A stationary point of an ODE is considered “stable” if, at the stationary point y=c, the function with initial condition. If you start near a stationary point, the function will either diverge t\to \infty to that stationary point, or converge to a stationary point. Whether the functions done that makes it “stable”/“unstable”. For an autonomous ODEs y’(t) = f(y(t)), suppose y(t) = c is a stationary solutiona: c is stable (i.e. t\to \infty, y \to c for y_0 \approx c) if the graph of f near c crosses from positive to negative; that is, when f’( c) < 0 c is unstable (i.e. t\to -\infty, y \to c for y_0 \approx c) if the graph of f near c crosses from negative to positive; that is, when f’(t) > 0 c is semi-stable (i.e. stable on one side, unstable on the other) if the graph of f near c has the same sign on both sides; meaning f’( c) = 0 and f’’( c)\neq 0 if f’( c) = 0 and f’’( c) \neq 0, we are sad and should investigate more away from zeros, the concavity of y(t) could be checked for f f’. when its positive, y(t) is concave up; when its negative y(t) is concave down.