Visualizing functions with a number of independent variables

For time-dependent and three-dimensional data, there exist a number of established programs already; among the most popular free and open source ones are Paraview and VisIt. Both support a variety of plots such as isosurfaces, volume rendering or (for vector-valued data) arrow or streamline plots.

They all have a bit of a learning curve, though, and you should not expect to be able to see a four- or five-dimensional function on a two-dimensional screen "at a glance". Visualizing of and and data-mining from high-dimensional data are very active research topics in computational science, though. I would thus suggest thinking about what kind of information about your functions you would like to see about your functions, and then ask on the Computational Science SE. (See for example this question.)

A picture is worth a thousand words, a movie is worth a thousand pictures, and an interactive app is worth a thousand movies. I would suggest making a picture that dynamically responds to your changing variables. For instance,

http://www.math.osu.edu/~fowler.291/phase/

lets you type in a function of two complex variables (z and mouse), and you get a phase plot of the function where the z domain is being colored.

I am currently working on an improved version of this which will use webGL, accept arbitrarily many variables (slide points around), and admit several view options (Riemann sphere, 3d graph of modulus colored by phase, ect). This will be used in an online introduction to Complex analysis that I will be helping to run next spring.

Hopefully these ideas help you!

If you have a function in three variables $f(x,y,z)$, you can try to plot surfaces solving the equation $f(x,y,z)=r_i$. Using transparency and different colours you might be able to visually grasp multiple of those surfaces at once (choose $r_0,\ldots r_4$ wisely and plot these $5$ surfaces (you also have to choose the perspective accordingly)). For $t$ use the time and animate your plot. But for five variables I have no idea.

I have no practical experience with it, Octave seems to have some support for such implicit surfaces, but I do know whether it will be suffiecient in your case.