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第1行： − 此词条暂由彩云小译翻译，翻译字数共71，未经人工整理和审校，带来阅读不便，请见谅。+ − + + + − Autonomous system may refer to:+ − 自治系统可参阅:+ − * [[Autonomous system (Internet)]], a collection of IP networks and routers under the control of one entity+ − * [[Autonomous system (mathematics)]], a system of ordinary differential equations which does not depend on the independent variable − * [[Autonomous robot]], robots which can perform desired tasks in unstructured environments without continuous human guidance − * [[Autonomous underwater vehicle]], a system that travels underwater without requiring input from an operator. − {{disambiguation}} − * Autonomous system (Internet), a collection of IP networks and routers under the control of one entity+ − * Autonomous system (mathematics), a system of ordinary differential equations which does not depend on the independent variable − * Autonomous robot, robots which can perform desired tasks in unstructured environments without continuous human guidance − * Autonomous underwater vehicle, a system that travels underwater without requiring input from an operator. + + − + − + − + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − +
Moved page from wikipedia:en:Autonomous system (mathematics) (history)
此词条暂由彩云小译翻译，翻译字数共1522，未经人工整理和审校，带来阅读不便，请见谅。
'''Autonomous system''' may refer to:
{{Short description|Math}}
[[File:Stability_Diagram.png|thumb|500px|[[Stability theory|Stability diagram]] classifying [[Poincaré map#Poincaré maps and stability analysis|Poincaré maps]] of linear '''autonomous system''' <math>x' = Ax,</math> as stable or unstable according to their features. Stability generally increases to the left of the diagram.<ref>[http://www.egwald.ca/linearalgebra/lineardifferentialequationsstabilityanalysis.php Egwald Mathematics - Linear Algebra: Systems of Linear Differential Equations: Linear Stability Analysis] Accessed 10 October 2019.</ref> Some sink, source or node are [[equilibrium point]]s.]]
[[File:Phase plane nodes.svg|thumb|300px|2-dimensional case refers to [[Phase plane]].]]
In [[mathematics]], an '''autonomous system''' or '''autonomous differential equation''' is a [[simultaneous equations|system]] of [[ordinary differential equation]]s which does not explicitly depend on the [[independent variable]]. When the variable is time, they are also called [[time-invariant system]]s.
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
在数学中，自治系统或自治微分方程是一个普通微分方程系统，它并不明确地依赖于独立变量。当变量为时间时，它们也称为时不变系统。
Many laws in [[physics]], where the independent variable is usually assumed to be [[time]], are expressed as autonomous systems because it is assumed the [[Physical law|laws of nature]] which hold now are identical to those for any point in the past or future.
Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future.
物理学中的许多定律，其中自变量通常被假定为时间，被表示为自治系统，因为它被假定为现在所持有的自然定律与过去或未来任何一点的自然定律是相同的。
* 自治系统(互联网) ，由一个实体控制的 IP 网络和路由器的集合
== Definition ==
* 自治系统(数学) ，一个不依赖于独立变量的常微分方程系统
* 自主机器人，能够在没有连续的人类引导的情况下在非结构化环境中执行所需任务的机器人
== Definition ==
* 自主水下载具，一个在水下旅行而不需要操作员输入的系统。
= = 定义 = =
An '''autonomous system''' is a [[system of ordinary differential equation]]s of the form
<math display="block">\frac{d}{dt}x(t)=f(x(t))</math>
where {{mvar|x}} takes values in {{mvar|n}}-dimensional [[Euclidean space]]; {{mvar|t}} is often interpreted as time.
An autonomous system is a system of ordinary differential equations of the form
\frac{d}{dt}x(t)=f(x(t))
where takes values in -dimensional Euclidean space; is often interpreted as time.
自治系统是一个常微分方程组，其形式为 frac { d }{ dt } x (t) = f (x (t)) ，取欧几里德空间中的数值，通常被解释为时间。
It is distinguished from systems of differential equations of the form
<math display="block">\frac{d}{dt}x(t)=g(x(t),t)</math>
in which the law governing the evolution of the system does ''not'' depend solely on the system's current state but also the parameter {{mvar|t}}, again often interpreted as time; such systems are by definition not autonomous.
It is distinguished from systems of differential equations of the form
\frac{d}{dt}x(t)=g(x(t),t)
in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous.
