General analytical method for determining whether a system of ODEs exhibits damped oscillations?

Question

Does the following method suffice for proving that a system of three first-order, non-linear ODEs always exhibits damped oscillations (as opposed to undamped), using the system's characteristic polynomial? If not, why?

Method

1. Calculate the system's non-trivial equilibrium (assuming there is one).

2. Calculate the characteristic polynomial corresponding to that equilibrium. This polynomial is third order and therefore has three eigenvalues

3. Show that the characteristic polynomial's discriminate is always negative (this shows that the equilibrium exhibits oscillations, right?)

4. Show that the real part of the leading eigenvalue is always negative (this shows that said oscillations decay in amplitude toward zero, right?)

Answer 1

I am not exactly sure about the method you are describing, but here is how I would do it (for a system $\dot{x} = f(x)$):

1. Compute a fixed point $x_0$ where $f(x_0)=0$.
2. Determine the eigenvalues of $∇f(x_0)$.
3. Show that the real parts of all eigenvalues are negative (then you got a stable fixed point).
4. Show that there is at least one eigenvalue with non-zero imaginary part. Those eigenvalues always come in complex conjugate pairs and correspond to damped oscillations. If all your eigenvalues are real, you get an overdamped convergence to your fixed point instead, i.e., no oscillation.

All of this follows from the general solution of matrix differential equations, which describe the linear approximation of the dynamics around $x_0$. Of course, the dynamics may exhibit damped oscillations far from the fixed point that you cannot detect with this method.

Some comments on your method as you describe it:

• If you have three second-order differential equations, you get six first-order ones and thus your characteristic polynomial would have degree six – not three.

• Since the polynomial has degree six, it does not have a clear discriminant.