5. Expansion to 3D under the slow-fast hypothesis

If R is very small then is very small too, so that z(t) is almost constant in time or, to be more precise, varies much slower than x(t) and y(t) . This means that, given a generic initial state (x(0),y(0),z(0)), the fast subsystem (x,y) evolves very fast to an attractor A(z(t)) (equilibrium or limit cycle) corresponding to the value of z(0) frozen as a parameter. Than, as z(t) slowly varies according to (2c), the state (x,y) of the fast subsystem follows the evolution of the attractor A(z(t)).

Let us now make a second assumption, namely that there exists a manifold defined by (2c), which is such, that it seperates for any z the nontrivial attractor of the subsystem (x,y) (equilibrium or limit cycle) from the trivial one. This manifold was already shown in the 4 diagrams in the (x,y,z)-spaceof the previous Section. That means that z(t) will be increasing above this manifold, and decreasing below. Feichtinger [FFP93] states, that this condition (the separation principle, see Muratori and Rinaldi [MuRi91]) can always be satisfied by taking G sufficiently small.

Under these two assumptions mentioned and explained above (R and G small), it is easy to understand that system (2) may have a limit cycle with peculiar geometrical characteristics, that is, it is composed by alternating slow and fast branches.
On the slow branch, z(t) evolves very slowly over time, while (x(t),y(t)) are trapped on the attractor A(z(t)). On the fast branch, a fast (catastrophic) transition takes place where the subsystem (x,y) suddenly moves from one attractor to another, while z(t) remains almost constant.

In the next diagrams we will show again the four last diagrams of the previous Section (where z was considered to be a parameter) and next to them the existence of limit cycles in the slow-fast case is shown.

bifurcation diagram for the value E1 (see last Section) in the space (x,y,z)
and slow-fast behaviour for comparision

In the above figure of the slow-fast behaviour, the cycle is composed by the fast branch 1 -> 2, followed by the slow branch 2 -> SN- where (x,y) are trapped by the nontrivial stable equilibrium, then by another fast branch SN- -> 3, and finally there is again a slow branch, namely 3 -> 1, which corresponds to the trivial equilibrium.

bifurcation diagram for the value E2 (see last Section) in the space (x,y,z)
and slow-fast behaviour for comparision

In this figure above of the slow-fast behaviour, the upper branch is charcterized by oscillations of (x,y) at relatively high frequancy, whose amplitude slowly decreases to zero as z(t) slowly increases.

bifurcation diagram for the value E3 (see last Section) in the space (x,y,z)
and slow-fast behaviour for comparision

In the above diagram, as well as in the next one below, the amplitude of the high-frequency oscillations that was explained above, is not tending to zero, as in the last mentioned example befor that diagrams. In contrast the period of the oscillations increases while z(t) is approaching the value .

bifurcation diagram for the value E4 (see last Section) in the space (x,y,z)
and slow-fast behaviour for comparision

In the following some visualisations for these cases are shown and explained.

First we will consider the case two, that is, where the parameter E has the value E2 (see above). When visualizing this case with a colored streamline, we obtain the following picture:

Obviously that gives not a very clear picture and is not very comprehensive, but when you rescale the x, y, and especially the z-values, then you get a much more comprehensive picture. In the figure below this is shown, and there is also an animation (~600K, Quicktime) available, that shows the process of the rescaling.

In the above picture the colorcoding gives the velocity, where red means high velocity, and blue is used for low velocity. Of course, one can also think of using the color of the streamline that shows the cycle, to code the time. This is done in the next picture, where blue is used for the begining of the streamline (oldest part) and the newest part is red colored. In both figures one can easily see the slow and fast parts of the cycle. In the first one because of the colorcoding itself, and the following one, wheter over a short part of the cycle in space the color changes a lot, or less.

There is also an animation (~1MB, Quicktime), that shows the generation of the above images step by step.

Another possibility to visualize the above case would be the following:
Instead of taking a startpoint and integrating from there and thus producing a single streamline showing the behavior of the dynamical system, one can take a startline (very similar to what you do, when you want to produce streansurfaces). On this startline you take a number of startpoints and for each of these you generate a streamline, thus getting a bunch of streamlines, all starting very close to each other. This technique is called "Rake", because if you don't take too many startpoints on the startline, you end up with a figure looking very similar to a rake. But you can also take very many startpoints and then you will end up with something that looks like a streamsurface (see next figure).

Now we will consider the case where the parameter E has the value E3 (see above). When visualizing this case with a colored streamline, we obtain the following picture:

In this picture above the color of the streamline is again used to code the velocity and this is done again exactly in the same way as described a little bit further up.

In the next few pictures the same case is studied (namely, that E is equal to E3), but a different visualization method has been added to the colored streamline, in orderto make the understanding of the behaviour of the dynamical system even more easy and comprehensible. For this purpose only a very short part in time of the above shown streamline is considered and shown, and this part is animated over time, so that this short part wanders along the long part of the former (longer) streamline. Furthermore the values of x, y, and z are shown in the lower part of the pictures, again ploted for the same short streamline shown in the upper half of the images, ploted over the same time as the streamlines. The red ploted line gives x over the time, the green one corresponds to y and the blue one shows z over the time.
By remembering that x, y and z correspond to farmers bandits and soldiers respectively, one can better understand the behaviour of the dynamics of this model.

There is an animation (~4.5M, Quicktime) available that shows the process of this short trajectory wandering along the original streamline and plotting the x, y, and z values. The next four pictures are screenshots from this animation for different timestamps.

When looking at the above images, one can clearly notice, that the cycle time is growing while walking along the streamline, although the z value is increasing only very, very slowly, nearly not visibly. In the last picture the behaviour is at the point where it changes suddenly. After this point there are no further oscillations for a long time, but x and y stay nearly constant, while z decreases again very, very slowly. When z gets small enough, the oscillations suddenly start again (and are as fast as in the beginning).

The same techniques with these timeplots for the x, y, and z-values over time can also be used in the first (further above) shown case. There is also an animation (~11MB, Quicktime), available, which shows the basically the same as the above video, but only for a value of E = E2.

All that has been done and shown in this section was under the assumption, that we have slow-fast dynamics, namely a fast farmers-bandits subsystem, and soldiers that are acting very slowly in response to the changes in this fast subsystem.
In the next Section we will relax this assumption, and have a look at what happens in the system, when it is supposed, that also the soldiers are reacting very fast.