Hopf Bifurcation  

Every dynamic system ist controlled by parameters. And when a parameter changes, then the state of the system may change as well. Such changes can occur drastically - the quality of the state may become completely different! The critical parameter values where the state changes are the bifurcations. A fascinating type of bifurcation is the Hopf bifurcation.

Roughly speaking, Hopf bifurcation ist a transition from stationary state to periodic state, or vice versa. The key role is stability of the state. For example, on one side of the bifurcation the state may be stable and stationary, and on the other side it may be stable and periodic. The transition can be continuous, or with a jump. Another situation would be stable stationary on one side, and no stable state on the other side! We present some examples. 

First example: an autocatalytic reaction (Example 7.15 in the book) 

This is a classic situation: stable stationary on one side of the bifurcation (flat in the figure), and stable periodic on the other side (wavy in the figure).

This figure shows time-dependent solutions for a sequence of parameter values. Here "solution" means the function that is solution of the underlying differential equation. "Stationary" on the left side, and "periodic" on the right side. And the bifurcation is the intermediate state in-between. Solutions as shown in the figure do not only exist at these discrete parameter values, rather there is a continuum of solutions. This is illustrated in the next figure, which extends the previous figure:

This dynamic and qualitative change at the bifurcation can be summarized by a bifurcation diagram, which looks quite simple for this example:

The horizontal axis of a bifurcation diagram depicts relevant parameter values. Here we choose an interval which includes the Hopf bifurcation. The vertical axis of a bifurcation diagram depicts a characteristic measure that describes the type of the solution. Su stands for stationary unstable, Ss for stationary stable, and Ps for periodic stable. Stable solutions attract nearby solutions. The figure of the bifurcation diagram shows how the dynamical behavior is organized. The red dot is the Hopf bifurcation, here for parameter value 1.30 (rounded). The blue lines represent stable states, the green line represents unstable states. The time period of the periodic solution close to the Hopf bifurcation is 6.03.

More exciting is the next example: A stirred-tank reactor  (Example 2.14 in the book):

The next figure shows the temporal behavior of solutions, when an external parameter is shifted linearly from the value 0.1 to the value 0.3. The horizontal axis of the following figure is the time, varying from 0 to 200. The vertical axis displays some characteristic variable of the solution, say, the temperature of the reactor. The reaction of the system to the varying parameter is shown in red color:  It starts stationary with low values, and all of a sudden jumps to periodic oscillations with large amplitude. For higher values of the parameter (at later time) the amplitude diminishes and dies out, culminating at a stationary state, now at higher level.

And the explaining bifurcation diagram is 

Again Hopf bifurcations! The left one goes along with a hard loss of stability, because of the jump in the solution. The right Hopf bifurcation goes along with a soft loss of stability, because there is a continuous transition between the neighboring solutions. 

Risks

As mentioned before, the quality of states change when a parameter passes the value of a bifurcation. Such changes are often not favorable! Therefore reaching a bifurcation often must be regarded a risk. To give a drastic example of this kind, we consider a voltage collapse of an electric power generator (Example 3.8 in the book). Here the parameter is the power demand of the generator. When the power demand reaches the critical value of a (Hopf) bifurcation, the voltage production collapses. Using this model [due to Dobson and Chiang], we shift the parameter linearly from the value 2.558 to 2.572 - that means, increasing power demand. For 2.558 the state is stable as desired. The chosen parameter interval includes a Hopf bifurcation at parameter value 2.559 with hard loss of stability. As a consequence, the stable stationary state loses its stability, begins to oscillate, and finally the power supply collapses. This is simulated in the following figure:

The question is, where does the state go when it collapses? Are there other attracators, maybe nearby? Some answers are given by the corresponding bifurcation diagram:

The Hopf bifurcation HB is shown. For parameter values very close to HB, the hard loss of stability may lead to a stable periodic state Ps, but these attractors cease to exist for larger parameter values. There exist many different kind of solutions, with unstable periodic solutions, and "period doubling" PD.  There is even a turning point (not shown). And the sequence of period doublings hints at the possibility of chaos. 

This is an interesting story. An analysis requires a bifurcation and stability analysis.