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Stability of Affine Difference Equation
If the absolute value of _{ } is less than 1 ( _{ } or _{ } ), the long run solution will be equal to the 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.
Even if the absolute value of _{ } is not less than 1 ( _{ } or _{ } ), the long run solution may be equal to the equilibrium if the value of _{ } is chosen properly to get a constant solution (i.e. _{ } ). This equilibrium value is unstable or repelling because any deviation, however slight, will prevent the solution from returning to its equilibrium value. For _{ } , if _{ } the solution will never be a constant (no equilibrium value, go without bound) and if _{ } then the solution is always a constant. In this later case the equilibrium is unstable (repelling) because if the solution is perturbed, it remains at its perturbed value and does not return to its original value.
See also: Application in Personal Finance
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Preferable reference for this tutorial is
Teknomo, Kardi (2015) Difference Equation Tutorial. https:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\