Ordinary differential equations Qualitative theory Example

Van der Pol oscillator

Builds on Liénard equation Liénard's theorem Harmonic oscillator Linearisation at an equilibrium

Example

Let $\mu > 0$ . The van der Pol oscillator is the equation $x'' - \mu (1 - x^2)\, x' + x = 0.$ It is a Liénard equation with $f(x) = -\mu (1 - x^2)$ and $g(x) = x$ , so $F(x) = \mu (x^3/3 - x)$ . The function $F$ has the single positive zero $\sqrt{3}$ , is negative on $(0, \sqrt{3})$ , and is increasing on $(1, \infty)$ . By Liénard's theorem, the equation has exactly one limit cycle, and it is stable.

Remarks

The only equilibrium is the origin. The linearisation there is $x'' - \mu x' + x = 0$ , whose characteristic roots $\frac{1}{2}\left(\mu \pm \sqrt{\mu^2 - 4}\right)$ have positive real part. So the origin is unstable, and every other solution tends to the limit cycle. For $\mu = 0$ the equation is the harmonic oscillator. For small $\mu > 0$ , the limit cycle is close to the circle of radius $2$ in the $(x, x')$ -plane. For large $\mu$ , the oscillation is a relaxation oscillation: slow drifts alternate with fast jumps.

Reading path

To understand this page from the ground up, read these 46 pages in order. The foundations come first.

Autonomous system concept
Complex numbers object
Continuously differentiable function concept
Differential equation concept
Even and odd functions concept
Function concept
Interval concept
Limit cycle concept
Euler's formula statement
Ordinary differential equation definition
Lipschitz condition definition
Grönwall's inequality statement
Order of a differential equation definition
Solution of an ordinary differential equation definition
Autonomous system definition
Initial value problem definition
Linear differential equation of order n definition
Equilibrium point definition
System of linear differential equations definition
Uniqueness of solutions for Lipschitz vector fields statement
Homogeneous linear equation definition
Linear equation with constant coefficients definition
Homogeneous linear system with constant coefficients definition
Maximal solution definition
Superposition principle statement
Characteristic equation definition
Matrix exponential as the solution of a linear system statement
Flow of a vector field definition
Solution space of a homogeneous linear equation has dimension n statement
Lyapunov stability definition
Trajectory definition
Fundamental system of solutions definition
Asymptotic stability definition
Periodic orbit definition
Phase plane concept
Solutions for distinct real roots statement
Stability of linear systems via eigenvalues statement
Limit cycle definition
Solutions for repeated roots statement
Linearisation at an equilibrium concept
Poincaré–Bendixson theorem statement
Solutions for complex conjugate roots statement
Harmonic oscillator example
Liénard's theorem statement
Liénard equation example
Van der Pol oscillator example

Tag	`WE89NFZ84Q`
URI	`m2://WE89NFZ84Q`
Kind	example
Writing	W0 stub
MSC	34C15, 34C26, 34C07
Illustrates	Limit cycle
Uses	Liénard equation, Liénard's theorem, Harmonic oscillator, Linearisation at an equilibrium
Changed	2 November 2026

Van der Pol oscillator

Example

Remarks

Reading path

Further reading

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