Periodic points of the logistic map with diffusion (Q2733868)

The authors deal with the so-called ``logistic map with diffusion'', that is NEWLINE\[NEWLINEu_{n+1}= F(u_n)\tag{1}NEWLINE\]NEWLINE with \(F(u)= (r-\Delta_x)^{-1}\mu h(u)\), \(h(u)= u(1- u)\), where \(\Delta_x\) is the Laplacian, \(r= {1\over \alpha k}\), \(\mu= {1+\lambda k\over\alpha k}\), \(\alpha> 0\), \(k>0\). The authors show that under some natural boundary condition (1) has periodic points of arbitrary period. A proof of this result relies on a compact perturbation of the one-dimensional logistic map.