Bifurcation Diagram and Stability Analysis for Nonlinear Differential Equation

Consider the differential equation $\frac{dX}{dt} = xe^{-ax} - x^2$. Here, $a$ is the bifurcation parameter.

Sketch a bifurcation diagram and indicate the stability of each section created by the bifurcation diagram. Additionally, identify any thresholds where a change in long-term behavior occurs.

This problem deals with sketching a bifurcation diagram, which is a powerful tool in the study of differential equations, particularly when analyzing systems where parameters can change over time, affecting stability and system behavior. In this context, the bifurcation parameter 'a' plays a critical role. Let's consider what a bifurcation diagram represents. Essentially, it plots the steady states or equilibrium points of a system as a function of a parameter—in this case, 'a'. As the parameter changes, we may observe qualitatively different behaviors, known as bifurcations. For this differential equation, we aim to understand how changes in 'a' might lead to sudden shifts in system dynamics—like transitions from stable to unstable equilibria.

Identifying the stability of these equilibrium points is crucial because it predicts the system's behavior in the long run. Stability is typically determined by the sign and size of the derivative of the function that describes the system. In this problem, analyzing the stability involves linearizing the system around the equilibrium points and determining whether solutions nearby are attracted to or repelled from these points. Such stability analysis often uses methods like the Jacobian matrix in more complex systems; for one-dimensional systems, simpler derivative tests suffice.

Finally, finding thresholds where changes in long-term behavior occur—often where bifurcations happen—is about recognizing the critical values of 'a' where the number or stability of equilibrium points change. These critical points can lead to insights about system dynamics under various conditions, helping predict and possibly control the behavior of complex systems rooted in fields like ecology, economics, and physics.