Current drop in black box
Inside, there's a microcontroller and a battery. The microcontroller is programmed to do exactly what you describe. Aside from Monday evenings: Every Monday at 7pm, it morses the text of the French national anthem out.

No, seriously, when trying to reason about what's inside a black box, you need to first describe a model of what it might be.

For beginner's problems, that usually restricts it to linear circuits, i.e. ones that are made of resistors, capacitors, inductors, and maybe transformers. 

Linearity is clearly not given in your case, so that means that your model would at least include one non-linear element. Maybe it's a single diode, or a single transistor, or maybe it's a combination of a couple thousand of these to actually make the microcontroller I jokingly introduced in the beginning. 

You'll need to do more testing:


is the system time-invariant? Time-invariant when only excited with harmonic signals?
is the system maybe exponential in its in/output behaviour? can you find any other model?
is there secondary information on what might be inside? Is it possible that it actually contains a power source (e.g. the battery mentioned above), or can we rule that out?