Samiksha Jaiswal (Editor)

Liénard equation

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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.

Contents

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.

Definition

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

d 2 x d t 2 + f ( x ) d x d t + 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 ) := ∫ 0 x f ( ξ ) d ξ x 1 := x x 2 := d x d t + F ( x )

then

[ x ˙ 1 x ˙ 2 ] = h ( x 1 , x 2 ) := [ x 2 − F ( x 1 ) − g ( x 1 ) ]

is called a Liénard system.

Alternatively, since Liénard equation itself is also an autonomous differential equation, the substitution v = d x d t leads the Liénard equation to become a first order differential equation:

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

which belongs to Abel equation of the second kind.

Example

The Van der Pol oscillator

d 2 x d t 2 − μ ( 1 − x 2 ) d x d t + x = 0

is a Liénard equation. The solution of 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. Van der Pol equation hasn’t exact, analytic solution. Such solution for limit cycle exists if f ( x ) is constant piece-wise function.

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;
  • lim x → ∞ F ( x ) := lim x → ∞ ∫ 0 x f ( ξ ) d ξ   = ∞ ;
  • 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.

    • References

    Liénard equation Wikipedia
    (Text) CC BY-SA


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