Chaotic functions and Deterministic function interaction

Using the movie Sliding Doors as an example we see sensitivity to initial conditions, Chaos. In each trajectory of life, the Chaotic conditions cause different interactions with the world around the person. Now consider everything as a function (everybody) with a trajectory. In one situation the initial function may interact with another function's trajectory or not. If it does, it changes the trajectory of that other function (person.) In other words, one trajectory of the Chaotic function with one set of intial conditions may cause it "bump" into the another function's trajectory (an interaction with a person) and cause it to change from it's "normal"or deterministic behavior. The question in my mind is this. Given the function space of deterministic functions, if we include in this space a Chaotic function and build a rule for the interaction of the Chaotic function with a deterministic function, what happens after x amountof time to the function space and the functions in that space?