它区别于形式为 frac { d }{ dt } x (t) = g (x (t) ，t)的微分方程系统，在这种微分方程系统中，控制系统进化的规律不仅取决于系统的当前状态，而且还取决于参数，这些参数也常常被解释为时间; 这种系统根据定义不是自治的。
== Properties ==
== Properties ==
= = 属性 = =
Solutions are invariant under horizontal translations:
Solutions are invariant under horizontal translations:
解在水平翻译下是不变的:
Let <math>x_1(t)</math> be a unique solution of the [[initial value problem]] for an autonomous system
<math display="block">\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(0)=x_0.</math>
Then <math>x_2(t)=x_1(t-t_0)</math> solves
<math display="block">\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(t_0)=x_0.</math>
Indeed, denoting <math>s=t-t_0</math> we have <math>x_1(s)=x_2(t)</math> and <math>ds=dt</math>, thus
<math display="block">\frac{d}{dt}x_2(t) = \frac{d}{dt}x_1(t-t_0)=\frac{d}{ds}x_1(s) = f(x_1(s)) = f(x_2(t)) .</math>
For the initial condition, the verification is trivial,
<math display="block">x_2(t_0)=x_1(t_0-t_0)=x_1(0)=x_0 .</math>
Let x_1(t) be a unique solution of the initial value problem for an autonomous system
\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(0)=x_0.
Then x_2(t)=x_1(t-t_0) solves
\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(t_0)=x_0.
Indeed, denoting s=t-t_0 we have x_1(s)=x_2(t) and ds=dt, thus
\frac{d}{dt}x_2(t) = \frac{d}{dt}x_1(t-t_0)=\frac{d}{ds}x_1(s) = f(x_1(s)) = f(x_2(t)) .
For the initial condition, the verification is trivial,
x_2(t_0)=x_1(t_0-t_0)=x_1(0)=x_0 .
设 x _ 1(t)是自治系统 frac { d }{ dt } x (t) = f (x (t)) ，四次 x (0) = x _ 0初值问题的唯一解。Then x_2(t)=x_1(t-t_0) solves
\frac{d}{dt}x(t)=f(x(t)) \, , \quad x(t_0)=x_0.事实上，表示 s = t-t _ 0我们有 x _ 1(s) = x _ 2(t)和 ds = dt，因此 frac { d }{ dt } x _ 2(t) = frac { d }{ dt } x _ 1(t-t _ 0) = frac { d }{ ds } x _ 1(s) = f (x _ 1(s)) = f (x _ 2(t))。对于初始条件，验证很简单，x _ 2(t _ 0) = x _ 1(t _ 0-t _ 0) = x _ 1(0) = x _ 0。
==Example==
The equation <math>y'= \left(2-y\right)y</math> is autonomous, since the independent variable, let us call it <math>x</math>, does not explicitly appear in the equation.
To plot the [[slope field]] and [[isocline]] for this equation, one can use the following code in [[GNU Octave]]/[[MATLAB]]
<syntaxhighlight lang="matlab">
Ffun = @(X, Y)(2 - Y) .* Y; % function f(x,y)=(2-y)y
[X, Y] = meshgrid(0:.2:6, -1:.2:3); % choose the plot sizes
DY = Ffun(X, Y); DX = ones(size(DY)); % generate the plot values
quiver(X, Y, DX, DY, 'k'); % plot the direction field in black
hold on;
contour(X, Y, DY, [0 1 2], 'g'); % add the isoclines(0 1 2) in green
title('Slope field and isoclines for f(x,y)=(2-y)y')
</syntaxhighlight>
One can observe from the plot that the function <math>\left(2-y\right)y</math> is <math>x</math>-invariant, and so is the shape of the solution, i.e. <math>y(x)=y(x-x_0)</math> for any shift <math>x_0</math>.
The equation y'= \left(2-y\right)y is autonomous, since the independent variable, let us call it x, does not explicitly appear in the equation.
To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB
Ffun = @(X, Y)(2 - Y) .* Y; % function f(x,y)=(2-y)y
[X, Y] = meshgrid(0:.2:6, -1:.2:3); % choose the plot sizes
DY = Ffun(X, Y); DX = ones(size(DY)); % generate the plot values
quiver(X, Y, DX, DY, 'k'); % plot the direction field in black
hold on;
contour(X, Y, DY, [0 1 2], 'g'); % add the isoclines(0 1 2) in green
title('Slope field and isoclines for f(x,y)=(2-y)y')
One can observe from the plot that the function \left(2-y\right)y is x-invariant, and so is the shape of the solution, i.e. y(x)=y(x-x_0) for any shift x_0.
