To classify the critical point of the plane autonomous system
corresponding to the second order equation:
x''+ μ(x^2-1)x' ̇+x = 0
in terms of μ where μ is a real valued constant. How to rewrite the equation in matrix notation to change it as a set of first order differential equations. Step by step.

Question

To classify the critical point of the plane autonomous  system
 corresponding  to the second order equation:
x''+ μ(x^2-1)x' ̇+x = 0        
   in terms of  μ where μ is a real valued constant.  How to rewrite the equation in matrix notation to change it as a set of first order differential equations. Step by step.

experimentx · Answer

Hell man ... this is quite difficult

anonymous · Answer

Not really here is how

experimentx · Answer

x^2-1 is bugging me

anonymous · Answer

Define new var y = x' then your eqn is equivalent to

anonymous · Answer

Wait one more variable
u = y' = x''

anonymous · Answer

Now you have a system
x'=y
y'=u
u +mu(x^2-1)y +x =0

anonymous · Answer

It is obviously a NON-linear system

experimentx · Answer

yeah that's the difficulty

anonymous · Answer

Now, by definition critical point is where the coefficients are all equal to zero

anonymous · Answer

Sorry, no u needed
x'=y
y'=-mu(x^2-1) - x

anonymous · Answer

Critical pont is where y=o and -Mu*(x^2-1) -x =0

anonymous · Answer

Many thanks Mikael. I will post another question on unstable critical points.

experimentx · Answer

sorry man ... can't help you at the moment. 
haven't worked with non linear system.