16.2 Stability of an equilibrium solution

The idea of stability of an equilibrium solution for a nonlinear system is intuitively similar to that of a linear system: the equilibrium is stable when all the phaseplane arrows point towards the equilibrium solution. For Equation (16.3), the equilibrium solution at \((1,1)\) is unstable because some of the arrows point towards the equilibrium solution, whereas others point away from it.

For the moment, we won’t go into more specifics of how to classify a stable versus unstable equilibrium solution. If you can recognize from the phaseplane equilibrium solutions that appear to be stable (versus those aren’t), then you have established some good intuition that can be confirmed with additional analyses. We will study how to do these analyses in the next few sections.