= = 示例 = = 方程 y’= left (2-y right) y 是自治的，因为自变量，我们称之为 x，没有明确出现在方程中。要绘制这个方程的斜率场和等倾线，可以使用 GNU Octave/MATLAB Ffun =@(X，Y)(2-Y)中的以下代码。* Y;% 函数 f (x，y) = (2-y) y [ X，Y ] = meshgrid (0: . 2:6，-1: . 2:3) ;% 选择阴谋大小 DY = Ffun (X，Y) ; DX = ones (size (DY)) ;% 生成阴谋值震动(X，Y，DX，DY，‘ k’) ;% 用黑色绘制方向域; 等高线(X，Y，DY，[012] ，‘ g’) ;% 在绿色标题(“斜率场和 f (x，y) = (2-y) y 的等斜线”)中加上等斜线(012)。从图中可以看出，函数 left (2-y right) y 是 x 不变的，解的形状也是如此。任意位移 x _ 0的 y (x) = y (x-x _ 0)。
Solving the equation symbolically in [[MATLAB]], by running
<syntaxhighlight lang="matlab">
syms y(x);
equation = (diff(y) == (2 - y) * y);
% solve the equation for a general solution symbolically
y_general = dsolve(equation);
</syntaxhighlight>
we obtain two [[Equilibrium point|equilibrium]] solutions, <math>y=0</math> and <math>y=2</math>, and a third solution involving an unknown constant <math>C_3</math>,
<syntaxhighlight lang="matlab" inline>-2 / (exp(C3 - 2 * x) - 1)</syntaxhighlight>.
Solving the equation symbolically in MATLAB, by running
syms y(x);
equation = (diff(y) == (2 - y) * y);
% solve the equation for a general solution symbolically
y_general = dsolve(equation);
we obtain two equilibrium solutions, y=0 and y=2, and a third solution involving an unknown constant C_3,
-2 / (exp(C3 - 2 * x) - 1).
在 MATLAB 中，通过运行系统 y (x) ，方程 = (diff (y) = = (2-y) * y) ,% 解一般解的方程，得到了两个平衡解，y = 0和 y = 2，第三个解涉及一个未知常数 C _ 3，-2/(exp (C3-2 * x)-1)。
Picking up some specific values for the [[initial condition]], we can add the plot of several solutions
[[File:Slop field with isoclines and solutions.png|thumb|Slope field with isoclines and solutions]]
<syntaxhighlight lang="matlab">
% solve the initial value problem symbolically
% for different initial conditions
y1 = dsolve(equation, y(1) == 1); y2 = dsolve(equation, y(2) == 1);
y3 = dsolve(equation, y(3) == 1); y4 = dsolve(equation, y(1) == 3);
y5 = dsolve(equation, y(2) == 3); y6 = dsolve(equation, y(3) == 3);
% plot the solutions
ezplot(y1, [0 6]); ezplot(y2, [0 6]); ezplot(y3, [0 6]);
ezplot(y4, [0 6]); ezplot(y5, [0 6]); ezplot(y6, [0 6]);
title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')
legend('Slope field', 'Isoclines', 'Solutions y_{1..6}');
text([1 2 3], [1 1 1], strcat('\leftarrow', {'y_1', 'y_2', 'y_3'}));
text([1 2 3], [3 3 3], strcat('\leftarrow', {'y_4', 'y_5', 'y_6'}));
grid on;
</syntaxhighlight>
Picking up some specific values for the initial condition, we can add the plot of several solutions
thumb|Slope field with isoclines and solutions
% solve the initial value problem symbolically
% for different initial conditions
y1 = dsolve(equation, y(1) == 1); y2 = dsolve(equation, y(2) == 1);
y3 = dsolve(equation, y(3) == 1); y4 = dsolve(equation, y(1) == 3);
y5 = dsolve(equation, y(2) == 3); y6 = dsolve(equation, y(3) == 3);
% plot the solutions
ezplot(y1, [0 6]); ezplot(y2, [0 6]); ezplot(y3, [0 6]);
ezplot(y4, [0 6]); ezplot(y5, [0 6]); ezplot(y6, [0 6]);
title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')
legend('Slope field', 'Isoclines', 'Solutions y_{1..6}');
text([1 2 3], [1 1 1], strcat('\leftarrow', {'y_1', 'y_2', 'y_3'}));
text([1 2 3], [3 3 3], strcat('\leftarrow', {'y_4', 'y_5', 'y_6'}));
grid on;
为初始条件选取一些特定的值，我们可以添加几个解的图具有等倾线和解的斜率场符号化地解决了不同初始条件下的初值问题: y1 = dsol (方程，y (1) = = 1) ; y2 = dsol (方程，y (2) = = 1) ; y3 = dsol (方程，y (3) = = 1) ; y4 = dsol (方程，y (1) = = 3) ;Y5 = dsol (方程，y (2) = = 3) ; y6 = dsol (方程，y (3) = = 3) ;% 绘制解 ezplot (y1，[06]) ;Ezplot (y2，[06]) ; ezplot (y3，[06]) ; ezplot (y4，[06]) ; ezplot (y5，[06]) ; ezplot (y6，[06]) ; title (“斜率场，等倾线和 f (x，y) = (2-y) y”)图例(“斜率场”，“等倾线”，“解 y _ {1。. 6}’) ; text ([123] ，[111] ，strcat (‘ left tarrow’，{‘ y _ 1’，‘ y _ 2’，‘ y _ 3’}) ; text ([123] ，[333] ，strcat (‘ left tarrow’，{‘ y _ 4’，‘ y _ 5’，‘ y _ 6’})) ;
