If we start at the red starting state we move toward one equilibrium state.If we start at the black starting point we move toward another.
If we start at the blue starting state we seem to head somewhere else altogether.
Long term predictions for these similar starting states would be very different.
We are not saying here that nearby starting states will always drift apart in time.We are just saying there is no guarantee that close starting states will stay that way as time goes by.
This is because it is usually impossible to be sure that measurements of the state of a system at a given time are exactly right.In some cases, our measurements are necessarily coarse. If such a system also has sensitive dependence, any long term predictions we make will be rather questionable.
This is probably why weather forecasts are so unreliable, even a few days in advance. Our capacity to measure the global state of the weather is limited, and weather systems almost certainly exhibit sensitive dependence most of the time.
We usually say that a slight quantitative difference in starting state can lead to major qualitative differences in long term behaviour.