Stability Analysis of Equilibria through Graphical Methods

Given a differential equation $\frac{dx}{dt} = F(x)$, determine the stability of its equilibria by graphical analysis.

When dealing with differential equations such as $\frac{dx}{dt} = F(x)$, understanding the stability of equilibria is fundamental in understanding the behavior of dynamical systems. Graphical analysis offers a visual approach to identifying where equilibria occur and whether they are stable, unstable, or semi-stable. By plotting the function F(x) against x, you can pinpoint the behavior of the system at various points. Equilibria occur where F(x) crosses the x-axis, that is, where F(x)=0. These are the points where the rate of change of x is momentarily zero, indicating a potential equilibrium. Once the equilibria are identified, the next step is to analyze their stability, which involves examining the sign of F(x) on either side of the equilibrium point. If F(x) changes from positive to negative as x increases through the equilibrium point, the equilibrium is stable since nearby points tend to move towards it over time. Conversely, if F(x) changes from negative to positive, the equilibrium is unstable since nearby points tend to move away. Graphical analysis can provide an intuitive understanding of the long-term behavior of solutions without requiring explicit solutions, especially when such solutions may be difficult to obtain analytically. This method is particularly useful in visualizing how small perturbations in initial conditions can affect the stability and long-term trends of the system.