I am trying to analyse the effect of adding a linear term to a nonlinear system. Specifically, for $\vec{x}=(x_1,...x_n)$, the original dynamical system is given by:
$\frac{d \vec{x}}{dt} = f(\vec{x}) \hspace{5cm} (1)$
where $f(\vec{x})$ is quadratic in the $x_i$. For this system, I know the system is neutrally stable at the fixed point. That is, it will oscillate in proportion to the initial magnitude of the disturbance.
Now I am adding a linear perturbation to this system, based on some linear operator $A$ that is full rank:
$\frac{d \vec{x}}{dt} = f(\vec{x}) + A\vec{x} \hspace{3.75cm} (2)$
I would like to try to understand what the necessary or sufficient conditions are for when this additional term damps the system, such that the oscillations decrease over time, i.e., I'd like to show under what conditions this linear term leads to stability (either local or global). Note that in general, the solution to Eqns. (1) and (2) will be different, since the original solution $\vec{x}_0$ need not be an eigenvector of $A$ with eigenvalue zero.
Are there any good tools to show that the inclusion of the linear $Ax$ term makes the fixed point stable? I have considered looking at perturbation theory, but I don't really care about showing that the perturbations have no effect on the value of the equilibrium point --- I really just want to show that adding the perturbations damp the oscillations and cause it to converge over time. Alternately, I have considered looking at damped oscillators, but I can't find any general theoretical results that apply broadly to general equations.
Anyone one have any suggestions of how to attack this or specific approached to take?
Thanks!
Edit: just to clarify, I'm not asking about general practical approaches to showing stability. My question is more theoretical in nature. Are there known results or theorems that can deterring under what conditions this inclusion of a linear term can stabilize a nonlibear system?
Linearize the system at the origin. If the resulting system with coefficient matrix $\hat{A}$ is asymptotically stable it is guaranteed by Lyapunovs converse theorem that you can find a Lyapunov function $V=x^TPx$ for the nonlinear system by solving the Lyapunov Equation
$P\hat{A}+\hat{A}^TP=-I,$
in which $I$ is the identity matrix.