Stability and Phase Diagram

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We can be determined graphically the equilibrium value of a difference equation, if it is exist, by plotting the value of   as horizontal axis and  as the vertical axis. The point of intersection of the graph of the difference equation with the line  is the equilibrium values.

Start from initial value , we take vertical line to the graph, and then take horizontal line to the line . From here we again take vertical line to the graph. Repeating this task will eventually lead us to the equilibrium point if the point exists.

Example below is for iterated function  for = 0.5 lead to zero equilibrium point.

Suppose at some point the solution of a difference equation deviates form the equilibrium value. Will the solution return to the equilibrium value? This problem is called stability problem of the difference equation.

Suppose we are studying the growth of a population and suppose the population has reached a point where for all intents and purposes it is not changing. We say that the population is in equilibrium with its surrounding and the value of the population is the population equilibrium value. Now suppose there is a disaster and 10% of the population is suddenly killed. Will the population return to its original equilibrium value? Will it oscillate? Will it become extinct? Will the population find a new equilibrium value?

We called the equilibrium value is attracting or stable. Regardless the choice of , the solution of a stable difference equation will stabilize itself even if it is temporarily perturbed from its course. The equilibrium is called unstable (repelling) if the solution is perturbed, it remains at its perturbed value and does not return to its original value.

See: Numerical example

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