更改

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.]]

拓扑动力学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。

拓扑动力学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 ==

== Applications 应用==

=== In human development ===

=== In human development 人类发展中的应用===

In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified:  early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).

In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified:  early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).

在人类发展方面，动态系统理论已经被用来增强和简化 Erik Erikson 的8个社会心理发展阶段，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的自组织性和分形特性。利用数学模型，人类发展的自然进程被确定为八个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、幼儿期(4-7岁)、中期儿童期(7-11岁)、青春期(11-18岁)、青年期(18-29岁)、中期成年期(29-48岁)和老年期(48-78岁以上)。

在人类发展方面，动态系统理论已经被用来增强和简化 Erik Erikson 的社会心理发展8阶段理论，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的自组织性（self-organizing）和分形特性。利用数学模型，人类发展的自然进程被分为8个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、童年早期(4-7岁)、童年中期(7-11岁)、青春期(11-18岁)、成年早期(18-29岁)、成年中期(29-48岁)和老年成年期(48-78岁及以上)。

According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males and the period of optimal fertility begins in females. Similarly, at age 30 optimal fertility begins to decline in females, and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs.

According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males and the period of optimal fertility begins in females. Similarly, at age 30 optimal fertility begins to decline in females, and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs.

根据这个模型，年龄间隔之间的阶段转换代表了多层次的自我组织过程(例如，分子、基因、细胞、器官、器官系统、生物体、行为和环境)。例如，在从青春期向成年早期过渡的阶段，以及在达到18岁这一关键点(成年早期)之后，男性的睾丸激素达到高峰，而女性的最佳生育期开始。同样，在30岁时，女性的最佳生育能力开始下降，在48岁这一阶段，从中年成年期过渡到老年成年期，开始绝经的平均年龄出现。

根据这个模型，年龄的阶段转换代表了多层次的自组织过程(例如，分子、基因、细胞、器官、器官系统、生物体、行为和环境)。例如，在从青春期向成年早期过渡的阶段中，在达到18岁这一关键年龄之后，男性的睾丸激素达到高峰，女性的最佳生育期开始。同样，在30岁时，女性的最佳生育能力开始下降；在从成年中期过渡到老年成年期时，48岁是绝经的平均年龄。

These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA and self-organizing properties of the Fibonacci numbers that converge on the golden ratio.

These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA and self-organizing properties of the Fibonacci numbers that converge on the golden ratio.

从斐波那契数学模型和动力系统理论的角度来看，这些事件是衰老的物理生物吸引子。实际上，人类发展的预测在同样的统计意义上成为可能，即一年中不同时间的平均气温或降水量可以用来预测2010年的天气预报。人类发展的每个预定阶段都遵循最佳的表观遗传生物模式。这种现象可以用生物 DNA 中斐波那契数的出现和收敛于黄金分割比的斐波那契数的自组织特性来解释。

从斐波那契数学模型和动力系统理论的角度来看，上述事件是衰老的物理生物吸引子（physical bioattractors）。实际上，正如一年中不同时间的平均气温和降水量可以用来预测天气，预测人类的发展在统计意义上同样是可能的。人类发展的每个预定阶段（predetermined stages）都遵循最佳的表观遗传生物模式（epigenetic biological pattern）。这种现象可以用 DNA 中的斐波那契数和收敛于黄金分割比的斐波那契数的自组织特性来解释。

=== In biomechanics ===

=== In biomechanics 在生物力学中的应用===

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.<ref>Paul S Glazier, Keith Davids, Roger M Bartlett (2003). [http://www.sportsci.org/jour/03/psg.htm "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research"]. in: Sportscience 7. Accessed 2008-05-08.</ref> 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.<ref>Paul S Glazier, Keith Davids, Roger M Bartlett (2003). [http://www.sportsci.org/jour/03/psg.htm "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for Performance-Oriented Sports Biomechanics Research"]. in: Sportscience 7. Accessed 2008-05-08.</ref> 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.

In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems. There is no research validation of any of the claims associated to the conceptual application of this framework.

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

在运动生物力学中，动力系统理论新兴地成为运动表现建模的可行框架。从动力系统的角度来看，人类的运动系统是一个高度复杂的相互依赖的子系统网络（如呼吸、循环、神经、骨骼肌系统和知觉系统等），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相关的任何主张。--[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 没有研究证实？那就是说不可信吗？

=== In cognitive science ===

=== In cognitive science 在认知科学中的应用===

Dynamical system theory has been applied in the field of [[neurodynamics|neuroscience]] and [[cognitive science|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 [[neurodynamics|neuroscience]] and [[cognitive science|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.

Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.

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

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(或者是)(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.

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

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具有新状态的结构，这个过程被称为“扇贝化（scalloping）”，也就是头脑的复杂性能的不断累积和崩溃的过程 --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]）扇贝化的翻译拿不准，而且这句话的描述比较抽象 。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.

动态系统理论最近被用来解释儿童发展中一个长期没有答案的问题，即 A-not-B 错误。

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===

===In second language development 在二语习得中的应用===

{{Main|Dynamic approach to second language development}}

{{Main|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.

动态系统理论在二语习得研究中的应用归功于 Diane Larsen-Freeman，她在1997年发表的一篇文章中声称，二语习得应该被看作是一个包括语言流失和语言习得在内的发展过程。她在文章中主张，语言应该被看作是一个动态的、复杂的、非线性的、混沌的、不可预知的、对初始条件敏感的、开放的、自组织的、反馈敏感的、适应性的动态系统。

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。