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Phase Diagram

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

Start from initial value Stability and Phase Diagram , we take vertical line to the graph, and then take horizontal line to the line Stability and Phase Diagram . 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 Stability and Phase Diagram for Stability and Phase Diagram = 0.5 lead to zero equilibrium point.

Stability and Phase Diagram


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 this kind of equilibrium value is attracting or stable. Regardless the choice of Stability and Phase Diagram , the solution of a stable difference equation will stabilize itself even if it is temporarily perturbed from its course. The equilibrium is called unstable or 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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Preferable reference for this tutorial is

Teknomo, Kardi (2015) Difference Equation Tutorial. https:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\