Chaotic orbits for differentiable maps near anti-integrable limits

In this work, we study the dynamical system which can be transformed into a difference equation of the form ϵg(xt−n2,…,xt,…,xt+n1;ϵ)+g0(xt)=0ϵg(xt−n2,…,xt,…,xt+n1;ϵ)+g0(xt)=0 where ϵ∈Rϵ∈R is a parameter, g:Rn1+n2+1→Rg:Rn1+n2+1→R and g0:R→Rg0:R→R are both C1C1 functions and n1,n2n1,n2 are both nonnegative integers. Suppose that the function g0g0 has k   simple zeros where k≥2k≥2. We give criteria of existence of some kinds of chaotic orbits, and construct infinite ones explicitly. As applications of these results, we establish snap-back repellers and heteroclinical repellers for the modified Mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the Hénon maps, the Arneodo–Coullet–Tresser maps and the quadratic volume preserving maps.