I have a system of equations that I have performed a linear stability analysis on. This linear stability analysis resulted in a two-dimensional system of nonautonomous ODEs that has the form
$\frac{\mathrm{d}}{\mathrm{d}t} \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{cc} A(t) & B(t) \\ C(t) & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right)$.
There is only one equilibrium point, namely $(x,y) = (0,0)$. I wish to know the stability of this equilibrium point. I have learned that the eigenvalues cannot alone be used to determine stability, as they can if the coefficients were only constants. What conditions must $A(t), B(t)$ and $C(t)$ meet in order for the solution to approach the equilibrium point?
The system can be solved explicitly, but the necessary and sufficient condition of stability/asymptotic stability expressed in terms of $A,B,C$ appears to be complicated. Let us derive some sufficient conditions that easy to check and use.
An explicit solution of the system is
$(1)$ $x(t) = e^{\int_0^tA(s)ds}(x_0 + y_0\int_0^tB(s)e^{\int_0^s(C(\tau)-A(\tau))d\tau}ds)$
$(2)$ $y(t) = y_0e^{\int_0^tC(s)ds}$
Here we suppose that all the functions are defined in $[0, \infty)$ without loss of generality.
All the stability conditions can be derived from the solution above. Let confine ourselves to the question of trivial solution asymptotic stability. The solution of linear system is asymptotically stable IFF any solution tends to zero. Obvious necessary conditions for that are
$(3)$ $\lim_{t\rightarrow\infty}\int_0^tC(s)ds = -\infty,$
$(4)$ $\lim_{t\rightarrow\infty}\int_0^tA(s)ds = -\infty $
Probably the most simple sufficient condition for $x(t) \rightarrow0$ from $(1)$ is
$(5)$ $\int^\infty|B(s)|ds < \infty$.
Indeed, $(3,4)$ imply that exponent in the second term of $(1)$ tends to zero and therefore is bounded. Thus, $(5)$ implies finiteness of whole second term in $(1)$, so $x(t) \rightarrow0$.
Finally we come to relatively simple sufficient conditions for asymptotical stability of the system in question: the system is stable if integrals of $A(t), B(t)$ are both $-\infty$ and $B(t)$ is absolutely integrable function (conditions $(3,4,5)$).