Here's a program you can type into your TI-85 to draw stairstep diagrams. It prompts for a value of rho and an initial condition u₀, and then draws the usual picture and starts putting in vertical and horizontal lines. It keeps going until you press the [ON] key to stop it.

:Input "rho=",rho
:Input "u0=",u0
:y1=rho*x*(1-x):y2=x
:FnOff:FnOn 1,2
:mar=0.2
:mar->xMin:1+mar->xMax
:mar->yMin:1+mar->yMax
:ClDrw:DispG
:While 1
:rho*u0*(1-u0)->u1
:Line(u0,u0,u0,u1)
:Line(u0,u1,u1,u1)
:u1->u0
:End

The string "->" indicates the [STO] key. You can find the variable names xMin, xMax, yMin, and yMax in the VARS list, or just type them in. The "Line" function is under GRAPH, then DRAW.

Here's what the screen looks like after the program has run for a few seconds with rho=3.65 and u₀=0.01.