Unstability test for non-autonomous nonlinear systems

Question

Consider the non-autonomous nonlinear dynamical system $$ \dot{x}=f(t,x) $$ It is a known result (Theorem 4.13 in Khalil (Nonlinear Systems)) that if the linearization $$ \dot{x}=A(t)x $$ where $A(t)=\frac{\partial f}{\partial x}(t,x) \rvert_{x=0}$, is such that the origin is an exponentially stable equilibrium point, then the origin is an exponentially stable equilibrium point for the nonlinear system. I am wondering whether an analogous result holds in the unstable case, i.e, if the linearized system is unstable and with diverging solutions then the nonlinear system is also unstable. Is anyone aware of such a result? I have not been able to find it in the literature or on the standard nonlinear systems textbooks.

Answer 1

As usual, eigenvalues of real part $0$ can make things complicated. Consider the system

$$ \eqalign{\dot{x} &= y \cr \dot{y} &= -x^3 \cr}$$ The linearized system (with an eigenvalue $0$ of algebraic multiplicity $2$) is unstable, but the nonlinear system is stable.