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动力系统理论和'''混沌理论 Chaos Theory'''是用来处理动力系统的<font color="red">长期定性行为</font><font color="blue"> 长周期性行为</font>的理论。寻找动力系统方程的精确解通常是很难达到的。<font color="blue"> 因此，</font>这两个理论的重点不在于找到精确解，而是回答如下的问题，如“系统长期来看是否会稳定下来，如果可以，那么可能的稳定状态是什么样的?”，或“系统长期的行为是否取决于其初始条件?”

动力系统理论和'''混沌理论 Chaos Theory'''是用来处理动力系统的<font color="red">长期定性行为</font><font color="blue"> 长周期性行为</font>的理论。寻找动力系统方程的精确解通常是很难达到的。<font color="blue"> 因此，</font>这两个理论的重点不在于找到精确解，而是回答如下的问题，如“系统长期来看是否会稳定下来，如果可以，那么可能的稳定状态是什么样的?”，或“系统长期的行为是否取决于其初始条件?”

The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient 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 spring in a lake.

The dynamical system concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient 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 spring in a lake.

 

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。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。<font color='red'>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。</font><font color='blue'>动力系统的演化规律是一种固定的规则，它描述了从当前状态得出的未来状态。</font>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===

Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues that differential equations are more suited to modelling cognition than more traditional computer models.

Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues that differential equations are more suited to modelling cognition than more traditional computer models.

'''动态主义 Dynamicism'''，又称动态假设，或称认知科学、动态认知中的动态假设，是以哲学家Tim van Gelder的著作为代表的认知科学的一种新取向。动态主义认为微分方程比传统的计算机模型更适合于建立认知模型。

'''动态主义 Dynamicism'''，也称动态假设，<font color='red'>或称认知科学、动态认知中的动态假设，</font><font color='blue'>或称认知科学的动态假设或动态认知，</font>是以哲学家Tim van Gelder的著作为代表的认知科学的一种新取向。动态主义认为微分方程比传统的计算机模型更适合于建立认知模型。

In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.

In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.

在数学中，'''非线性系统 Nonlinear System'''是指系统不是线性的——也就是说，一个不满足叠加原理的系统。更通俗地说，非线性系统是待求解变量不能被写成其独立分量的线性和的系统。非齐次系统根据定义严格来说是非线性的，除了它的自变量包含函数以外其他部分都是线性的。但非齐次系统通常与线性系统一起研究，因为只要知道特解，它们就可以转化为线性系统。

在数学中，'''非线性系统 Nonlinear System'''是指系统不是线性的——也就是说，一个不满足叠加原理的系统。更通俗地说，非线性系统是待求解变量不能被写成其独立分量的线性和的系统。非齐次系统根据定义严格来说是非线性的，除了它的自变量<font del>包含</font>函数以外其他部分都是线性的。但非齐次系统通常与线性系统一起研究，因为只要知道特解，它们就可以转化为线性系统。

== Related fields 相关领域==

== Related fields 相关领域==