Can universal continuity be experimentally falsified? It is an unresolved question whether the universe is discrete or continuous in its intricate quantum level structure.
See for example:
Is the universe finite and discrete?
How could spacetime become discretised at the Planck scale?
Is time continuous or discrete?
It is often stated that it is beyond our reach to resolve this issue. See for example:
Is time continuous or discrete?
Is this however really true? Consider a simple dynamical system, such as the Lorenz attractor. When you solve this system numerically it quickly becomes evident that the solutions found depend heavily on the numerical precision. The number of revolutions around one attractor point before the evolving curve moves to the other attractor point varies with numerical precision. At some point you can wonder if you are really studying general behavior rather than a near exact solution.
Would it be possible to set up an actual experiment with a highly non-linear system, exhibiting long term iteration, to show whether or not the real-world solution at some point deviates from high-precision numerical simulation? 
 A: I don't have an answer for the discrete/continuous question, but experiments like the one you propose won't resolve it.
What you are describing is what is often called 'sensitive dependence on initial conditions' or SDIC, and is very common, as you know.  The problem with such systems is that they are really sensitive to initial conditions, and not just to the convenient subset of initial conditions that you'd like them to be sensitive to: the ones you might stand a chance of measuring.  As the system evolves its state starts depending on where you are standing in the lab, then on what kind of shoes you are wearing, then on where everyone is standing, then on the position of the Mars rovers, then of the positions of all the dust in the Solar System, and so on.  No simulation can take the initial conditions into account in any useful way.
This is not a joke: there is a lovely thought-experiment called 'the electron at the edge of the universe': in this experiment you are asked to consider a perfect billiard-ball classical gas in a perfect box which insulates everything but gravity.  Everything is Newtonian physics.   The universe outside the box is empty, except that there is a single electron 'at the edge' of it -- billions of light years away somewhere, but you don't know where it is.  You know all the initial conditions apart from where this wretched electron is, and so you don't know its gravitational influence on the gas.
So the question is: how far can you predict the microstate of this system forward? About how many collisions does each particle undergo before the first particle leaves a collision 90° from where you predict?  (After that point you clearly can't say anything useful.)
The answer is about 50.
A: Naturally, we are only operating in our current level of awareness about the universe. We cannot argue about what we don't know yet. Right now, it appears that indeed the universe is not continuous, that there is a quantum of energy, time, and so on. But as you also probably know, relativity does not agree very well with quantum mechanics, so we do not seem to know the full story. There are unification theories out there, but none of them is complete enough to be accepted as "the truth". 
On the other hand, you are mixing math with physics in you argument about the Lorenz attractor. Solving such a differential equation as the Lorenz attractor is done mathematically with continuous functions. So of course, the mathematical language used in this problem does not give you any discrete effects equivalent to Planck's scale. You also talk about numerical precision: that is a computational problem that has nothing to do with the real, physical system, or the mathematical solution to it.
Finally, and going back to my first argument about our current scientific awareness of the universe, quantum mechanics tells us that the principle of uncertainty would render impossible the thought experiment you are describing. Such an experiment would require not only huge numerical precision (that is, in principle, achievable), but also near-perfect detection precision which is quantum-mechanically unfeasible.
