The Van der Pol equation

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The Van der Pol equation is the second-order non-linear differential equation …

x” + εx'(x2-1)+x=0, where x=x(t), tεR, and ε>0 is a constant.

If y2(t)=tan2(t)/(atan2(t)+btan(t)+c), where a and c are >0, 0<b2<4ac, and b<0,

and if we let z(t) be a ‘test’ function defined by z(t) =|y(t)| for 0<=t<=π

and z(t) = -|y(t)| for π<=t<=2π etc. then the 2π-periodic function z(t) for certain values of a, b, and c ‘seems’ to have the right kind of shape for the periodic solution to the Van der Pol equation – ignoring any scaling in the ‘t’ direction.

For example try a=0.28, b=-0.02 and c=0.004. This gives the following shape …


Note that the ‘first’ turning point in the first quadrant has t=tan -1(-2c/b) while the z coordinate is z(t)=2√c/(√(4ac-b2).

If the amplitude is 2 then b2 = c(4a-1). This requires that a>1/4.

Note also that if b=0 and a=c>0 then z(t) is just a sine curve.

The expression for y(t) suggests that the Van der Pol equation might be written as

x² = 1 -(x + x”)/(εx’) for the purposes of obtaining a sequence of approximations to x(t).

If in the RHS of x² = 1 – (x + x”)/(εx’) we let  x = λsin(μt) then the LHS does not give an expression like that for y2(t) as above.

However, if we re-write the Van der Pol equation as x² = (εxx’ – xx”)/(1 + εxx’)

and let x=λsin(μt) in the RHS, then the resulting expression,

(εxx’ – xx”)/(1 + εxx’) = (λ²μ²tan²(μt) + ελ²μtan(μt))/(tan²(μt) + ελ²μtan(μt) +1),

is very like that for y2(t) as above.

If a = 1, b = 0.1 and c = 1 we get …


If a = 0.25, b = 0.2 and c = 0.05 we get …


If a = 100, b = 0.2 and c = 0.05 we get …


These shapes are very similar to those obtain by numerical solutions of the Van der Pol equation.

The Van der Pol equation in the form x² = (εxx’ – xx”)/(1 + εxx’) can be transformed a little by introducing a non-zero constant k (say).

So, it is easily checked that x² = (kεxx’ – kxx” – (k-1)x²)/(1 + kεxx’) is essentially equivalent to the Van der Pol equation and if we let x=λsin(μt) in the RHS of this equation then x² still takes the form of the ratio of 2 quadratics of the tan() function.

If x=λsin(μt) then x² = λ²tan²(μt))/(1+tan²(μt))).

Transformation of x” + εx'(x2-1)+x=0.  First multiply through by x to get  xx” + εx’x(x2-1)+x²=0.

If we let y = x² then y’ = 2xx’ and so (y’)² = 4y(x’)².  Also, y” = 2(xx” + (x’)²).  Hence, xx’ = y’/2 and xx” = y”/2 – (x’)² = y”/2 -(y’)²/(4y).

Substituting into xx” + εx’x(x2-1)+x²=0 we get 2yy” – (y’)² +2yy’ε(y – 1) +4y² = 0.

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