An Introduction to Hopf Bifurcation

Introduction to Bifurcations

Consider some equilibrium point of a one-dimensional system . Bifurcation occurs when the hyperbolicity of that equilibrium point is violated. Hence one can see that this violation can occur in two ways: (1) the eigenvalues become 0, which corresponds to the fold bifurcation, and (2) the eigenvalues become purely imaginary, i.e. , which is the focus of today – the Hopf bifurcation. Hopf bifurcation is used in fields such as semi-classical and quantum physics as well as neuroscience.

Hopf Bifurcation

If we consider a coupled nonlinear system with one parameter :

It’s dynamics are clearer by decoupling the system, and we can do so by introducing a new variable , so that the system can be rewritten in its complex form:

And by transforming it onto polar coordinates, i.e. let , we obtain the normal modes of the system:

The angular component of the system is obviously just performing a constant-speed rotation. The interesting part lies in its radial part. By considering the regimes for , we first observe that it has an equilibrium point at . The Jacobian of the system near that equilibrium point yields eigenvalues . This means that if then this equilibrium point is stable, and if it becomes unstable. Next, we see that if , the part contained within the parenthesis will always be less than zero, therefore will be the sole equilibrium point. However, if , there exists another stable equilibrium point . Coupled with the angular component, its phase plane for different values are shown in fig 3.5 of the original book. More concisely, the space graph is shown in fig 3.6.

The circle that forms is called a limit cycle, and in this system it is a stable one, meaning that all initial conditions except those starting at the origin will converge towards it.

Subcritical and Supercritical Hopf Bifurcation

If we observe a second system whose nonlinear terms differ from the previous one in terms of sign:

Going through the same analysis as above shows that when there is a stable equilibrium point at the origin and an unstable limit cycle with . When , there is only one unstable equilibrium point at the origin. This second system has a so-called subcritical Hopf bifurcation, and the first a supercritical Hopf bifurcation. In the case of supercritical Hopf bifurcation, the loss of stability “after” the bifurcation (we define “after” as going from negative to positive, as there is no real direction for bifurcation) is somewhat contained as there is an finite attractive limit cycle for finite . That is not true in the subcritical case, and that gives us a catastrophic loss of stability – that is, if returns negative, the system may not return to the stable equilibrium point as it may have moved out of its basin of attraction.

Original book: Kuznet︠s︡ov, I. A. (2011). Elements of applied bifurcation theory. New York: Springer.

Summary written by: Pei-Hsien Liu