== Qualitative analysis ==
Autonomous systems can be analyzed qualitatively using the [[phase space]]; in the one-variable case, this is the [[phase line (mathematics)|phase line]].
Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line.
自治系统可以用相空间进行定性分析; 在单变量情况下，这是相线。
== Solution techniques ==
== Solution techniques ==
= = 解决方法 = =
The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order <math>n</math> is equivalent to an <math>n</math>-dimensional first-order system (as described in [[Ordinary differential equation#Reduction to a first-order system|reduction to a first-order system]]), but not necessarily vice versa.
The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order n is equivalent to an n-dimensional first-order system (as described in reduction to a first-order system), but not necessarily vice versa.
以下技巧适用于一维自治微分方程。任何一维 n 阶方程都等价于 n 维一阶系统(如一阶系统的简化所描述的) ，但不一定相反。
=== First order ===
=== First order ===
= = 第一顺序 = = =
The first-order autonomous equation
<math display="block">\frac{dx}{dt} = f(x)</math>
is [[Examples of differential equations#Separable first order linear ordinary differential equations|separable]], so it can easily be solved by rearranging it into the integral form
<math display="block">t + C = \int \frac{dx}{f(x)}</math>
The first-order autonomous equation
\frac{dx}{dt} = f(x)
is separable, so it can easily be solved by rearranging it into the integral form
t + C = \int \frac{dx}{f(x)}
一阶自治方程 frac { dx }{ dt } = f (x)是可分的，因此可以通过将它重新排列成积分形式 t + C = int frac { dx }{ f (x)}来求解
=== Second order ===
=== Second order ===
= = 第二阶段 = = =
The second-order autonomous equation
<math display="block">\frac{d^2x}{dt^2} = f(x, x')</math>
is more difficult, but it can be solved<ref>{{cite book |last=Boyce |first=William E. |author2=Richard C. DiPrima | title=Elementary Differential Equations and Boundary Volume Problems |edition=8th |year=2005 |publisher=John Wiley & Sons | isbn=0-471-43338-1 |page=133}}</ref> by introducing the new variable
<math display="block">v = \frac{dx}{dt}</math>
and expressing the [[second derivative]] of <math>x</math> via the [[chain rule]] as
<math display="block">\frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx}</math>
so that the original equation becomes
<math display="block">v\frac{dv}{dx} = f(x, v)</math>
which is a first order equation containing no reference to the independent variable <math>t</math>. Solving provides <math>v</math> as a function of <math>x</math>. Then, recalling the definition of <math>v</math>:
The second-order autonomous equation
\frac{d^2x}{dt^2} = f(x, x')
is more difficult, but it can be solved by introducing the new variable
v = \frac{dx}{dt}
and expressing the second derivative of x via the chain rule as
\frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx}
so that the original equation becomes
v\frac{dv}{dx} = f(x, v)
which is a first order equation containing no reference to the independent variable t. Solving provides v as a function of x. Then, recalling the definition of v:
二阶自治方程 frac { d ^ 2x }{ dt ^ 2} = f (x，x’)比较难,但是可以通过引入新的变量 v = frac { dx }{ dt } ，并通过链式规则将 x 的二阶导数表示为 frac { d ^ 2x }{ dt ^ 2} = frac { dv }{ dt } = frac { dx }{ dt } frac { dv }{ dx } = v 来求解使得原方程变成 v，这是一个一阶方程，不包含独立变量 t，求解提供 v 作为 x 的函数。然后，回顾 v 的定义:
<math display="block">\frac{dx}{dt} = v(x) \quad \Rightarrow \quad t + C = \int \frac{d x}{v(x)} </math>
\frac{dx}{dt} = v(x) \quad \Rightarrow \quad t + C = \int \frac{d x}{v(x)}
Frac { dx }{ dt } = v (x)四边形右边四边形 t + C = int frac { dx }{ v (x)}
which is an implicit solution.
which is an implicit solution.
