Why is it that in complex systems things move very slowly and then everything changes? visualization 
Nothing much changes. Then, everything changes.
There's a tool that can help us understand this: attractor landscapes. Or, in less fancy words: "a ball rolling down some hills". This tool was first created in the field of physics, but has since been used to help us understand genetics, neuroscience, political alliances, and more.
I'll explain attractors using an environmental example. Let's say you're fishing on a small, sucky pond. You can exhaust your natural resources of fish pretty easily...
If you don't catch too much fish, the ecosystem stays "attracted" to Population=70. You're sustainable! However, catch too much, and the ecosystem's "attracted" to Population=0. You've killed them all!