Chaotic sub-dynamics in coupled logistic maps

We study the dynamics of Laplacian-type coupling induced by logistic family fμ(x)=μx(1−x), where μ∈[0,4], on a periodic lattice, that is the dynamics of maps of the form F(x,y)=((1−ε)fμ(x)+εfμ(y),(1−ε)fμ(y)+εfμ(x)) where ε>0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.