A parametric plot with branches for which restricting the range with xmin, xmax doesn't work

edit

For some reason, the plot is not correct.

You mean that xmin and xmax are not honored ? See Trac#16686 and the ticket-chain referenced...

Workaround :

sage: def ininterval(x,l,u):return float(x) if x>l and x<u else float("NaN")
sage: plot([lambda t:ininterval(8*(6+t)/(t^2-16),-3,3),lambda t:ininterval(2*(8+ 3 *t)/(t^2-16),-3,3)], (t,-6,6), axes_labels=["x","y"], parametric=True)
verbose 0 (3839: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 130 points.
verbose 0 (3839: plot.py, generate_plot_points) Last error message: 'Unable to compute f(6.0)'
Launched png viewer for Graphics object consisting of 2 graphics primitives

has three branches.

Nope.

What are the possible values of x and y for $t\in[-6~6]$ ? An analytical solution is possible by solving x and y for t then solving the inequations for $t\in|-5~6]$, but is somewhat painful... A graphical solution is easier to grasp :

sage: plot([lambda t:ininterval(8*(6+t)/(t^2-16),-30,30),lambda t:ininterval(2*(8+ 3 *t)/(t^2-16),-30,30)], (t,-6,6), axes_labels=["t",""], legend_label=["x","y"])+plot([-3, 3],(-6,6),color=["green","red"])
verbose 0 (3839: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 14 points.
verbose 0 (3839: plot.py, generate_plot_points) Last error message: 'Unable to compute f(4.325615967306306)'
Launched png viewer for Graphics object consisting of 8 graphics primitives

The most limiting variable is x, which is positive and <3 for t>-6, t> -9/2 approximately, and slightly superior to -3 for t>-3, t<0 approximately.

How to plot the branches in different colors ?

The painful analytical solution I studiously avoided above is necessary to define the different regions; the color= argument can then be built around this definition (left tothe reader as an exercise;-)...

EDIT : after addition of another question...

A static version of your chronogram could be :

var("t")
fx(t)=8*(6+t)/(t^2-16)
fy(t)=2*(8+ 3 *t)/(t^2-16)
def ininterval(x,l,u):return float(x) if x>l and x<u else float("NaN")
P=plot([lambda t:ininterval(fx(t),-3,3), lambda t:ininterval(fy(t),-3,3)],
       (t,-6,6), axes_labels=["x","y"], parametric=True)
Pts=[]
T=[]
xmin=P.xmin()
xmax=P.xmax()
ymin=P.ymin()
ymax=P.ymax()
for t in srange(-6,6.1,0.1):
    px=fx(t)
    py=fy(t)
    if px>=xmin and px<=xmax and py>=xmin and py<=ymax:
        Pts+=[(px,py)]
        T+=[text("%4.1f"%t,(px,py),
                 vertical_alignment="bottom",
                 horizontal_alignment="left",
                 color="black")]
sage: show(P+points(Pts)+sum(T))
Launched png viewer for Graphics object consisting of 43 graphics primitives

An animated/interactive version is left to the reader of the documentation as an exercise...

HTH,