Consider the van der Pol equation below:
$(x'')+a(x^2-1)(x')+(x)=0$
I need to :
-
Find an equilibrium point and linearize this equation near it
-
Find solutions of the linearized equation depending on a
ordinary differential equations
Consider the van der Pol equation below:
$(x'')+a(x^2-1)(x')+(x)=0$
I need to :
Find an equilibrium point and linearize this equation near it
Find solutions of the linearized equation depending on a
Best Answer
$$ \frac{d^2x}{dt^2} - a(1 - x^2)\frac{dx}{dt}+ x = 0 $$
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.