李纳德方程

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In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard.

In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equationLiénard, A. (1928) "Etude des oscillations entretenues," Revue générale de l'électricité 23, pp. 901–912 and 946–954. is a second order differential equation, named after the French physicist Alfred-Marie Liénard.

在数学中，更具体地说在动力系统和微分方程的研究中，一个 Liénard 方程，a。(1928) "Etude des oscillations entretenues," Revue générale de l'électricité 23, pp.901-912和946-954。是一种二阶微分方程，以法国物理学家阿弗雷-玛丽·黎纳命名。

During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system.

During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system.

在无线电和真空管技术的发展过程中，由于利纳德方程可以用来模拟振荡电路，因此对其进行了深入的研究。在一定的附加假设下，Liénard 定理保证了该系统极限环的唯一性和存在性。

= 定义 =

Let f and g be two continuously differentiable functions on R, with g an odd function and f an even function. Then the second order ordinary differential equation of the form

Let f and g be two continuously differentiable functions on R, with g an odd function and f an even function. Then the second order ordinary differential equation of the form

设 f 和 g 是 r 上的两个连续可微函数，g 是奇函数，f 是偶函数。然后是表单的二阶常微分方程

[math]\displaystyle{ {d^2x \over dt^2}+f(x){dx \over dt}+g(x)=0 }[/math]

{d^2x \over dt^2}+f(x){dx \over dt}+g(x)=0

{ d ^ 2x/dt ^ 2} + f (x){ dx/dt } + g (x) = 0

is called the Liénard equation.

is called the Liénard equation.

叫做 Liénard 方程。

Liénard system

The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define

[math]\displaystyle{ F(x) := \int_0^x f(\xi) d\xi }[/math]

[math]\displaystyle{ x_1:= x }[/math]

[math]\displaystyle{ x_2:={dx \over dt} + F(x) }[/math]

then

The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define

F(x) := \int_0^x f(\xi) d\xi

x_1:= x

x_2:={dx \over dt} + F(x)

then

该方程可以转化为等价的二维常微分方程组。我们定义: f (x) : = int _ 0 ^ x f (xi) d xi: x _ 1: = x: x _ 2: = { dx over dt } + f (x) then

[math]\displaystyle{ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \mathbf{h}(x_1, x_2) := \begin{bmatrix} x_2 - F(x_1) \\ -g(x_1) \end{bmatrix} }[/math]

\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \mathbf{h}(x_1, x_2)

=

\begin{bmatrix} x_2 - F(x_1) \\ -g(x_1) \end{bmatrix}

开始{ bmatrix }点{ x }1点{ x }2 end { bmatrix } = mathbf { h }(x _ 1，x _ 2) : = begin { bmatrix } x _ 2-f (x _ 1)-g (x _ 1) end { bmatrix }

is called a Liénard system.

is called a Liénard system.

被称为 Liénard 系统。

Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution [math]\displaystyle{ v = {dx \over dt} }[/math] leads the Liénard equation to become a first order differential equation:

Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution v = {dx \over dt} leads the Liénard equation to become a first order differential equation:

另外，由于 Liénard 方程本身也是一个自治微分方程，置换 v = { dx/dt }导致 Liénard 方程成为一阶微分方程:

[math]\displaystyle{ v{dv \over dx}+f(x)v+g(x)=0 }[/math]

v{dv \over dx}+f(x)v+g(x)=0

v { dv over dx } + f (x) v + g (x) = 0

which belongs to Abel equation of the second kind.

which belongs to Abel equation of the second kind.Liénard equation at eqworld.Abel equation of the second kind at eqworld.

它属于方程中的第二类 Abel 方程。方程中的 liénard 方程。方程中的第二类 Abel 方程。

= 实例 =

The Van der Pol oscillator

The Van der Pol oscillator

范德波尔振荡器

[math]\displaystyle{ {d^2x \over dt^2}-\mu(1-x^2){dx \over dt} +x= 0 }[/math]

{d^2x \over dt^2}-\mu(1-x^2){dx \over dt} +x= 0

{ d ^ 2x/dt ^ 2}-mu (1-x ^ 2){ dx/dt } + x = 0

is a Liénard equation. The solution of a Van der Pol oscillator has a limit cycle. Such cycle has a solution of a Liénard equation with negative [math]\displaystyle{ f(x) }[/math] at small [math]\displaystyle{ |x| }[/math] and positive [math]\displaystyle{ f(x) }[/math] otherwise. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if [math]\displaystyle{ f(x) }[/math] is a constant piece-wise function.

is a Liénard equation. The solution of a Van der Pol oscillator has a limit cycle. Such cycle has a solution of a Liénard equation with negative f(x) at small |x| and positive f(x) otherwise. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if f(x) is a constant piece-wise function.Pilipenko A. M., and Biryukov V. N. «Investigation of Modern Numerical Analysis Methods of Self-Oscillatory Circuits Efficiency», Journal of Radio Electronics, No 9, (2013). http://jre.cplire.ru/jre/aug13/9/text-engl.html

是一个 Liénard 方程。范德波尔振荡器的解决方案有一个极限循环。这样的循环有一个 Liénard 方程的解，f (x)在小 | x | 处是负的，f (x)是正的。方程没有精确的解析解。如果 f (x)是常数分段函数，则存在这样的极限环解。和 Biryukov v. n. “现代数值分析自振荡电路效率的研究”，《无线电电子学杂志》 ，第9期(2013)。Http://jre.cplire.ru/jre/aug13/9/text-engl.html

Liénard's theorem

A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:

• g(x) > 0 for all x > 0;
• [math]\displaystyle{ \lim_{x \to \infty} F(x) := \lim_{x \to \infty} \int_0^x f(\xi) d\xi\ = \infty; }[/math]
• F(x) has exactly one positive root at some value p, where F(x) < 0 for 0 < x < p and F(x) > 0 and monotonic for x > p.

A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:For a proof, see

• g(x) > 0 for all x > 0;
• \lim_{x \to \infty} F(x) := \lim_{x \to \infty} \int_0^x f(\xi) d\xi\ = \infty;
• F(x) has exactly one positive root at some value p, where F(x) < 0 for 0 < x < p and F(x) > 0 and monotonic for x > p.

如果 Liénard 系统满足下列附加性质，那么它在原点周围有唯一稳定的极限环: 对于证明，见

• g (x) > 0对于所有 x > 0;
• lim { x to infty } f (x) : = lim { x to infty } int _ 0 ^ x f (xi) d xi = infty;
• f (x)在某个值 p 处有一个正根，其中 f (x) < 0对于0 < x < p，f (x) > 0，p > p > p > p > p > 0。

See also

• Autonomous differential equation
• Abel equation of the second kind
• Biryukov equation

• Autonomous differential equation
• Abel equation of the second kind
• Biryukov equation

= = =

• 第二类 Biryukov 方程的自治微分方程
• Abel 方程

= 脚注 =

External links

• 模板:Springer
• 模板:PlanetMath

= = 外部链接 =

Category:Dynamical systems Category:Differential equations Category:Theorems in dynamical systems

范畴: 动力系统范畴: 微分方程范畴: 动力系统中的定理

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