Implementing "wild magic" with the logistic map

[Alister]

I was listening to someone today describe their idea for a magic system. Players were able to 'charge' spells with extra mana to magnify their effects. If spells were charged too much, they might instead backfire and become unpredictable. The amount a player could charge a spell while still keeping it predictable was determined by their stat value in 'magic control'.

I commented that it would be interesting if players could exploit this some how. For example, if the amount they charged a spell was just above the amount they could control, they'd have extra or augmented effects that would be predictable (or at least semi-predictable). However if they went any further, the effects would be completely unpredictable. Effectively, players could have the option of riding this dangerous balance of power vs. control and exploit 'wild magic' to power up their spells.

One way we talked about doing this was using an iterated map, like the logistic map. It's the map, x_(n+1) = r * x_n * (1 - x_n). What's really interesting about this map is that its behavior changes as the control parameter, r, is varied between [0, 4]. For r's in the range [0, 3], the iterated map converges to a single value. For r's in the range (3, 3.45], the map begins oscillating between two values, at 3.57 it begins oscillating between 4 values, then 8, 16, at later values of r. At approximately 3.83, the map becomes chaotic, forming a deterministic but unpredictable series. At various values beyond for r, the map shifts in and out of chaotic behavior. A visual depiction of the behavior of the logistic map for various values of r can be seen here.

Now consider using the logistic map to determine some 'special' outcome of a spell. The output of the map is always in the range [0, 1]. You could assign specific special outcomes to specific intervals over this range. You could iterate the map a number of times determined by how much you charge the spell plus some small random number. The control parameter, r, would be the ratio: r = K * charge/control where K is some scaling constant. When a spell is cast, the map is iterated the appropriate number of times with the specified value for r, and then the special outcome is determined by the output.

When the ratio of charge/control is on the cusp of a transition from predictable to chaotic behavior, casters can flexibly and reliably augment their spells by charging them because of the predictable nature of the map. But if they charge spells too much, the map goes into a chaotic regime, no longer with predictable outcomes (but also more varied outcomes). If charging a spell also increases the magnitude of the outcome, maybe something like this could result in a fun little magic system.

[KaVir]

This sounds like a really fun idea with some interesting potential, but I'd like to make a couple of observations: