The behaviour of a one-dimensional dynamical system may depend on the value of a certain parameter. As the parameter value passes through a critical value the system dynamics can change substantially. This is termed a bifurcation. Consider the following example.

• Worked Example 6

Maple Solution

It can be shown that the polar form of the system is

from which it is clear that there is a single equilibrium point at the origin. The first polar equation can be written as

Clearly there is a limit cycle of radius in this case. Also the origin is unstable since for small r

which is positive.

If then and the origin is globally asymptotically stable. This behaviour is illustrated below for the cases and respectively.
 
 

We see that as the parameter passes through zero from negative to positive a stable focus gives way to an unstable focus surrounded by a stable limit cycle whose radius increases with . This is an example of a Hopf bifurcation.

In order to gain further insight into the behaviour of this system and to set the scene for the introduction of the principal result of this section let us carry out a linear stability analysis of the cartesian form of the equations. The linearization about the origin is obviously

The Jacobian matrix is therefore

The eigenvalues are given by

Solving gives

Thus if we have a stable focus, if an unstable focus and if we cannot decide (although we know from the polar form that we have a stable focus). As the bifurcation from a stable focus to an unstable focus surrounded by a stable limit cycle occurs the real part of the eigenvalues changes from negative to positive being zero at the bifurcation point. This is typical of this type of Hopf bifurcation which is said to be supercritical. The following theorem generalises these results.

Theorem - The Hopf Bifurcation Theorem

Suppose that the system

has an equilibrium point at the origin for all . In addition suppose that the eigenvalues of the linearization are purely imaginary when . If the real part of the eigenvalues satisfy

and the origin is asymptotically stable when  then

(i)  is a bifurcation point of the system;

(ii) for some such that the origin is a stable focus;

(iii) for some such that the origin is an unstable focus        surrounded by a stable limit cycle whose size increases with.

and hence the theorem predicts the existence of the stable limit cycle.

• Worked Example 7

as the parameteris varied.

Maple Solution

This system has a single equilibrium point at the origin as you should verify. The linearization is clearly

which is identical to that found in Worked Example 6. This means that all the conditions of the Hopf bifurcation theorem are satisfied apart from the asymptotic stability of the origin at . This suggests that there may be a Hopf bifurcation. However the polar form of the equation for r is

which shows that the origin is unstable. Hence the theorem does not apply. Using Maple we obtain the following plots for and respectively.

The first plot shows the existence of an unstabe limit cycle surrounding the stable focus at the origin The second plot shows the unstable focus which occurs when is positive. We have a different type of Hopf bifurcation which is referred to as subcritical.

• Worked Example 8

undergoes a subcritical Hopf bifurcation at for a certain range of values of the parameter a.

Clearly an E.P. at the origin.

Transforming to polar coordinates

Thus for ‘small’ values of r

and hence the equilibrium point is a stable focus if and an unstable focus if .

For 

for ‘small’ r. Thus the origin is unstable.

Now we can factorise the expression for to obtain

If then are both real and positive so we can further factorise to obtain

and thus we obtain two limit cycles of radii and . Examining the signs of the terms in the factorisation

Thus is unstable and is stable.

If then is real and positive but is real and negative. In this case we can only factorise into the form

and thus there is a single limit cycle of radius . Examining the signs of the terms in the factorisation

and thus the limit cycle is stable.

Summarising these results we see that if the origin is a stable focus surrounded by an unstable limit cycle of radius and a stable limit cycle of radius . If the origin is an unstable focus surrounded by a stable limit cycle of radius .

Thus there is a subcritical Hopf bifurcation. The existence of the second stable limit cycle does not affect this conclusion.