这是一个隐含的解决方案。
====Special case: {{math|1=''x''″ = ''f''(''x'')}}====
====Special case: ====
= = = = 特殊情况: = = = =
The special case where <math>f</math> is independent of <math>x'</math>
The special case where f is independent of x'
F 独立于 x’的特殊情况
<math display="block">\frac{d^2 x}{d t^2} = f(x)</math>
\frac{d^2 x}{d t^2} = f(x)
Frac { d ^ 2 x }{ d t ^ 2} = f (x)
benefits from separate treatment.<ref>{{cite web |title=Second order autonomous equation |url=https://eqworld.ipmnet.ru/en/solutions/ode/ode0301.pdf |website=[[Eqworld]] |access-date=28 February 2021 |language=en |format=pdf}}</ref> These types of equations are very common in [[classical mechanics]] because they are always [[Hamiltonian system]]s.
benefits from separate treatment. These types of equations are very common in classical mechanics because they are always Hamiltonian systems.
分开治疗的好处。这类方程在经典力学中很常见，因为它们总是哈密顿系统。
The idea is to make use of the identity
The idea is to make use of the identity
这个想法就是利用这个身份
<math display="block">\frac{d x}{d t} = \left(\frac{d t}{d x}\right)^{-1}</math>
\frac{d x}{d t} = \left(\frac{d t}{d x}\right)^{-1}
Frac { d x }{ d t } = left (frac { d t }{ d x } right) ^ {-1}
which follows from the [[chain rule]], barring any issues due to [[division by zero]].
which follows from the chain rule, barring any issues due to division by zero.
它遵循链式规则，排除任何因除以零而引起的问题。
By inverting both sides of a first order autonomous system, one can immediately integrate with respect to <math>x</math>:
By inverting both sides of a first order autonomous system, one can immediately integrate with respect to x:
通过反转一阶自治系统的两边，我们可以立即对 x 进行积分:
<math display="block">\frac{d x}{d t} = f(x) \quad \Rightarrow \quad \frac{d t}{d x} = \frac{1}{f(x)} \quad \Rightarrow \quad t + C = \int \frac{dx}{f(x)}</math>
\frac{d x}{d t} = f(x) \quad \Rightarrow \quad \frac{d t}{d x} = \frac{1}{f(x)} \quad \Rightarrow \quad t + C = \int \frac{dx}{f(x)}
Frac { d x }{ d t } = f (x) quad 右箭头四边形{ d t }{ d x } = frac {1}{ f (x)}四边形右箭头四边形 t + C = int frac { dx }{ f (x)}
which is another way to view the separation of variables technique. Can we do something like this with higher order equations? The answer is yes for second order equations, but there's more work to do. The second derivative must be expressed as a derivative with respect to <math>x</math> instead of <math>t</math>:
which is another way to view the separation of variables technique. Can we do something like this with higher order equations? The answer is yes for second order equations, but there's more work to do. The second derivative must be expressed as a derivative with respect to x instead of t:
这是另一种看待分离变量法技术的方式。我们能用高阶方程做这样的事情吗？对于二阶方程，答案是肯定的，但是还有更多的工作要做。二阶导数必须表示为关于 x 的导数，而不是 t:
<math display="block">\begin{align}
\frac{d^2 x}{d t^2} &= \frac{d}{d t}\left(\frac{d x}{d t}\right)
= \frac{d}{d x}\left(\frac{d x}{d t}\right) \frac{d x}{d t} \\[4pt]
&= \frac{d}{d x}\left(\left(\frac{d t}{d x}\right)^{-1}\right) \left(\frac{d t}{d x}\right)^{-1} \\[4pt]
