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'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续，或者像cantor（康托尔）集一样在任意的时间集合上运行时，人们就能得到时间尺度上的动力方程。

'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续，或者像cantor集一样在任意的时间集合上运行时，人们就能得到时间尺度上的动力方程。

'''算子 Operators'''是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。动力系统的有些情况也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。

'''算子 Operators'''是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。动力系统的有些情况也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space.  Small changes in the state of the system correspond to small changes in the numbers.  The numbers are also the coordinates of a geometrical space—a manifold.  The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state.  The rule may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space.  Small changes in the state of the system correspond to small changes in the numbers.  The numbers are also the coordinates of a geometrical space—a manifold.  The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state.  The rule may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

动力系统的状态是由一组实数决定的，更广泛地说，是由适当的状态空间中的一组点决定的。系统状态的微小变化对应于数字的变化。这些数字也是几何空间——'''流形（Manifold）'''——的坐标组（coordinates）。动力系统的演化是描述了在当前状态之后出现的未来状态的固定规则。这个规则可以是确定性的(在给定的时间间隔内，有且仅有一个未来状态在当前状态之后出现)，或随机性的(状态的演化受到随机因素的影响)。

动力系统的状态是由一组实数决定的，更广泛地说，是由适当的状态空间中的一组点决定的。系统状态的微小变化对应于数字的变化。这些数字也是几何空间——'''流形 Manifold'''——的坐标组。动力系统的演化是描述了在当前状态之后出现的未来状态的固定规则。这个规则可以是确定性的(在给定的时间间隔内，有且仅有一个未来状态在当前状态之后出现)，或随机性的(状态的演化受到随机因素的影响)。

=== Dynamicism 动态主义===

=== Dynamicism 动态主义===

'''控制理论 Control Theory'''是工程和数学的一个交叉学科，它的其中一部分研究影响动力系统行为的各种因素。

'''控制理论 Control Theory'''是工程和数学的一个交叉学科，它的其中一部分研究影响动力系统行为的各种因素。

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

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

'''投影动力系统 Projected Dynamical Systems'''是研究解在一个约束集内的动力系统行为的数学理论。这门学科与静态世界中的最优化和平衡问题以及动态世界中的常微分方程都有联系，并且都有相互联系的应用。一个投影动力系统是由投影微分方程的'''流行(flow)'''给定的。

'''投影动力系统 Projected Dynamical Systems'''是研究解在一个约束集内的动力系统行为的数学理论。这门学科与静态世界中的最优化和平衡问题以及动态世界中的常微分方程都有联系，并且都有相互联系的应用。一个投影动力系统是由投影微分方程的'''流形 flow'''给定的。

通过对投影微分方程的流分析，给出了一个投影动力系统的表达式：

通过对投影微分方程的流分析，给出了一个投影动力系统的表达式：

:<math>

:<math>

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，动力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 Eight Stages of Psychosocial Development'''，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(Fractal)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)。

在人类发展方面，动力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 Eight Stages of Psychosocial Development'''，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 Fractal''' 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)。

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，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。

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。

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

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

动力系统理论已经被应用于神经科学和认知发展领域，特别是在认知发展的'''新皮亚杰学派(neo-Piagetian)'''中。人们相信，物理学理论比句法学理论和人工智能理论更能代表认知发展。人们还相信微分方程是人类行为建模最合适的工具。人们认为微分方程可以解释为通过状态空间代表一个主体的认知轨迹的算式。换句话说，动力学家认为心理学应该(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。

动力系统理论已经被应用于神经科学和认知发展领域，特别是在认知发展的'''新皮亚杰学派 neo-Piagetian'''中。人们相信，物理学理论比句法学理论和人工智能理论更能代表认知发展。人们还相信微分方程是人类行为建模最合适的工具。人们认为微分方程可以解释为通过状态空间代表一个主体的认知轨迹的算式。换句话说，动力学家认为心理学应该(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|扇贝化的翻译拿不准，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。

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|扇贝化的翻译拿不准，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。

===集智课程推荐===

===集智课程推荐===

====[https://campus.swarma.org/course/1655 圣塔菲课程：Introduction to Dynamical Systems and Chaos]====

====[https://campus.swarma.org/course/1655 圣塔菲课程：Introduction to Dynamical Systems and Chaos]====

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