As we have seen in the preliminary analysis of 1D systems, there aren't many possibilities for their behavior. Trajectories either decay to fixed points or go off to infinity. However, there are some very important considerations regarding a system's dependence on parameters. Often, in real models, we cannot determine parameters with infinite precision, and sometimes the parameters can change for a multitude of reasons. Therefore, we must understand the connection between a system's behavior and its parameters. This is the study of bifurcations: qualitative changes in a system's dynamics due to a change in the system's parameter(s).
What is a Bifurcation?
A bifurcation occurs when a small smooth change in a parameter of a system causes a sudden qualitative change in its behavior. More concretely, consider the parameterized system
where \(\mu\) is a parameter. A bifurcation happens at a value \(\mu = \mu_c\) if the phase portrait of the system for \(\mu < \mu_c\) is not topologically equivalent to that for \(\mu > \mu_c\). In practice, this often means that fixed points appear, disappear, or change stability.
Bifurcations in the Logistic Equation
A classical example is the logistic equation
where the parameter \(r\) represents the growth rate. The fixed points of this system are obtained by setting \(\dot{x} = 0\):
The stability of each fixed point depends on the parameter \(r\). We compute
At \(x^* = 0\), we have \(f'(0) = r\). Thus the fixed point is stable if \(r < 0\) and unstable if \(r > 0\). At \(x^* = 1\), we have \(f'(1) = -r\). So the fixed point is stable if \(r > 0\).
At \(r = 0\), the stability of the fixed points changes: the fixed point at \(x=0\) changes from stable to unstable, while the fixed point at \(x=1\) emerges as stable. This is an example of a transcritical bifurcation.
Types of Bifurcations
In one-dimensional systems, the most common local bifurcations are:
- Saddle-node bifurcation: two fixed points collide and annihilate each other.
- Transcritical bifurcation: two fixed points exchange stability.
- Pitchfork bifurcation: one fixed point splits into three (supercritical or subcritical).
Each type of bifurcation has a characteristic "normal form" equation, which captures its essential behavior.