1. MATH10001 Project 3 Difference Equations 2
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2. Non-Linear Difference Equations
yn+1 = yn 2
yn+2 = yn+1yn
yn+2 = sin(yn )
There is no standard method for solving non-linear difference equations.
Instead, we investigate properties of the difference equation.

3. Fixed points and linear stability
A fixed point Y of a difference equation is such that
yn = Y for all n  N for some N.
A fixed point Y is linearly stable if the sequence moves back to Y after it is slightly perturbed.
A fixed point Y is linearly unstable if the sequence moves away from Y after it is slightly perturbed.
e.g. a pendulum
unstable stable

4. Let yn = Y + ŷn where ŷn is a small perturbation.
We consider what happens to the sequence as n .
In a linear stability analysis we neglect terms in (ŷn)m, for m  2, assuming ŷn is small.
Example yn+1 = yn2.

5. Other methods for analysing non-linear difference equations
Other properties that can be considered are
(1) Boundedness – for which values of y0 does the sequence (yn)n0 remain within a certain range for all n.
(2) Periodicity – for which values of y0 does the sequence cycle through a finite set of values?
Example The Tent Map
yn+1 = { r yn if yn < ½
{ r(1 - yn) if yn  ½
where r > 0 is a constant.