&= - \left(\frac{d t}{d x}\right)^{-2} \frac{d^2 t}{d x^2} \left(\frac{d t}{d x}\right)^{-1}
= - \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} \\[4pt]
&= \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right)
\end{align}</math>
\begin{align}
\frac{d^2 x}{d t^2} &= \frac{d}{d t}\left(\frac{d x}{d t}\right)
= \frac{d}{d x}\left(\frac{d x}{d t}\right) \frac{d x}{d t} \\[4pt]
&= \frac{d}{d x}\left(\left(\frac{d t}{d x}\right)^{-1}\right) \left(\frac{d t}{d x}\right)^{-1} \\[4pt]
&= - \left(\frac{d t}{d x}\right)^{-2} \frac{d^2 t}{d x^2} \left(\frac{d t}{d x}\right)^{-1}
= - \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} \\[4pt]
&= \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right)
\end{align}
\begin{align}
\frac{d^2 x}{d t^2} &= \frac{d}{d t}\left(\frac{d x}{d t}\right)
= \frac{d}{d x}\left(\frac{d x}{d t}\right) \frac{d x}{d t} \\[4pt]
&= \frac{d}{d x}\left(\left(\frac{d t}{d x}\right)^{-1}\right) \left(\frac{d t}{d x}\right)^{-1} \\[4pt]
&= - \left(\frac{d t}{d x}\right)^{-2} \frac{d^2 t}{d x^2} \left(\frac{d t}{d x}\right)^{-1}
= - \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} \\[4pt]
&= \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right)
\end{align}
To reemphasize: what's been accomplished is that the second derivative with respect to <math>t</math> has been expressed as a derivative of <math>x</math>. The original second order equation can now be integrated:
To reemphasize: what's been accomplished is that the second derivative with respect to t has been expressed as a derivative of x. The original second order equation can now be integrated:
再强调一下，已经完成的是，关于 t 的二阶导数已经表示为 x 的导数。原来的二阶方程现在可以积分了:
<math display="block">\begin{align}
\frac{d^2 x}{d t^2} &= f(x) \\
\frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) &= f(x) \\
\left(\frac{d t}{d x}\right)^{-2} &= 2 \int f(x) dx + C_1 \\
\frac{d t}{d x} &= \pm \frac{1}{\sqrt{2 \int f(x) dx + C_1}} \\
t + C_2 &= \pm \int \frac{dx}{\sqrt{2 \int f(x) dx + C_1}}
\end{align}</math>
\begin{align}
\frac{d^2 x}{d t^2} &= f(x) \\
\frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) &= f(x) \\
\left(\frac{d t}{d x}\right)^{-2} &= 2 \int f(x) dx + C_1 \\
\frac{d t}{d x} &= \pm \frac{1}{\sqrt{2 \int f(x) dx + C_1}} \\
t + C_2 &= \pm \int \frac{dx}{\sqrt{2 \int f(x) dx + C_1}}
\end{align}
\begin{align}
\frac{d^2 x}{d t^2} &= f(x) \\
\frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) &= f(x) \\
\left(\frac{d t}{d x}\right)^{-2} &= 2 \int f(x) dx + C_1 \\
\frac{d t}{d x} &= \pm \frac{1}{\sqrt{2 \int f(x) dx + C_1}} \\
t + C_2 &= \pm \int \frac{dx}{\sqrt{2 \int f(x) dx + C_1}}
\end{align}
This is an implicit solution. The greatest potential problem is inability to simplify the integrals, which implies difficulty or impossibility in evaluating the integration constants.
This is an implicit solution. The greatest potential problem is inability to simplify the integrals, which implies difficulty or impossibility in evaluating the integration constants.
