sup > 是 t 对 m 的作用。其中，φ < sup > 0 </sup > 是同一函数，φ < sup > s + t </sup > 是 φ < sup > s </sup > 和 φ < sup > t </sup > 的合成。这是一个半群作用，它不要求 t 的负值存在，也不要求 φ < sup > t </sup > 可逆。 − − − − === Examples === − − The evolution function Φ<sup> ''t''</sup> is often the solution of a ''differential equation of motion'' − − The evolution function Φ<sup> t</sup> is often the solution of a differential equation of motion − − 进化函数 φ < sup > t </sup > 通常是运动微分方程的解 − − − − : <math> \dot{x} = v(x). </math> − − <math> \dot{x} = v(x). </math> − − < math > dot { x } = v (x).数学 − − − − The equation gives the time derivative, represented by the dot, of a trajectory ''x''(''t'') on the phase space starting at some point ''x''<sub>0</sub>. The [[vector field]] ''v''(''x'') is a smooth function that at every point of the phase space ''M'' provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space ''M'', but in the [[tangent space]] ''T<sub>x</sub>M'' of the point ''x''.) Given a smooth Φ<sup> ''t''</sup>, an autonomous vector field can be derived from it. − − The equation gives the time derivative, represented by the dot, of a trajectory x(t) on the phase space starting at some point x<sub>0</sub>. The vector field v(x) is a smooth function that at every point of the phase space M provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space M, but in the tangent space T<sub>x</sub>M of the point x.) Given a smooth Φ<sup> t</sup>, an autonomous vector field can be derived from it. − − 该方程给出了从某点 x < sub > 0 </sub > 开始的相空间中的轨迹 x (t)的时间导数，用点表示。向量场 v (x)是一个光滑函数，它在相空间 m 的每一点上都提供了动力系统的速度矢量。(这些向量不是相空间 m 中的向量，而是点 x 的切空间 t < sub > x </sub > m 中的向量)给定一个光滑的 φ < sup > t </sup > ，可以由它导出一个自治向量场。 − − − − There is no need for higher order derivatives in the equation, nor for time dependence in ''v''(''x'') because these can be eliminated by considering systems of higher dimensions. Other types of [[differential equations]] can be used to define the evolution rule: − − There is no need for higher order derivatives in the equation, nor for time dependence in v(x) because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule: − − 在方程中不需要高阶导数，在 v (x)中也不需要时间依赖性，因为这些可以通过考虑高维系统而消除。其他类型的微分方程可以用来定义演化规则: − − − − : <math> G(x, \dot{x}) = 0 </math> − − <math> G(x, \dot{x}) = 0 </math> − − < math > g (x，dot { x }) = 0 </math > − − − − is an example of an equation that arises from the modeling of mechanical systems with complicated constraints. − − is an example of an equation that arises from the modeling of mechanical systems with complicated constraints. − − 是一个例子的方程式，从建模的机械系统与复杂的约束。 − − − − The differential equations determining the evolution function Φ<sup> ''t''</sup> are often [[ordinary differential equation]]s; in this case the phase space ''M'' is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally [[Banach space]]s—in which case the differential equations are [[partial differential equation]]s. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. − − The differential equations determining the evolution function Φ<sup> t</sup> are often ordinary differential equations; in this case the phase space M is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. − − 确定发展函数 φ < sup > t </sup > 的微分方程通常是常微分方程，在这种情况下，相空间 m 是一个有限维流形。动力系统中的许多概念可以推广到无限维流形——那些局部的 Banach 空间——在这种情况下，微分方程是偏微分方程。20世纪后期，动力系统的偏微分方程观点开始流行。 − − − − === Further examples === − − {{Div col|colwidth=25em}} − − * [[Arnold's cat map]] − − * [[Baker's map]] is an example of a chaotic [[piecewise linear function|piecewise linear]] map − − * [[Dynamical billiards|Billiards]] and [[Dynamical outer billiards|outer billiards]] − − * [[Bouncing ball dynamics]] − − * [[Circle map]] − − * [[Complex quadratic polynomial]] − − * [[Double pendulum]] − − * [[Dyadic transformation]] − − * [[Hénon map]] − − * [[Irrational rotation]] − − * [[Kaplan–Yorke map]] − − * [[List of chaotic maps]] − − * [[Lorenz attractor|Lorenz system]] − − * [[Complex quadratic polynomial#Map|Quadratic map simulation system]] − − * [[Rössler map]] − − * [[Swinging Atwood's machine]] − − * [[Tent map]] − − {{Div col end}} − − − − − − ==Linear dynamical systems== − − {{Main|Linear dynamical system}} − − Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the ''N''-dimensional Euclidean space, so any point in phase space can be represented by a vector with ''N'' numbers. The analysis of linear systems is possible because they satisfy a [[superposition principle]]: if ''u''(''t'') and ''w''(''t'') satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will ''u''(''t'') + ''w''(''t''). − − Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t). − − 线性动力系统可以用简单的函数来求解，并且可以对所有轨道的行为进行分类。在线性系统中，相空间是 n 维欧氏空间，因此相空间中的任何点都可以用 n 个数字的向量来表示。线性系统的分析是可能的，因为它们满足一个叠加原理: 如果 u (t)和 w (t)满足向量场的微分方程(但不一定满足初始条件) ，那么 u (t) + w (t)也满足。 − − − − ===Flows=== − − For a [[flow (mathematics)|flow]], the vector field v(''x'') is an [[affine transformation|affine]] function of the position in the phase space, that is, − − For a flow, the vector field v(x) is an affine function of the position in the phase space, that is, − − 对于流动，向量场 v (x)是相空间位置的仿射函数，即, − − :<math> \dot{x} = v(x) = A x + b,</math> − − <math> \dot{x} = v(x) = A x + b,</math> − − (数学)点{ x } = v (x) = a x + b，(数学) − − with ''A'' a matrix, ''b'' a vector of numbers and ''x'' the position vector. The solution to this system can be found by using the superposition principle (linearity). − − with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). − − 一个矩阵，一个数的向量，x 是位置向量。该系统的解决方案可以通过使用叠加原理(线性)来找到。 − − The case ''b'' ≠ 0 with ''A'' = 0 is just a straight line in the direction of ''b'': − − The case b ≠ 0 with A = 0 is just a straight line in the direction of b: − − A = 0的情况下，b ≠0是 b 方向的一条直线: − − − − : <math>\Phi^t(x_1) = x_1 + b t. </math> − − <math>\Phi^t(x_1) = x_1 + b t. </math> − − < math > Phi ^ t (x _ 1) = x1 + b t </math > − − − − When ''b'' is zero and ''A'' ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if ''x''<sub>0</sub> = 0, then the orbit remains there. − − When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x<sub>0</sub> = 0, then the orbit remains there. − − 当 b 为零且 a ≠0时，原点是流动的平衡点(或奇异点) ，即如果 x < sub > 0 </sub > = 0，轨道保持不变。 − − For other initial conditions, the equation of motion is given by the [[matrix exponential|exponential of a matrix]]: for an initial point ''x''<sub>0</sub>, − − For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x<sub>0</sub>, − − 对于其他初始条件，运动方程是由矩阵的指数给出的: 对于初始点 x < sub > 0 </sub > , − − − − : <math>\Phi^t(x_0) = e^{t A} x_0. </math> − − <math>\Phi^t(x_0) = e^{t A} x_0. </math> − − < math > Phi ^ t (x _ 0) = e ^ { t a } x _ 0.数学 − − − − When ''b'' = 0, the [[eigenvalue]]s of ''A'' determine the structure of the phase space. From the eigenvalues and the [[eigenvector]]s of ''A'' it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. − − When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin. − − 当 b = 0时，a 的本征值决定了相空间的结构。从 a 的特征值和特征向量可以确定一个初始点是收敛还是发散到原点的平衡点。 − − − − The distance between two different initial conditions in the case ''A'' ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for [[chaos theory|chaotic behavior]]. − − The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior. − − 在大多数情况下，两个不同初始条件之间的距离在 a ≠0时将呈指数变化，要么指数快速收敛到一个点，要么指数快速发散。在发散的情况下，线性系统表现出对初始条件的敏感依赖性。对于非线性系统，这是混沌行为的(必要但不充分)条件之一。 − − − − [[File:LinearFields.png|thumb|500px|center|Linear vector fields and a few trajectories.]] − − Linear vector fields and a few trajectories. − − 线性向量场和一些轨迹。 − − {{-}} − − − − ===Maps=== − − A [[Discrete-time dynamical system|discrete-time]], [[Affine transformation|affine]] dynamical system has the form of a [[matrix difference equation]]: − − A discrete-time, affine dynamical system has the form of a matrix difference equation: − − 一个离散的仿射动力系统具有矩阵差分方程的形式: − − : <math> x_{n+1} = A x_n + b, </math> − − <math> x_{n+1} = A x_n + b, </math> − − [ math > x { n + 1} = a x _ n + b，</math > − − with ''A'' a matrix and ''b'' a vector. As in the continuous case, the change of coordinates ''x'' → ''x'' + (1 − ''A'')<sup> –1</sup>''b'' removes the term ''b'' from the equation. In the new [[coordinate system]], the origin is a fixed point of the map and the solutions are of the linear system ''A''<sup> ''n''</sup>''x''<sub>0</sub>. − − with A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (1 − A)<sup> –1</sup>b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system A<sup> n</sup>x<sub>0</sub>. − − A 是矩阵，b 是向量。和连续情况一样，坐标 x → x + (1-a) < sup > -- 1 </sup > b 的变化使方程中的 b 项消失。在新的坐标系中，原点是映射的一个不动点，解是线性系统 a < sup > n </sup > x < sub > 0 </sub > 。 − − The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. − − The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. − − 映射的解不再是曲线，而是在相空间中跳跃的点。轨道被组织成曲线，或者说纤维，它们是在地图的作用下映射到它们自己的点的集合。 − − − − As in the continuous case, the eigenvalues and eigenvectors of ''A'' determine the structure of phase space. For example, if ''u''<sub>1</sub> is an eigenvector of ''A'', with a real eigenvalue smaller than one, then the straight lines given by the points along ''α'' ''u''<sub>1</sub>, with ''α'' ∈ '''R''', is an invariant curve of the map. Points in this straight line run into the fixed point. − − As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For example, if u<sub>1</sub> is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u<sub>1</sub>, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point. − − 在连续的情况下，a 的特征矢量决定了相空间的结构。例如，如果 u < sub > 1 </sub > 是 a 的实特征值小于1的特征向量，那么沿着 α u < sub > 1 </sub > 的点给出的直线，α ∈ r，就是映射的不变曲线。点在这条直线上运行到定点。 − − − − There are also many [[List of chaotic maps|other discrete dynamical systems]]. − − There are also many other discrete dynamical systems. − − 还有许多其他的离散动力系统。 − − − − ==Local dynamics== − − The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a ''singular point'' of the vector field (a point where ''v''(''x'') = 0) will remain a singular point under smooth transformations; a ''periodic orbit'' is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible. − − The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible. − − 动力系统的定性性质在平滑坐标变化下不发生变化(这有时被认为是定性的定义) : 向量场的奇点(v (x) = 0)在平滑变换下仍然是奇点; 周期轨道是相空间中的一个回路，相空间的平滑变形不能改变它是一个回路。正是在奇异点和周期轨道的附近，才能很好地理解动力系统的相空间结构。在动力系统的定性研究中，这种方法是为了表明坐标的变化(通常是不确定的，但是可计算的)使得动力系统变化尽可能简单。 − − − − ===Rectification=== − − A flow in most small patches of the phase space can be made very simple. If ''y'' is a point where the vector field ''v''(''y'') ≠ 0, then there is a change of coordinates for a region around ''y'' where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem. − − A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem. − − 在相空间的大多数小块中的流动可以变得非常简单。如果 y 是向量场 v (y)≠0的一点，那么在 y 周围的一个区域的坐标会发生变化，在这个区域中，向量场变成了一系列相同大小的平行向量。这就是所谓的整流定理。 − − − − The ''rectification theorem'' says that away from [[Mathematical singularity|singular points]] the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space ''M'' the dynamical system is ''integrable''. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where ''v''(''x'') = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. − − The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. − − 校正定理说，离开奇异点，一个小块中的一个点的动力学是一条直线。有时可以通过将几个补丁缝合在一起来扩大补丁，当这在整个相空间 m 中得到解决时，补丁动力系统是可积的。在大多数情况下，补丁不能扩展到整个相空间。在向量场中可能存在奇异点(其中 v (x) = 0) ，或者随着某个点的逼近，曲面可能变得越来越小。更微妙的原因是一个全局约束，它的轨迹从一个补丁开始，在访问了一系列其他补丁之后返回到最初的那个补丁。如果下一次轨道以不同的方式环绕相空间，则不可能对整个系列的矢量场进行校正。 − − − − ===Near periodic orbits=== − − In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point ''x''<sub>0</sub> in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to ''v''(''x''<sub>0</sub>). These points are a [[Poincaré section]] ''S''(''γ'', ''x''<sub>0</sub>), of the orbit. The flow now defines a map, the [[Poincaré map]] ''F'' : ''S'' → ''S'', for points starting in ''S'' and returning to ''S''. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes ''x''<sub>0</sub>. − − In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x<sub>0</sub> in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to v(x<sub>0</sub>). These points are a Poincaré section S(γ, x<sub>0</sub>), of the orbit. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and returning to S. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes x<sub>0</sub>. − − 一般来说，在周期轨道附近不能使用整流定理。庞加莱开发了一种方法，将周期轨道附近的分析转化为对地图的分析。在轨道 γ 上选择一个 x < sub > 0 </sub > 的点，考虑相空间中垂直于 v (x < sub > 0 </sub >)的点。这些点是轨道的 poincaré 截面 s (γ，x < sub > 0 </sub >)。这个流现在定义了一个地图，Poincaré 地图 f: s → s，用于从 s 开始返回 s 的点。并不是所有这些点都需要同样的时间返回，但是时间将接近 x < sub > 0 </sub > 。 − − − − The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map ''F''. By a translation, the point can be assumed to be at ''x'' = 0. The Taylor series of the map is ''F''(''x'') = ''J'' · ''x'' + O(''x''<sup>2</sup>), so a change of coordinates ''h'' can only be expected to simplify ''F'' to its linear part − − The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x<sup>2</sup>), so a change of coordinates h can only be expected to simplify F to its linear part − − 周期轨道与 Poincaré 截面的交点是 poincaré 映射 f 的一个不动点。该映射的泰勒级数为 f (x) = j x + o (x < sup > 2 </sup >) ，因此坐标 h 的变化只能被期望将 f 简化为它的线性部分 − − − − : <math> h^{-1} \circ F \circ h(x) = J \cdot x.</math> − − <math> h^{-1} \circ F \circ h(x) = J \cdot x.</math> − − = j cdot x − − − − This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If ''λ''<sub>1</sub>, ..., ''λ''<sub>''ν''</sub> are the eigenvalues of ''J'' they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form ''λ''<sub>''i''</sub> – ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function ''h'', the non-resonant condition is also known as the small divisor problem. − − This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ<sub>1</sub>, ..., λ<sub>ν</sub> are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λ<sub>i</sub> – ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem. − − 这就是共轭方程。寻找该方程成立的条件一直是动力系统研究的主要任务之一。庞加莱首先假定所有函数都是解析函数，在此过程中发现了非共振条件。如果 j 的特征值是 λ < sub > 1 </sub > ，... ，λ < sub > ν </sub > ，则 j 的特征值是共振的，如果其中一个特征值是两个或两个以上其他特征线性组合的整数。当函数 h 的项的分母出现 λ < sub > i </sub > -- ∑(其它特征值的倍数)时，非共振条件也称为小除数问题。 − − − − ===Conjugation results=== − − The results on the existence of a solution to the conjugation equation depend on the eigenvalues of ''J'' and the degree of smoothness required from ''h''. As ''J'' does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of ''J'' are not in the unit circle, the dynamics near the fixed point ''x''<sub>0</sub> of ''F'' is called ''[[Hyperbolic fixed point|hyperbolic]]'' and when the eigenvalues are on the unit circle and complex, the dynamics is called ''elliptic''. − − The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x<sub>0</sub> of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic. − − 共轭方程解的存在性依赖于 j 的特征值和 h 所需的光滑度。