mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, more specifically in the study of

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

s and differential equations, a Liénard equation is a type of second-order ordinary differential equation named after the French physicist

Alfred-Marie Liénard Alfred-Marie Liénard (2 April 1869 in Amiens Amiens (English: or ; ; , or ) is a city and Communes of France, commune in northern France, located north of Paris and south-west of Lille. It is the capital of the Somme (department), Somm ...

. During the development of

radio Radio is the technology of communicating using radio waves. Radio waves are electromagnetic waves of frequency between 3 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmitter connec ...

and

vacuum tube A vacuum tube, electron tube, thermionic 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 voltage, potential difference has been applied. It ...

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 orbit In the study of dynamical systems, a homoclinic orbit is a path through phase space which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold o ...

Definition

Let and be two

continuously differentiable In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...

functions on with an

even function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

and an

odd function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...

. Then the second order

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

of the form

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

is called a Liénard equation.

Liénard system

The equation can be transformed into an equivalent two-dimensional

system of ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives o ...

s. 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 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 ...

, 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 (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives o ...

: :

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

which is an

Abel equation of the second kind In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form :y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y+f_0(x) \ ...

Example

The

Van der Pol oscillator In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-Conservative force, conservative, oscillating system with non-linear damping. It evolves in time according to the second-order ...

-\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 in general no exact, analytic solution. Such a solution for a limit cycle does exist if

f(x)

is a constant piece-wise function.

Liénard's theorem

A Liénard system has a unique and

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

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