an example of a bifurcation point?

Question

A bifurcation point is an attractor... like a node or event that looks (on a graph of its mathematics) like a hole in the bottom of a basin.
A dissipative structure (imagine the butterfly effect) tends toward an attractor, which in turn acts like a precipitant or irritant around which crystals begin to form in a supersaturated solution.
Once the crystallization process begins, it will continue as long as the system remains open to more input.

What 'meme' do you suppose fits these descriptions?
And given that variable, is there a benefit to the crystallization?
If not, how do we eradicate the crystallization? (metaphorically speaking of course).

Answer 1

I'm suddenly reminded of the old equation (x)(U)(1-U being between 0.0000 and 1.0000, inclusive) where X = 3.0000000... is a bifurcation point for the chaotic equation.

I once did an animation where I took 10000 frames, each one ranging from X=0.0000 to X=4.0000, in increments of 0.0004, and showed how the bifurcations occured. Wish I still had that thing.

Answer 2

Bifurcation diagram

A one-parameter bifurcation diagram summarizes the qualitative behavior corresponding to different values of a parameter. A state variable, or some function of the state variables, is usually plotted on the y -axis and the parameter on the x -axis. The positions and local stabilities of equilibrium points, as well as periodic orbits, are indicated using different line types. Solid curves are used to represent locally stable equilibria and dotted curves are used for locally unstable equilibria. Maxima and minima of periodic orbits are indicated using circles: solid for stable orbits and open for unstable orbits.

It is important to note that bifurcation diagrams summarize the behavior associated with a range of parameter values. They do not represent the dynamics corresponding to a continually varying parameter (Wiggins 1990). To read a bifurcation diagram, fix the parameter at a particular value and mentally draw a vertical line at that value. Each crossing of this line with a curve in the diagram corresponds to an equilibrium point or a periodic orbit (limit cycle). The local stability properties of a particular phenomenon are given by the type of curve: solid, dotted, or open or closed circles. For example, the phase portraits in Fig. 14b (i) and (ii) were obtained by mentally drawing vertical lines at the parameter values µ = µ1 and µ = µ2, respectively, in Fig. 14a.

A two-parameter bifurcation diagram shows how the positions of bifurcation points change as two parameters are varied. For example, if a bifurcation point is encountered in a one-parameter bifurcation diagram, a second parameter may be varied to see how it affects the position of the bifurcation point.

Answer 3

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Answer 4

Go cave hiking and come back and ask the same question. Don't fool with Mother Nature....she is tooo big!

Answer 5

I think you're being a little too clever for your own good. If you have a point, spit it out.

Answer 6

I saw 'bi' and put my head back under the covers.