# 李纳德方程

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.

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.

# = 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

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

## Liénard system

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

$\displaystyle{ F(x) := \int_0^x f(\xi) d\xi }$
$\displaystyle{ x_1:= x }$
$\displaystyle{ x_2:={dx \over dt} + F(x) }$

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

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

\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}

is called a Liénard system.

is called a Liénard system.

Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution $\displaystyle{ v = {dx \over dt} }$ 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:

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

# = 实例 =

The Van der Pol oscillator

$\displaystyle{ {d^2x \over dt^2}-\mu(1-x^2){dx \over dt} +x= 0 }$
{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 $\displaystyle{ f(x) }$ at small $\displaystyle{ |x| }$ and positive $\displaystyle{ f(x) }$ otherwise. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if $\displaystyle{ f(x) }$ 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

## 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;
• $\displaystyle{ \lim_{x \to \infty} F(x) := \lim_{x \to \infty} \int_0^x f(\xi) d\xi\ = \infty; }$
• 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.

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.

• 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。

• 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 =

1. Liénard, A. (1928) "Etude des oscillations entretenues," Revue générale de l'électricité 23, pp. 901–912 and 946–954.
2. 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
3. For a proof, see Perko, Lawrence (1991). Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 254–257. ISBN 0-387-97443-1.