In mathematics, more specifically in the study of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

s and

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s, a Liénard equation is a second order differential equation, named after the French physicist

Alfred-Marie Liénard Alfred-Marie Liénard (2 April 1869 in Amiens – 29 April 1958 in Paris), was a French physicist and engineer. He is most well known for his derivation of the Liénard–Wiechert potentials. From 1887 to 1889 Liénard was a student at the Éco ...

. During the development of

radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmi ...

and

vacuum tube A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied. The type known as ...

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 In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...

for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits.

Definition

Let ''f'' and ''g'' be two continuously differentiable functions on R, with ''g'' an

odd function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...

and ''f'' an even function. Then the second order

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

of the form :

+f(x)+g(x)=0

is called the Liénard equation.

Liénard system

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:= + F(x)

then :

\begin 
\dot_1 \\
\dot_2 
\end
= 
\mathbf(x_1, x_2) 
:= 
\begin 
x_2 - F(x_1) \\
-g(x_1)
\end

is called a Liénard system. Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution

v =

leads the Liénard equation to become a

first order differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

: :

v+f(x)v+g(x)=0

which belongs to Abel equation of the second kind.

Example

The

Van der Pol oscillator In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: :-\mu(1-x^2)+x= 0, where ''x'' is the position coordinate—which is a f ...

-\mu(1-x^2) +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

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.

Liénard's theorem

A Liénard system has a unique and stable

surrounding the origin if it satisfies the following additional properties:For a proof, see * ''g''(''x'') > 0 for all ''x'' > 0; *

\lim_ F(x) := \lim_ \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''.

Footnotes

External links