这是一个隐式的解决方案。最大的潜在问题是无法简化积分，这意味着在求积分常数方面存在困难或不可能。
====Special case: {{math|1=''x''″ = ''x''′<sup>''n''</sup> ''f''(''x'')}}====
====Special case: ====
= = = = 特殊情况: = = = =
Using the above approach, we can extend the technique to the more general equation
Using the above approach, we can extend the technique to the more general equation
利用上述方法，我们可以将该技术推广到更一般的方程
<math display="block">\frac{d^2 x}{d t^2} = \left(\frac{d x}{d t}\right)^n f(x)</math>
\frac{d^2 x}{d t^2} = \left(\frac{d x}{d t}\right)^n f(x)
Frac { d ^ 2 x }{ d t ^ 2} = left (frac { d x }{ d t } right) ^ n f (x)
where <math>n</math> is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of <math>x'</math>. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:
where n is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of x'. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:
其中 n 是一个不等于2的参数。这将工作，因为二阶导数可以写成一个包含 x’的幂的形式。重写第二个导数，重新排列，并将左边表示为一个导数:
<math display="block">\begin{align}
&- \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \left(\frac{d t}{d x}\right)^{-n} f(x) \\[4pt]
&- \left(\frac{d t}{d x}\right)^{n - 3} \frac{d^2 t}{d x^2} = f(x) \\[4pt]
&\frac{d}{d x}\left(\frac{1}{2 - n}\left(\frac{d t}{d x}\right)^{n - 2}\right) = f(x) \\[4pt]
&\left(\frac{d t}{d x}\right)^{n - 2} = (2 - n) \int f(x) dx + C_1 \\[2pt]
&t + C_2 = \int \left((2 - n) \int f(x) dx + C_1\right)^{\frac{1}{n - 2}} dx
\end{align}</math>
\begin{align}
&- \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \left(\frac{d t}{d x}\right)^{-n} f(x) \\[4pt]
&- \left(\frac{d t}{d x}\right)^{n - 3} \frac{d^2 t}{d x^2} = f(x) \\[4pt]
&\frac{d}{d x}\left(\frac{1}{2 - n}\left(\frac{d t}{d x}\right)^{n - 2}\right) = f(x) \\[4pt]
&\left(\frac{d t}{d x}\right)^{n - 2} = (2 - n) \int f(x) dx + C_1 \\[2pt]
&t + C_2 = \int \left((2 - n) \int f(x) dx + C_1\right)^{\frac{1}{n - 2}} dx
\end{align}
开始{ Alliance } &-left (frac { d t }{ d x } right) ^ {-3} frac { d ^ 2 t }{ d x ^ 2} = left (frac { d t }{ d x } right) ^ {-n } f (x)[4 pt ] &-left (frac { d t }{ d x } right) ^ { n-3} frac { d ^ 2 t }{ d x ^ 2} = f (x)[4 pt ] & frac{ d }{ d x } left (frc {1}{2-n } left (frc { d t }{ d x } right)(frc { d t }{ d x } right) ^ { n-2} right) = f (x)[4 pt ]& left (frac { d t }{ d x } right) ^ { n-2} = (2-n) int f (x) dx + C _ 1[2 pt ] & t + C _ 2 = int left ((2-n) int f (x) dx + C _ 1 right) ^ { frac {1}{ n-2} dx end { lig}
The right will carry +/− if <math>n</math> is even. The treatment must be different if <math>n = 2</math>:
The right will carry +/− if n is even. The treatment must be different if n = 2:
如果 n 是偶数，则右边将进位 +/-。如果 n = 2，则处理方法必须不同:
<math display="block">\begin{align}
- \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2} &= f(x) \\
-\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) &= f(x) \\
\frac{d t}{d x} &= C_1 e^{-\int f(x) dx} \\
t + C_2 &= C_1 \int e^{-\int f(x) dx} dx
\end{align}</math>
\begin{align}
- \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2} &= f(x) \\
-\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) &= f(x) \\
\frac{d t}{d x} &= C_1 e^{-\int f(x) dx} \\
t + C_2 &= C_1 \int e^{-\int f(x) dx} dx
\end{align}
\begin{align}
- \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2} &= f(x) \\
-\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) &= f(x) \\
\frac{d t}{d x} &= C_1 e^{-\int f(x) dx} \\
t + C_2 &= C_1 \int e^{-\int f(x) dx} dx
\end{align}
=== Higher orders ===
=== Higher orders ===
= = 更高的命令 = = =
There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance [[linear differential equation|linearity]] or dependence of the right side of the equation on the dependent variable only<ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0503.pdf Third order autonomous equation] at [[eqworld]].</ref><ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0506.pdf Fourth order autonomous equation] at [[eqworld]].</ref> (i.e., not its derivatives). This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly [[chaos theory|chaotic]] behavior such as the [[Lorenz attractor]] and the [[Rössler attractor]].
There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable onlyThird order autonomous equation at eqworld.Fourth order autonomous equation at eqworld. (i.e., not its derivatives). This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.
