Many nonlinear differential equations cannot be solved explicitly as functions of time. Phase diagrams, however, offer qualitative information about the stability of equations that is helpful in determining whether the equations will converge to an intertemporal (steady-state) equilibrium or not.
If the arrows of motion point towards a steady-state solution, the solution is stable; if the arrows of motion point away from a steady-state solution, the solution is unstable.
The stability of the steady-state equilibrium points can now be read from the graph. Since the arrows of motion point away from the first intertemporal equilibrium is an unstable equilibrium. With the arrows of motion pointing toward the second intertemporal equilibrium is a stable equilibrium.
References:
Pena-Levano, Luis Moises. Schaum's Outline of Calculus for Business, Economics and Finance, Fourth Edition (p. 368). McGraw Hill LLC. Kindle Edition.
Python code for graph generated using Grok.
