[Math] van der pol equation

ordinary differential equations

Consider the van der Pol equation below:

$(x'')+a(x^2-1)(x')+(x)=0$

I need to :

  1. Find an equilibrium point and linearize this equation near it

  2. Find solutions of the linearized equation depending on a

Best Answer

enter image description here $$ \frac{d^2x}{dt^2} - a(1 - x^2)\frac{dx}{dt}+ x = 0 $$

let $\frac{dx}{dt} = y $ . which yields the two dimensional first order system:

$$\frac{dx}{dt} = y $$

$$\frac{dy}{dt} = -x + a(1 - x^2)y $$

The linearized system is easy to write down in this case:

$$\frac{dx}{dt} = y $$

$$\frac{dy}{dt} = -x + ay $$

clearly (0,0) is the equilibrium point

a plot of the equation near the origin with a as parameter . (You can play around with this quite a bit). The red solution curve is the Van der Pol Equation, the blue solution curve is the linearized system. It is spiraling out from the origin , but without a limit cycle.

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