Posted March 30, 2013 by Dr. Henri Montandon in mathematics
 
 

Will the real bifurcation diagram please stand up!

Netcraft reports as of March 2013 there are 631,521,198 active websites on the internet. In Cool Runnings, I try to filter the wheat from the chaff, the nuggets from the pebbles, in particular, to find those sites which are formed by pedagogic prodigies. These special sites seem to me to demonstrate knowledge in ways that make that knowledge more knowable. In this simple observation lurks the whole grand endeavor of the transmission of what we know from the mothers and fathers to the daughters and sons. As a society, we do not sufficiently honor great or even good teachers.

So far, I have presented in this blog nine sites which I qualify as especially meritorious presentations of knowledge. That’s 0.00000001425 % of what Netcraft has cataloged in the visible internet. I suppose this percentage will rise with time, but I have no idea how to estimate what number it might approach.

Today’s site concerns the topic of bifurcation, a term of art in dynamical systems theory. In DYNAMICS THE GEOMETRY OF BEHAVIOR, an illustrated introduction to dynamics by Ralph Abraham and Christopher Shaw, bifurcation diagrams are explored as they emerge out of one-parameter periodic systems on a plane mapping response amplitudes against the driving frequencies of the system.

It turns out there is an Ubiquitous Imposter hiding in plain sight in the literature of dynamic systems. The UI is often called a bifurcation diagram, but it is not and cannot be. Professors Chip Ross and Jody Sorensen tell you all you want to know about how to identify the UI and distinguish it from the Real Bifurcation Diagrams. And they do it with clarity and enthusiasm. Three cheers for Chip and Jody!!!

Take a look at the two diagrams below. Can you fathom which is the Ubiquitous Imposter and which is the Real Bifurcation Diagram?

 

The authors state: The bifurcation diagram nicely complements the more commonly discussed orbit diagram. Together the two diagrams show attracting and repelling periodic orbits, along with possible locations of chaotic behavior. When combined they present the whole picture of the dynamic behavior for a one-parameter family of functions.

In a future blog, I will label the diagrams above and explain why the difference is important.

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Dr. Henri Montandon