由于 j 不需要有任何特殊的对称性，它的特征值通常是复数。当 j 的特征值不在单位圆内时，f 的不动点 x < sub > 0 </sub > 附近的动力学称为双曲线，当 j 的特征值在单位圆和复数上时，动力学称为椭圆。 − − − − In the hyperbolic case, the [[Hartman–Grobman theorem]] gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map ''J'' · ''x''. The hyperbolic case is also ''structurally stable''. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of ''J'' in the complex plane, implying that the map is still hyperbolic. − − In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic. − − 在双曲情形下，Hartman-Grobman 定理给出了一个连续函数存在的条件，这个连续函数将映射不动点的邻域映射到线性映射 jx。双曲线情况在结构上也是稳定的。向量场的微小变化只会在 Poincaré 地图上产生微小的变化，而这些微小的变化将反映在 j 的特征值在复杂平面上的位置的微小变化中，这意味着该地图仍然是双曲线的。 − − − − The [[Kolmogorov–Arnold–Moser theorem|Kolmogorov–Arnold–Moser (KAM)]] theorem gives the behavior near an elliptic point. − − The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point. − − Kolmogorov-Arnold-Moser (KAM)定理给出了椭圆点附近的行为。 − − − − ==Bifurcation theory== − − {{Main|Bifurcation theory}} − − When the evolution map Φ<sup>''t''</sup> (or the [[vector field]] it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the [[phase space]] until a special value ''μ''<sub>0</sub> is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. − − When the evolution map Φ<sup>t</sup> (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value μ<sub>0</sub> is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. − − 当演化映射 φ < sup > t </sup > (或它所导出的向量场)依赖于参数 μ 时，相空间的结构也依赖于这个参数。在达到一个特殊值 μ < sub > 0 </sub > 之前，相空间中的微小变化可能不会产生质的变化。在这一点上，相空间发生了质的变化，据说动力系统已经经历了一个分岔。 − − − − Bifurcation theory considers a structure in phase space (typically a [[Fixed point (mathematics)|fixed point]], a periodic orbit, or an invariant [[torus]]) and studies its behavior as a function of the parameter ''μ''. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems. − − Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems. − − 分岔理论考虑相空间中的结构(通常是固定点、周期轨道或不变环面) ，并将其行为作为参数 μ 的函数进行研究。在分叉点，结构可能改变其稳定性，分裂成新的结构，或与其他结构合并。通过使用泰勒级数近似的地图和理解的差异，可以消除坐标的变化，它是可能的目录分岔的动力系统。 − − − − The bifurcations of a hyperbolic fixed point ''x''<sub>0</sub> of a system family ''F<sub>μ</sub>'' can be characterized by the [[eigenvalues]] of the first derivative of the system ''DF''<sub>''μ''</sub>(''x''<sub>0</sub>) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of ''DF<sub>μ</sub>'' on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on [[Bifurcation theory]]. − − The bifurcations of a hyperbolic fixed point x<sub>0</sub> of a system family F<sub>μ</sub> can be characterized by the eigenvalues of the first derivative of the system DF<sub>μ</sub>(x<sub>0</sub>) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of DF<sub>μ</sub> on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory. − − 双曲型不动点 x < sub > 0 </sub > 系统 f < sub > μ </sub > 的分岔拥有属性可能是在分岔点计算得到的 DF < sub > μ </sub > (x < sub > 0 </sub >)系统的一阶导数的特征值。对于映射，当单位圆上存在 DF < sub > μ </sub > 的特征值时，将发生分叉。对于一个流，当在虚轴上有特征值时，就会发生这种情况。欲了解更多信息，请参阅分岔理论的主要文章。 − − − − Some bifurcations can lead to very complicated structures in phase space. For example, the [[Ruelle–Takens scenario]] describes how a periodic orbit bifurcates into a torus and the torus into a [[strange attractor]]. In another example, [[Bifurcation diagram|Feigenbaum period-doubling]] describes how a stable periodic orbit goes through a series of [[period-doubling bifurcation]]s. − − Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations. − − 一些分叉会导致相空间中非常复杂的结构。例如，Ruelle-Takens 假设描述了一个周期轨道如何分叉为一个环面和一个环面，形成一个奇怪的吸引子。在另一个例子中，Feigenbaum 倍周期描述了稳定的周期轨道是如何通过一系列倍周期分岔的。 − − − − ==Ergodic systems== − − {{Main|Ergodic theory}} − − In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset ''A'' into the points Φ<sup> ''t''</sup>(''A'') and invariance of the phase space means that − − In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ<sup> t</sup>(A) and invariance of the phase space means that − − 在许多动力学系统中，可以选择系统的坐标，使得相空间中的体积(实际上是一个 v 维体积)是不变的。对于从牛顿定律推导出来的机械系统，只要坐标是位置、动量和体积的单位(位置) × (动量) ，就会发生这种情况。流取子集 a 的点为 φ < sup > t </sup > (a) ，相空间的不变性意味着 − − : <math> \mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ). </math> − − <math> \mathrm{vol} (A) = \mathrm{vol} ( \Phi^t(A) ). </math> − − (a) = mathrm { vol }(Phi ^ t (a)).数学 − − In the [[Hamiltonian mechanics|Hamiltonian formalism]], given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the [[Liouville's theorem (Hamiltonian)|Liouville measure]]. − − In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure. − − 在哈密顿形式中，给定一个坐标系，就有可能导出适当的(广义的)动量，使得相关的体积被流保留下来。体积被称为由刘维尔量度计算。 − − − − In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. − − In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. − − 在哈密顿系统中，并非所有可能的位置和动量构型都能从初始条件得到。由于能量守恒，只有与初始条件能量相同的状态才可以得到。具有相同能量的状态构成能量壳 ω，是相空间的子流形。用刘维测量法计算的能量壳的体积在进化过程中保持不变。 − − − − For systems where the volume is preserved by the flow, Poincaré discovered the [[Poincaré recurrence theorem|recurrence theorem]]: Assume the phase space has a finite Liouville volume and let ''F'' be a phase space volume-preserving map and ''A'' a subset of the phase space. Then almost every point of ''A'' returns to ''A'' infinitely often. The Poincaré recurrence theorem was used by [[Ernst Zermelo|Zermelo]] to object to [[Ludwig Boltzmann|Boltzmann]]'s derivation of the increase in entropy in a dynamical system of colliding atoms. − − For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms. − − 对于体积被流保持的系统，poincaré 发现了递归定理: 假设相空间有一个有限的 Liouville 体积，f 是相空间保持体积的映射，a 是相空间的子集。然后 a 的几乎每一个点经常返回到 a。波尔兹曼推导出了在原子碰撞的一个庞加莱始态复现定理动力系统中熵增加的现象，Zermelo 用波尔兹曼模型对此进行了反驳。 − − − − One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the [[ergodic hypothesis]]. The hypothesis states that the length of time a typical trajectory spends in a region ''A'' is vol(''A'')/vol(Ω). − − One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω). − − 的工作提出的一个问题是时间平均值和空间平均值之间可能的平等，他称之为遍历假设。该假设指出，一个典型的轨道在 a 区的时间长度是 vol (a)/vol (ω)。 − − − − The ergodic hypothesis turned out not to be the essential property needed for the development of [[statistical mechanics]] and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. [[Bernard Koopman|Koopman]] approached the study of ergodic systems by the use of [[functional analysis]]. An observable ''a'' is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ<sup> t</sup>. This introduces an operator ''U''<sup> ''t''</sup>, the [[transfer operator]], − − The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ<sup> t</sup>. This introduces an operator U<sup> t</sup>, the transfer operator, − − 事实证明，遍历假设并不是统计力学系统发展所需的基本属性，而是引入了一系列其他遍历性的属性来捕捉物理系统的相关方面。库普曼利用泛函分析方法研究遍历系统。可观测值 a 是一个函数，它与相空间的每个点相关联一个数字(比如瞬时压力，或平均高度)。一个观测值可以在另一个时间通过使用演化函数 φ < sup > t </sup > 来计算。这里介绍了一个操作符 u </sup > ，即转移操作符, − − − − : <math> (U^t a)(x) = a(\Phi^{-t}(x)). </math> − − <math> (U^t a)(x) = a(\Phi^{-t}(x)). </math> − − (u ^ t a)(x) = a (Phi ^ {-t }(x)).数学 − − − − By studying the spectral properties of the linear operator ''U'' it becomes possible to classify the ergodic properties of Φ<sup> ''t''</sup>. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ<sup> ''t''</sup> gets mapped into an infinite-dimensional linear problem involving ''U''. − − By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ<sup> t</sup>. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ<sup> t</sup> gets mapped into an infinite-dimensional linear problem involving U. − − 通过研究线性算子 u 的谱性质，可以对 φ < sup > t </sup > 的遍历性质进行分类。利用考虑流对可观测函数作用的 Koopman 方法，将含有 φ < sup > t </sup > 的有限维非线性问题映射为含 u 的无限维线性问题。 − − − − The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in [[Statistical mechanics|equilibrium statistical mechanics]]. An average in time along a trajectory is equivalent to an average in space computed with the [[Statistical mechanics#Canonical ensemble|Boltzmann factor exp(−β''H'')]]. This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. [[SRB measure]]s replace the Boltzmann factor and they are defined on attractors of chaotic systems. − − The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems. − − 局限于能量面 ω 的刘维测度是计算平衡统计力学的平均值的基础。沿着轨道的平均时间等于用玻尔兹曼因子乘数(&-; βh)计算的空间平均值。这个观点已经被 Sinai，Bowen 和 Ruelle (SRB)推广到包含耗散系统的更大的动力系统类中。SRB 测量取代了玻尔兹曼因子，它们被定义在混沌系统的吸引子上。 − − − − ===Nonlinear dynamical systems and chaos=== − − {{Main|Chaos theory}} − − Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called ''[[chaos theory|chaos]]''. [[Anosov diffeomorphism|Hyperbolic systems]] are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the ''stable manifold'') and another of the points that diverge from the orbit (the ''unstable manifold''). − − Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold). − − 简单的非线性动力系统，甚至分段线性系统都可能表现出完全不可预测的行为，这可能看起来是随机的，尽管事实上他们是基本上确定的。这种看似不可预知的行为被称为混乱。双曲系统是一类具有混沌特性的动力学系统。在双曲系统中，垂直于轨道的切空间可以很好地分为两部分: 一部分是收敛于轨道的点(稳定流形) ，另一部分是离开轨道的点(不稳定流形)。 − − − − This branch of [[mathematics]] deals 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 [[attractor]]s?" or "Does the long-term behavior of the system depend on its initial condition?" − − This branch of mathematics deals 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 attractors?" or "Does the long-term behavior of the system depend on its initial condition?" − − 这个数学分支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组的精确解(这通常是无望的) ，而是回答诸如“系统是否会长期稳定下来，如果是这样，可能的吸引子是什么? ”或者“系统的长期行为是否取决于其初始条件? ” − − − − Note that the chaotic behavior of complex systems is not the issue. [[Meteorology]] has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The [[logistic map]] is only a second-degree polynomial; the [[horseshoe map]] is piecewise linear. − − Note that the chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear. − − 请注意，复杂系统的混沌行为并不是问题所在。多年来，气象学一直被认为涉及复杂甚至混乱的行为。混沌理论之所以如此令人惊讶，是因为混沌可以在几乎微不足道的系统中发现。Logistic 映射只是一个二次多项式，马蹄映射是分段线性的。 − − − − === Geometrical definition === − − A dynamical system is the tuple <math> \langle \mathcal{M}, f , \mathcal{T}\rangle </math>, with <math>\mathcal{M}</math> a manifold (locally a Banach space or Euclidean space), <math>\mathcal{T}</math> the domain for time (non-negative reals, the integers, ...) and ''f'' an evolution rule ''t'' → ''f''<sup> ''t''</sup> (with <math>t\in\mathcal{T}</math>) such that ''f<sup> t</sup>'' is a [[diffeomorphism]] of the manifold to itself. So, f is a mapping of the time-domain <math> \mathcal{T}</math> into the space of diffeomorphisms of the manifold to itself. In other terms, ''f''(''t'') is a diffeomorphism, for every time ''t'' in the domain <math> \mathcal{T}</math> . − − A dynamical system is the tuple <math> \langle \mathcal{M}, f , \mathcal{T}\rangle </math>, with <math>\mathcal{M}</math> a manifold (locally a Banach space or Euclidean space), <math>\mathcal{T}</math> the domain for time (non-negative reals, the integers, ...) and f an evolution rule t → f<sup> t</sup> (with <math>t\in\mathcal{T}</math>) such that f<sup> t</sup> is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain <math> \mathcal{T}</math> into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain <math> \mathcal{T}</math> . − − 一个动力系统是一个元组 < math > langle mathcal { m } ，f，mathcal { t } rangle </math > ，具有 < math > mathcal { m } </math > 一个流形(局部为 Banach 空间或欧几里德空间) ，< math > mathcal { t } </math > 时间域(非负数，整数，...)和一个进化规则 t→ f < sup > t </sup > (在数学上是 < math > t)使得 f < sup > t </sup > 是流形自身的一个微分同胚。因此，f 是流形到它自身的微分同胚空间的时域映射。换句话说，f (t)是区域 < math > mathcal { t } </math > 中的每个时间 t 的一个微分同胚。 − − − − === Measure theoretical definition === − − {{main|Measure-preserving dynamical system}} − − A dynamical system may be defined formally, as a measure-preserving transformation of a [[sigma-algebra]], the quadruplet (''X'', Σ, μ, τ). Here, ''X'' is a [[set (mathematics)|set]], and Σ is a [[sigma-algebra]] on ''X'', so that the pair (''X'', Σ) is a measurable space. μ is a finite [[measure (mathematics)|measure]] on the sigma-algebra, so that the triplet (''X'', Σ, μ) is a [[measure space|probability space]]. A map τ: ''X'' → ''X'' is said to be [[measurable function|Σ-measurable]] if and only if, for every σ ∈ Σ, one has <math>\tau^{-1}\sigma \in \Sigma</math>. A map τ is said to '''preserve the measure''' if and only if, for every σ ∈ Σ, one has <math>\mu(\tau^{-1}\sigma ) = \mu(\sigma)</math>. Combining the above, a map τ is said to be a '''measure-preserving transformation of ''X'' ''', if it is a map from ''X'' to itself, it is Σ-measurable, and is measure-preserving. The quadruple (''X'', Σ, μ, τ), for such a τ, is then defined to be a '''dynamical system'''. − − A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X, Σ, μ, τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X, Σ, μ) is a probability space. A map τ: X → X is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has <math>\tau^{-1}\sigma \in \Sigma</math>. A map τ is said to preserve the measure if and only if, for every σ ∈ Σ, one has <math>\mu(\tau^{-1}\sigma ) = \mu(\sigma)</math>. Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X, Σ, μ, τ), for such a τ, is then defined to be a dynamical system. − − 动力系统可以被正式定义为 sigma-代数的四元组(x，σ，μ，τ)的保测度变换。这里，x 是一个集合，σ 是 x 上的 sigma 代数，所以这对(x，σ)是一个可测空间。μ 是 sigma-代数上的一个有限测度，因此三重态(x，σ，μ)是概率空间。映射 τ: x → x 是 σ 可测的当且仅当，对于每个 σ ∈ σ，在 Sigma </math > 中有 < math > tau ^ {-1} σ。当且仅当，对于每个 σ ∈ σ，存在一个 < math > mu (tau ^ {-1} sigma) = mu (sigma) </math > 。综上所述，一个映射 τ 被称为 x 的一个保测度变换，如果它是一个从 x 到自身的映射，它是 σ 可测的，是保测度的。对于这样的 τ，四元组(x，σ，μ，τ)被定义为动力系统。 − − − − The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the [[iterated function|iterates]] <math>\tau^n=\tau \circ \tau \circ \cdots\circ\tau</math> for integer ''n'' are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated. − − The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates <math>\tau^n=\tau \circ \tau \circ \cdots\circ\tau</math> for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated. − − 地图 τ 体现了动力系统的时间演变。因此，对于离散动力系统，我们研究了整数 n 的 t ^ n = t ^ n = t ^ 圈 c ^ 圈 c ^ 圈 c ^ 圈 c ^ 圈 n 的迭代。对于连续动态系统，映射 τ 被理解为一个有限时间演化映射，其构造比较复杂。 − − − − == Multidimensional generalization == − − Dynamical systems are defined over a single independent variable, usually thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called [[multidimensional systems]]. Such systems are useful for modeling, for example, [[image processing]]. − − Dynamical systems are defined over a single independent variable, usually thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing. − − 动力系统是在一个单独的独立变量上定义的，通常被认为是时间。一类更一般的系统定义在多个独立变量上，因此被称为多维系统。这样的系统对于建模很有用，例如，图像处理。 − − − − == See also == − − {{Portal|Systems science}} − − {{Div col|colwidth=25em}} − − * [[Behavioral modeling]] − − * [[Cognitive model#Dynamical systems|Cognitive modeling]] − − * [[Complex dynamics]] − − * [[Dynamic approach to second language development]] − − * [[Feedback passivation]] − − * [[Infinite compositions of analytic functions]] − − * [[List of dynamical system topics]] − − * [[Oscillation]] − − * [[People in systems and control]] − − * [[Sharkovskii's theorem]] − − * [[System dynamics]] − − * [[Systems theory]] − − * [[Principle of maximum caliber]] − − {{Div col end}} − − − − ==References== − − {{reflist}} − − − − == Further reading == − − {{refbegin|2}} − − Works providing a broad coverage: − − Works providing a broad coverage: − − 涵盖范围广泛的工程: − − * {{cite book | author=[[Ralph Abraham (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]] | title= Foundations of mechanics | publisher= Benjamin–Cummings | year= 1978 | isbn=978-0-8053-0102-1}} (available as a reprint: {{ISBN|0-201-40840-6}}) − − * ''Encyclopaedia of Mathematical Sciences'' ({{ISSN|0938-0396}}) has a sub-series on dynamical systems with reviews of current research. − − * {{cite book |author1=Christian Bonatti |author2=Lorenzo J. Díaz |author3=Marcelo Viana | title= Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective| publisher= Springer | year= 2005 | isbn=978-3-540-22066-4}} − − * {{cite journal | doi=10.1090/S0002-9904-1967-11798-1 | author=Stephen Smale | title= Differentiable dynamical systems | journal= Bulletin of the American Mathematical Society | year= 1967 |volume= 73 |pages= 747–817 | issue=6| author-link=Stephen Smale | doi-access= free }} − − − − Introductory texts with a unique perspective: − − Introductory texts with a unique perspective: − − 具有独特视角的介绍性文本: − − * {{cite book | author=V. I. Arnold | title=Mathematical methods of classical mechanics | publisher=Springer-Verlag | year=1982 | isbn=978-0-387-96890-2 | author-link=Vladimir Arnold | url-access=registration | url=https://archive.org/details/mathematicalmeth0000arno }} − − * {{cite book | author=[[Jacob Palis]] and [[Welington de Melo]] | title=Geometric theory of dynamical systems: an introduction | publisher=Springer-Verlag | year=1982 | isbn=978-0-387-90668-3 | url-access=registration | url=https://archive.org/details/geometrictheoryo0000pali }} − − * {{cite book | author=David Ruelle | title=Elements of Differentiable Dynamics and Bifurcation Theory | publisher=Academic Press | year=1989 | isbn=978-0-12-601710-6| author-link=David Ruelle }} − − * {{cite book | author=Tim Bedford, Michael Keane and Caroline Series, ''eds.'' | title= Ergodic theory, symbolic dynamics and hyperbolic spaces | publisher= Oxford University Press | year= 1991 | isbn= 978-0-19-853390-0 }} − − * {{cite book | author= [[Ralph Abraham (mathematician)|Ralph H. Abraham]] and [[Robert Shaw (Physicist)#Illustrations|Christopher D. Shaw]] | title= Dynamics—the geometry of behavior, 2nd edition | publisher= Addison-Wesley | year= 1992 | isbn= 978-0-201-56716-8 }} − − − − Textbooks − − Textbooks − − 教科书 − − * {{cite book | author= Kathleen T. Alligood, Tim D. Sauer and [[James A. Yorke]] | title= Chaos. An introduction to dynamical systems | publisher= Springer Verlag | year= 2000 | isbn=978-0-387-94677-1}} − − * {{cite book | author= Oded Galor | title= ''Discrete Dynamical Systems'' | publisher= Springer | year= 2011 | isbn=978-3-642-07185-0}} − − * {{cite book | author= [[Morris Hirsch|Morris W. Hirsch]], [[Stephen Smale]] and [[Robert L. Devaney]] | title= Differential Equations, dynamical systems, and an introduction to chaos | publisher= Academic Press | year= 2003 | isbn=978-0-12-349703-1}} − − * {{cite book |author1=Anatole Katok |author2=Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=978-0-521-57557-7}} − − * {{cite book | author= Stephen Lynch | title= Dynamical Systems with Applications using Maple 2nd Ed.| publisher= Springer | year= 2010|isbn = 978-0-8176-4389-8 }} − − * {{cite book | title= Dynamical Systems with Applications using MATLAB 2nd Edition | publisher= Springer International Publishing | year= 2014|isbn = 978-3319068190 | author= Stephen Lynch }} − − * {{cite book | author= Stephen Lynch | title= Dynamical Systems with Applications using Mathematica 2nd Ed.| publisher= Springer | year= 2017|isbn = 978-3-319-61485-4 }} − − * {{cite book | title= Dynamical Systems with Applications using Python | publisher= Springer International Publishing | year= 2018|isbn = 978-3-319-78145-7 | author= Stephen Lynch }} − − * {{cite book | author= James Meiss | title= Differential Dynamical Systems | publisher= SIAM | year= 2007|isbn = 978-0-89871-635-1 }} − − * {{cite book | author= David D. Nolte | title= Introduction to Modern Dynamics: Chaos, Networks, Space and Time | publisher= Oxford University Press | year= 2015 | isbn=978-0199657032 }} − − * {{cite book | author= Julien Clinton Sprott | title= ''Chaos and time-series analysis'' | publisher= Oxford University Press | year= 2003 | isbn=978-0-19-850839-7}} − − * {{cite book | author=Steven H. Strogatz | title= Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering | publisher= Addison Wesley | year= 1994|isbn = 978-0-201-54344-5 | author-link= Steven Strogatz }} − − * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}} − − * {{cite book | author= Stephen Wiggins | title= Introduction to Applied Dynamical Systems and Chaos | publisher= Springer | year= 2003 | isbn= 978-0-387-00177-7 }} − − − − Popularizations: − − Popularizations: − − 流行语: − − * {{cite book | author=[[Florin Diacu]] and [[Philip Holmes]] | title= Celestial Encounters | publisher= Princeton | year= 1996 | isbn= 978-0-691-02743-2}} − − * {{cite book | author=James Gleick | title= Chaos: Making a New Science | publisher= Penguin | year= 1988 | isbn= 978-0-14-009250-9| author-link= James Gleick | title-link= Chaos: Making a New Science }} − − * {{cite book | authorlink=Ivar Ekeland | author=Ivar Ekeland | title= Mathematics and the Unexpected (Paperback) | publisher= University Of Chicago Press | year= 1990 | isbn= 978-0-226-19990-0}} − − * {{cite book | author=Ian Stewart | year = 1997 | title = Does God Play Dice? The New Mathematics of Chaos | publisher = Penguin | isbn = 978-0-14-025602-4}} − − {{refend}} − − − − ==External links== − − {{Commonscat|Dynamical systems}} − − *[http://www.arxiv.org/list/math.DS/recent Arxiv preprint server] has daily submissions of (non-refereed) manuscripts in dynamical systems. − − *[http://www.scholarpedia.org/article/Encyclopedia_of_Dynamical_Systems Encyclopedia of dynamical systems] A part of [[Scholarpedia]] — peer reviewed and written by invited experts. − − *[http://www.egwald.ca/nonlineardynamics/index.php Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens − − *[http://amath.colorado.edu/faculty/jdm/faq-Contents.html Sci.Nonlinear FAQ 2.0 (Sept 2003)] provides definitions, explanations and resources related to nonlinear science − − − − ;Online books or lecture notes − − Online books or lecture notes − − 网上书籍或课堂讲稿 − − *[https://arxiv.org/abs/math.HO/0111177 Geometrical theory of dynamical systems]. Nils Berglund's lecture notes for a course at [[ETH]] at the advanced undergraduate level. − − *[https://archive.org/details/dynamicalsystems00birk Dynamical systems]. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems. − − *[http://chaosbook.org/ Chaos: classical and quantum]. An introduction to dynamical systems from the periodic orbit point of view. − − *[http://www.cs.brown.edu/research/ai/dynamics/tutorial/home.html Learning Dynamical Systems]. Tutorial on learning dynamical systems. − − *[https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ Ordinary Differential Equations and Dynamical Systems]. Lecture notes by [[Gerald Teschl]] − − − − ;Research groups − − Research groups − − 研究小组 − − *[http://www.math.rug.nl/~broer/ Dynamical Systems Group Groningen], IWI, University of Groningen. − − *[http://www-chaos.umd.edu/ Chaos @ UMD]. Concentrates on the applications of dynamical systems. − − *[http://www.math.stonybrook.edu/dynamical-systems], SUNY Stony Brook. Lists of conferences, researchers, and some open problems. − − *[http://www.math.psu.edu/dynsys/ Center for Dynamics and Geometry], Penn State. − − *[http://www.cds.caltech.edu/ Control and Dynamical Systems], Caltech. − − *[https://web.archive.org/web/20061018031023/http://lanoswww.epfl.ch/ Laboratory of Nonlinear Systems], Ecole Polytechnique Fédérale de Lausanne (EPFL). − − *[https://web.archive.org/web/20070208153906/http://www.math.uni-bremen.de/ids.html Center for Dynamical Systems], University of Bremen − − *[https://web.archive.org/web/20070406053155/http://www.eng.ox.ac.uk/samp/ Systems Analysis, Modelling and Prediction Group], University of Oxford − − *[http://sd.ist.utl.pt/ Non-Linear Dynamics Group], Instituto Superior Técnico, Technical University of Lisbon − − *[http://www.impa.br/ Dynamical Systems], IMPA, Instituto Nacional de Matemática Pura e Applicada. − − *[http://ndw.cs.cas.cz/ Nonlinear Dynamics Workgroup], Institute of Computer Science, Czech Academy of Sciences. − − *[https://dynamicalsystems.upc.edu/ UPC Dynamical Systems Group Barcelona], Polytechnical University of Catalonia. − − *[https://www.ccdc.ucsb.edu/ Center for Control, Dynamical Systems, and Computation], University of California, Santa Barbara. − − − − {{Systems}} − − {{Chaos theory}} − − {{Authority control}} − − − − {{DEFAULTSORT:Dynamical System}} − − [[Category:Dynamical systems| ]] − − [[Category:Physical systems]] − − Category:Physical systems − − 分类: 物理系统 − − [[Category:Systems theory]] − − Category:Systems theory − − 范畴: 系统论 − − [[Category:Mathematical and quantitative methods (economics)]] − − Category:Mathematical and quantitative methods (economics) − − 类别: 数学和定量方法(经济学) − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Dynamical system]]. Its edit history can be viewed at [[非线性动力系统/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
重定向页面至动力系统
#重定向 [[动力系统]]