After a saddle-node bifurcation (for example) a system is left with a region of critical slowing down. It can be thought of as the ghost of the fix point that used to exist in that region. Thus observing local phenomena in the system can make it look like the system is still stable while in reality it isn't.
I guess the most famous example of this is global warning. As our emissions of greenhouse gases increased during the industrial revolution, the environment still seemed stable. But in reality we had already deleted the stable point around with our planets temperature oscillated.