Equilibrium solutions in double well potential duffing oscillator

[Ratpigeon]

Homework Statement

I am trying to show that for a duffing oscillator described by
x''+2g x'+ax+bx^3=0
with a<0, b>0
the equilibria at $x=+- \sqrt{-a/b}$ are stable

Homework Equations

I used coupled equations, and the characteristic equation of a linear system

The Attempt at a Solution

Coupled equations, x and x' related by
$d/dt(x,x')=[[0,1],[-a-bx^2, -2g]](x,x')$

Setting x=+-$\sqrt{-a/b}$ gives the characteristic is

$\lambda^2+2g \lambda+a+b(-a/b)=0$
But this can't be right since, i know from literature that the eigens are
$\lambda 1=-g + \sqrt{g^2+2a}$
$\lambda 2=-g- \sqrt{-g^2+2a}$
and this gives eigens 0 and -2g

 

[voko]

After you have reduced the equation to the system and found the equilibrium point, you have to linearize the system about the equilibrium point, and then find the eigenvalue of the linear system.

 

[TSny]

Since no one has chimed in yet, I'll throw in my 2 cents (and it might not be worth even that). I believe that one way to examine the behavior near an equilibrium point is to "linearize" the equation in the neighborhood of the equilibrium point. So, you could let z = x-x_o where x_o is an equilibrium point. Rewrite the differential equation in terms of z but keep only terms linear in z. Then find the characteristic equation for this linearized differential equation.

[Edit: Wow, I did not see voko's response. Sorry.]

 

[Ratpigeon]

I'd done all the linearising stuff - I found my problem; I needed to use a Jacobean matrix, instead of just a system

 

[paranormal]

which are not the same as above
Your attempt at a solution is on the right track. However, there are a few errors in your calculations. Firstly, the characteristic equation should be:

\lambda^2+2g \lambda+a+b(-a/b)=0

This can be solved using the quadratic formula to give:

\lambda_{1,2} = -g \pm \sqrt{g^2-2a}

This matches the literature you have referenced.

Additionally, the eigenvalues of a matrix are not necessarily the same as the solutions to a characteristic equation. The eigenvalues of the matrix you have used are:

\lambda_{1,2} = -g \pm \sqrt{g^2+2a}

This is incorrect and does not match the characteristic equation.

To show that the equilibria at x=+- \sqrt{-a/b} are stable, you can use the Routh-Hurwitz stability criterion. This states that if all the coefficients in the characteristic equation have the same sign, then all the roots have negative real parts and the system is stable. In this case, all the coefficients are positive, so the equilibria are stable. Additionally, you can also use the Lyapunov stability theorem to prove stability at these equilibria.