I find Catastrophe theory to be a source of compelling examples of the type you seem to be looking for. The applications aren't technological toys (at least, as far as I know)- rather, they're an understanding of how and why some systems can change suddenly.
Here's an interesting application of a cubic: put a bar of soft iron in a mild magnetic field. A slight magnetism is induced in the iron. As you increase the strength of the magnetic field slowly, the magnetism of the iron will increase slowly, but then suddenly jump up after which, as you still increase the strength of the magnetic field, it increases slowly again. If you now DECREASE the strength of the magnetic field, the magnetization will, of course, decrease slowly- PAST the strength at which is had suddenly jumped up. It will then suddenly jump down but at a point where the magnetic field is weaker than when it jumped up. E.e.s call that a "hysteresis loop".
Similar application. You wind the propellor of a toy rubber-band airplane and the rubber-band winds around itself. Suddenly, the entire rubber-band will twist into a spiral. If you now reverse the winding, the force on the rubber-band will go down PAST the point at which it had suddenly kinked before it "unkinks". That is referred to as the "rubber-band catastrophe" (anyone remember "catastrophe theory"?)
Why? Because the equilibrium solutions for magnetic field as a function of induced magnetization and for the force on the propellor as a function of "twist" of the rubber-band is a cubic. Notice the way those functions are going! Induced magnetization is not a FUNCTION of magnetic field (nor is "twist" a function of force) because the cubic would be "lying on its side" and we would have 3 values of induced magnetization for some values of magnetic field. Think of it as $x= y^3- 6y^2+ 9y$. The "switchback" section is between the two extrema for x, 4 and 18. In that region, the "switchback" section that connects the other two is an unstable equilibrium while the other two are stable. As you start increasing the magnetic field, you stay on the lower branch until you are past the local maximum x (in the example above, $x= 18$) and now the value jumps to the other branch. Reducing the magnetic field, you stay on the "upper" stable branch until you hit the local minimum x (in the example above, $x= 4$).
These and similar examples are magical to an audience- you actually see the catastrophe happen, and it's completely mysterious why. Until you explain it mathematically (and the explanation is simple enough for a general audience to understand), and then it becomes obvious.
Different but related magic is the double pendulum. As a young teenager I went to a popular talk on chaos in which this was explained, and I found it to be absolutely magical. Again, the mathematics behind it is accessible, and it's not difficult to convince people, at least implicitly, that this is something which humanity as a whole is richer for understanding.