Is the 'branching out' pattern always present in bifurcation diagrams?

[Benny]

I have another question about stability, it kind of leads on from the other topic that I posted. On bifurcation diagrams, you typically see a series of 'forks'(or branches) and the number of forks for a given interval usually increases as you move from end to the other doesn't it?

I'm wondering if, in general, it is possible for something like that to not occur. That is, as you move from one end to the other of the bifurcation diagram you don't necessarily see an increase in the number of 'forks'. I'm inclined to think that the number of branches doesn't necessarily increase as you from move from one interval to the next(ie. I don't think that all iterative maps have corresponding bifurcation diagrams where there is a continual increae in the number of branches). That is what I think because of some weird results I've been getting for some of the things I've been working on. Can someone shed some light on whether or not the 'branching out' pattern is a characteristic of all bifurcation maps?

An example of a bifurcation diagram is available via the following link to illustrate what I mean by forks/branches: http://mathforum.org/advanced/robertd/bifurcation.html

 

[saltydog]

The quadratic iterator is a "model" bifurcation diagram. Most non-linear dynamic systems have "deformed" diagrams: nonsymmetrical, the routes go from stable to chaotic and then back to stable, some routes seem to disappear entirely only to re-appear again further out (this last one however may be a reflection of round-off error, not sure). Try studying some ODE models like the Lorenz or Rossler or the full-quadratic iterator:

$$x_{i+1}=a+bx_i+cx_i^2+dx_i y_i+ey_i+fy_i^2$$

$$y_{i+1}=g+hx_i+jx_i^2+kx_i y_i+ly_i+my_i^2$$

To generate Figenbaum plots of these, run the iterator with initial starting values and just vary one of the constants such as c and then plot x vs c or y vs. c.

Edit: I'll post an example sometimes today. You know about J. Sprott?

Web address:http://sprott.physics.wisc.edu/fractals.htm

 

[Benny]

Ok thanks for your help Saltydog. I have another question about bifurcation maps. I think they're usually done by using a program but for an assignment I'm doing there is a simple(simple in that there aren't a very large number of cycles, or at least I don't think there are) I need to sketch the bifurcation diagram. Say for instance I have the relevant fixed cycles/periods and fixed points for a map like the quadratic map(I assume the bifurcation diagram in my link corresponds to what you referred to as being the 'model' diagram).

I would take them and plot the stable points vs the parameter as they(in the link I provided) have done. But how would I get the curve shapes that they've obtained? I am referring to the connections between successive branches on their diagram. I'm just wondering if there is a way to deduce the 'shape' of the branches.

Just one more question. How can you tell, from looking at a bifurcation diagram when the "routes" go from stable to chaotic? Do random things like paths crossing on the bifurcation diagram occur when the routes go to 'chaotic regions?' Any help would be good, thanks again.

 

[HallsofIvy]

Paths will NEVER cross on a bifurcation diagram, assuming only that your d.e. or iteration scheme or whatever is defining the process is not so horribly bad as to violate the "existence and uniqueness" theorem. "Chaos" is typically signaled by an "explosive" increase in the number of bifurcations over a very short interval. True "chaos", by the way, is NEVER random. It is extremely complex but always determinate.

 

[Benny]

Oh ok, thanks for enlightening me regarding the nature of chaos.

I would still like to know how the curves(the things which connect the points of interest on the bifurcation map) on the quadratic map(my link) are produced though. Is it arbitrary(I doubt it) or is there a reason why the connecting branches of that map are curved.