没有类似的方法来求解三阶或更高阶的自治方程。这类方程只有在具有其他简化性质时才能精确求解，例如方程的右侧在等式世界中只有三阶自治方程与因变量的线性关系或依赖关系。等式世界上的四阶自治方程。(即不是其衍生工具)。考虑到三维非线性自治系统可以产生真正的混沌行为，如洛伦兹吸引子和若斯叻吸引子，这不足为奇。
With this mentality, it also isn't too surprising that general non-autonomous equations of second order can't be solved explicitly, since these can also be chaotic (an example of this is a periodically forced pendulum<ref>{{cite book | author = Blanchard | author2 = Devaney | author2-link = Robert L. Devaney | author3 = Hall | title = Differential Equations | publisher = Brooks/Cole Publishing Co | year = 2005 | pages = 540–543 | isbn = 0-495-01265-3}}</ref>).
With this mentality, it also isn't too surprising that general non-autonomous equations of second order can't be solved explicitly, since these can also be chaotic (an example of this is a periodically forced pendulum).
按照这种思路，一般的二阶非自治方程不能明确地求解也就不足为奇了，因为这些方程也可能是混沌的(周期性强迫摆就是一个例子)。
===Multivariate case===
{{main|Matrix differential equation}}
Now we have <math>\mathbf x'(t) = A \mathbf x(t)</math>, where <math>\mathbf x(t)</math> is an <math>n</math>-dimensional column vector dependent on <math>t</math>.
Now we have \mathbf x'(t) = A \mathbf x(t), where \mathbf x(t) is an n-dimensional column vector dependent on t.
= = = 多变量情况 = = = 现在我们有 mathbf x’(t) = A mathbf x (t) ，其中 mathbf x (t)是依赖于 t 的 n 维列向量。
The solution is <math>\mathbf x(t) = e^{A t} \mathbf c</math> where <math>\mathbf c</math> is an <math>n \times 1</math> constant vector.<ref>{{cite web |title=Method of Matrix Exponential |url=https://www.math24.net/method-matrix-exponential/ |website=Math24 |access-date=28 February 2021}}</ref>
The solution is \mathbf x(t) = e^{A t} \mathbf c where \mathbf c is an n \times 1 constant vector.
解是 mathbf x (t) = e ^ { A t } mathbf c 其中 mathbf c 是 n 乘以1的常量向量。
=== Solutions of Finite Duration ===
=== Solutions of Finite Duration ===
= = 有限持续时间的解
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<ref>{{cite journal |author = Vardia T. Haimo |title = Finite Time Differential Equations |year = 1985 |doi = 10.1109/CDC.1985.268832 |url=https://ieeexplore.ieee.org/document/4048613}}</ref> meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. This finite-duration solutions can't be analytical functions on the whole real line, and because they will being non-Lipschitz function at the ending time, they don´t stand uniqueness of solutions of Lipschitz differential equations.
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. This finite-duration solutions can't be analytical functions on the whole real line, and because they will being non-Lipschitz function at the ending time, they don´t stand uniqueness of solutions of Lipschitz differential equations.
对于非线性自治微分方程，在某些条件下有可能发展出有限持续时间的解，这意味着从系统自身的动力学出发，系统将在结束时达到零值，并在结束后永远保持在零值。这类有限持续时间解不能是整条实线上的解析函数，由于它们在结束时是非 Lipschitz 函数，所以它们不具有 Lipschitz 微分方程解的唯一性。
As example, the equation:
:<math>y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math>
Admits the finite duration solution:
:<math>y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2</math>
As example, the equation:
:y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1
Admits the finite duration solution:
:y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2
例如，方程: : y’=-text { sgn }(y) sqrt { | y | } ，，y (0) = 1承认有限持续时间解: : y (x) = frac {1}{4} left (1-frac { x }{2} + left | 1-frac { x }{2} right | right) ^ 2
== See also ==
== See also ==
= = 另见 = =
* [[Time-invariant system]]
* [[Non-autonomous system (mathematics)]]
* Time-invariant system
* Non-autonomous system (mathematics)
* 时不变系统
* 非自主系统(数学)
== References ==
{{Reflist}}
{{Differential equations topics}}
{{Authority control}}
{{DEFAULTSORT:Autonomous System (Mathematics)}}
[[Category:Differential equations]]
[[Category:Dynamical systems]]
[[Category:Ordinary differential equations]]
Category:Differential equations
Category:Dynamical systems
Category:Ordinary differential equations
类别: 微分方程类别: 动力系统类别: 常微分方程
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<noinclude>
<small>This page was moved from [[wikipedia:en:Autonomous system]]. Its edit history can be viewed at [[自治系统/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Autonomous system (mathematics)]]. Its edit history can be viewed at [[自治系统/edithistory]]</small></noinclude>
[[Category:待整理页面]]
[[Category:待整理页面]]