Sensitive Dependence

Sometimes different starting states will "drift apart" after a while. This may lead to quite different long term behaviour.

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.

In some systems, no matter how close our starting states are we can never be sure that they won't drift apart after a while.

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.

In such systems, accurate long term predictions are impossible from a practical point of view.

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.

The future course of a system like this is sensitive to even the slightest change in the starting state.

We usually say that a slight quantitative difference in starting state can lead to major qualitative differences in long term behaviour.

This classic feature of chaotic systems is called "sensitive dependence".