There are two main mechanisms:

* chaotic dynamics; and
* multistability.

In the first case, chaos, your $S_X$ diverges trivially, from the definition itself $-$ in the chaotic regime nearby trajectories in phase space diverge exponentially. That's probably the explanation most closely related to the weather "instability".

Multistability means that more then one outcome is allowed by the dynamics, and it's the initial condition that determines which one is eventually attained. This situation can lead to sudden changes when the system finds itself close to the boundary between regions in phase space that lead to different outcomes. And when the region in phase space leading to the desired outcome is relatively small, it's hard *not* to be close to the boundary. Not falling, in the rock climbing example is the desired region in phase space, and it's rather small $-$ one typically has a safety margin of a few millimeters or less.