Stability of Affine Difference Equation

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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.

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