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# Bifurcations

Next, we need to understand how fixed points and periodic orbits change as parameters are varied. It is possible that at some particular set of parameters a fixed point or periodic orbit is stable, but at a different set it is unstable. Such a change in stability under a change in parameters is an example of a bifurcation. Another example of a bifurcation is when, as parameters are changed, new fixed points or periodic orbits come into existence. Loosely speaking, a bifurcation is a qualitative change in the dynamics of the system of ODEs as a parameter varies. We now give several examples of bifurcations. We show the flow of the vector fields in phase space, that is, the space of the variables . The bifurcation diagrams summarize the behavior near the bifurcation; solid lines show stable solutions, while dashed lines show unstable solutions.

Saddlenode bifurcation: Consider the one-dimensional differential equation

 (2)

It is readily shown that no fixed points exist for , but fixed points exist for . is stable, while is unstable. The qualitative change in behavior at is called a saddlenode bifurcation.

Hopf bifurcation: A Hopf bifurcation involves the change in stability of a fixed point of a dynamical system together with the birth of a periodic orbit. For example, for the Hopf bifurcation shown in Figure 2, for there is a stable fixed point, while for there is an unstable fixed point and a stable periodic orbit.

Homoclinic bifurcation: A homoclinic orbit is a trajectory which approaches a fixed point both as and as . The formation of a homoclinic orbit as a parameter is varied, called a homoclinic bifurcation, can lead to the creation or destruction of a periodic orbit. For the homoclinic bifurcation shown in Figure 3, a homoclinic orbit forms at , and a periodic orbit exists for but not for . As , the period of the periodic orbit diverges to infinity.

Next: A Simple Model Showing Up: APC591 Tutorial 3: The Previous: Fixed Points and Periodic
Jeffrey M. Moehlis 2001-10-03