Suppose we are numerically integrating a chaotic system and

Suppose we are numerically integrating a chaotic system and two nearby initial conditions lead to very different trajectories on our first attempt. If we use small enough time steps, will the trajectories stay close together when we ?

Solution

When we consider small time steps, initially the trajectories will be close, but as the number of time steps increase the distance between the initial condition will tend to be the same as in our first attempt and hence the trajectories will eventually get tend to the shapes that we got in our first attempt. Therefore, initially the trajectories will stay closer but eventually they will move away.