8.1: 8.1 What are Bifurcations?

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One of the important questions you can answer by mathematically analyzing a dynamical system is how the system’s long-term behavior depends on its parameters. Most of the time, you can assume that a slight change in parameter values causes only a slight quantitative change in the system’s behavior too, with the essential structure of the system’s phase space unchanged. However, sometimes you may witness that a slight change in parameter values causes a drastic, qualitative change in the system’s behavior, with the structure of its phase space topologically altered. This is called a bifurcation, and the parameter values at which a bifurcation occurs are called the critical thresholds.

Bifurcation

Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds.

Bifurcations play important roles in many real-world systems as a switching mechanism. Examples include excitation of neurons, pattern formation in morphogenesis (this will be discussed later), catastrophic transition of ecosystem states, and binary information storage in computer memory, to name a few.

There are two categories of bifurcations. One is called a local bifurcation,which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point. In this textbook, we focus only onthe local bifurcations, as they can be easily analyzed using the concepts of linear stability that we discussed in the previous chapters.

Local bifurcations occur when the stability of an equilibrium point changes between stable and unstable. Mathematically, this condition can be written down as follows:

Local bifurcations occur when the eigenvalues \(λ_{i}\) of the Jacobian matrix at an equilibrium point satisfy the following:

For discrete-time models: \(|λ_{i}| = 1\) for some \(i\), while\(|λ_{i}| < 1\) for the rest.
For continuous-time models: \(Re(λ_{i}) = 0\) for some \(i\), while \(Re(λ_{i}) < 0\) for the rest.

Theseconditionsdescribeacriticalsituationwhentheequilibriumpointisabouttochange its stability. We can formulate these conditions in equations and then solve them in terms of the parameters, in order to obtain their critical thresholds. Let’s see how this analysis can be done through some examples below.