Part 2.3: Linearization

The van der Pol equaionis a second order nonlinear differential equation and an ordinary differentialequation with nonlinear damping. It is defined as

\[ \frac{{\text d}^2 y}{{\text d}t^2} -\mu \left( 1 - y^2 \right) \frac{{\text d}y}{{\text d}t} + y =0 \]
Here μ is a constant that determines the strength of the damping on the oscillations. 

with(DEtools): 
  DE1 := diff(y(t),t) = z(t);
  DE2 := diff(z(t),t) = -y(t)*cos(y(t));
phaseportrait([DE1,DE2],[y,z],t=-5..5,[[y(0)=1,z(0)=0],[y(0)=0,z(0)=2],[y(0)=0,z(0)=-2]],y=-Pi..Pi,z=-3..3,color=aquamarine,linecolor=blue);


with(DEtools): 
  DE1 := diff(y(t),t) = z(t);
  DE2 := diff(z(t),t) = -y(t)*cos(y(t));
 phaseportrait([DE1,DE2],[y,z],t=-5..5,[[y(0)=1,z(0)=0],[y(0)=0,z(0)=2],[y(0)=0,z(0)=-2]],y=-Pi..Pi,z=-3..3,color=aquamarine,linecolor=blue);