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Dynamical systems theory is an area of mathematics used to describe the behavior of the  complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

Dynamical systems theory is an area of mathematics used to describe the behavior of the  complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

动态系统理论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，是直接假定运动方程的推广，不受最小作用原理的欧拉-拉格朗日方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续运行，或者是任意的时间集合，例如 Cantor 集合，得到时间尺度上的动力学方程。有些情况也可以用混合算子来模拟，如微分差分方程。

动态系统理论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，是直接假定运动方程的推广，不受最小作用原理的欧拉-拉格朗日方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量在一个离散的集合上运行，在另一个离散的集合上连续运行，或者是任意的时间集合，如 Cantor 集合，就得到了时间尺度上的动力学方程。有些情况也可以用混合算子来模拟，如微分差分方程。

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"

Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"

动态系统理论和混沌理论处理动力系统的长期定性行为。在这里，重点不在于找到定义动力系统的方程的精确解(这通常是没有希望的) ，而是回答诸如“系统是否会长期稳定下来，如果是的话，可能的稳定状态是什么? ”？或者“系统的长期行为是否取决于它的初始条件? ”

动态系统理论和混沌理论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的精确解(这通常是没有希望的) ，而是回答诸如“系统是否会长期稳定下来，如果是的话，可能的稳定状态是什么? ”？或者“系统的长期行为是否取决于它的初始条件? ”

Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.

Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.

Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.

Some excellent presentations of mathematical dynamic system theory include , , , and .

Some excellent presentations of mathematical dynamic system theory include , , , and .

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.

动力系统的概念是一个数学形式化的任何固定的“规则” ，描述了一个点的位置在其环境空间的时间依赖性。例子包括描述钟摆摆动的数学模型、管道中的水流量以及每年春天湖中鱼的数量。

动力系统的概念是一个数学形式化的任何固定的“规则” ，描述了一个点的位置在其环境空间的时间依赖性。这些例子包括描述钟摆摆动的数学模型、管道中的水流量以及每年春天湖中鱼的数量。

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).

动力系统状态是由一组实数决定的，或者更广泛地说是由一组适当状态空间中的点决定的。系统状态的微小变化对应于数字的微小变化。这些数字也是几何空间ーー流形ーー的坐标。动力系统的演变规则是一个固定的规则，描述了当前状态下的未来状态。该规则可以是确定性的(在给定的时间间隔内，只有一个未来状态从当前状态跟随)或随机性的(状态的演变受到随机冲击)。

动力系统的状态是由一组实数决定的，或者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微小变化。这些数字也是几何空间ーー流形ーー的坐标。动力系统的演变规则是一个固定的规则，描述了当前状态下的未来状态。该规则可以是确定性的(在给定的时间间隔内，只有一个未来状态从当前状态跟随)或随机性的(状态的演变受到随机冲击)。