given system, be expressed as:
where λ is the “Lyapunov exponent,” and quantifies the rate of divergence. The time-scales at which chaotic effects come to dominate the dynamics of the system, then depend on two factors: the value of the Lyapunov exponent, and how much divergence we’re willing to allow between two trajectories before we’re willing to consider it significant. For systems with a relatively small Lyapunov exponent, divergence at short timescales will be very small, and will thus likely play little role in our treatment of the system (unless we have independent reasons for requiring very great precision in our predictions). Likewise, there may be cases when we care only about whether the trajectory of the system after a certain time falls into one or another region of state space, and thus can treat some amount of divergence as irrelevant.
This point is not obvious but it is very important; it is worth considering some of the mathematics in slightly more detail before we continue on. In particular, let’s spend some time thinking about what we can learn by playing around a bit with the definition of a chaotic system given above.
To begin, let be some neighborhood on such that all pairs of points
That is, let be some neighborhood in an n-dimensional space such that for all pairs of points that are in , the distance between those two points is less than or equal to some small value epsilon. If is the state space of some dynamical system with Lyapunov exponent , then