Stability Criteria for Nonlinear Ordinary Differential Equations

The main results of this work are three sufficient conditions for the (1) stability, (2) uniform asymptotic stability in the large and (3) instability, of the equilibrium point $x = 0$ of the system of differential equations: $\dot x = f(t,x)$, $f(t,0) = 0$. Stated roughly these conditions are: The point $x = 0$ is (1) stable if $x'f(t,x)$ is a concave function of x, (2) uniformly asymptotically stable in the large if $x'f(t,x)$ is a concave function of x is a strictly concave function of x, and (3) unstable if $x'f(t,x)$ is a strictly convex function of x. These results are obtained by using the stability and instability criteria of Liapunov and properties of concave and convex functions.