Paper 2, Section I, B

Differential Equations | Part IA, 2018

Consider the following difference equation for real unu_{n} :

un+1=aun(1un2)u_{n+1}=a u_{n}\left(1-u_{n}^{2}\right)

where aa is a real constant.

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

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