Paper 2, Section I, B

Consider the following difference equation for real $u_{n}$ :

$u_{n+1}=a u_{n}\left(1-u_{n}^{2}\right)$

where $a$ is a real constant.

For $-\infty<a<\infty$ find the steady-state solutions, i.e. those with $u_{n+1}=u_{n}$ for all $n$, and determine their stability, making it clear how the number of solutions and the stability properties vary with $a$. [You need not consider in detail particular values of $a$ which separate intervals with different stability properties.]

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