Math 20 Lab #5 Imperfect bifurcations: tricritical bifurcation example in fluid flow

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Monday 4pm.

Make sure your bifurcation diagram show the correct solid versus dashed lines.  For example, a previous lab looked at the DE (dx/dt)= r-x-(e^(-x)), whose bifurcation diagram can be constructed with the following Show command.

Show[ParametricPlot[{x+Exp[-x],x},{x,0,3},PlotStyle->{Blue,Thickness[.005]}],ParametricPlot[{x+Exp[-x],x},{x,-2,0},PlotStyle->{Red,Dashing[{0.02,0.02}],Thickness[.005]}],PlotRange->All,AspectRatio->1,AxesOrigin->{0,0}]

Be sure to plot the stable branches with a solid line and the unstable branches with a dashed line (or you can use colors like blue for stable and red for unstable).

Consider the differential equation (dx/dt)= h+ϵx-(gx^3)-(kx^5), that describes Taylor-Couetter vortex flow. 

In an experimental set-up, physicists can vary the parameter g to be positive or negative. 

Let h=0.001 be a small imperfection and let k=0.2.

Exercise 1. Plot the bifurcation diagram in x and ϵ for g=0.1.

Exercise 2. Plot the bifurcation diagram in x and ϵ for g=-0.04.

Exercise 3. Plot the bifurcation diagram in x and ϵ for g=-0.11.

Compare your bifurcation diagrams to the experimentally derived bifurcation diagrams given below, taken from Aitta et al,  Physical Review Letters (1986)--the original article is available on the Math 260 schedule page if you would like to read it.