Local Analysis of Weakly Connected Maps

In previous chapters we studied dynamics of weakly connected networks governed by a system of ordinary differential equations. It is also feasible to consider weakly connected networks of difference equations, or mappings, of the form 1$$ X_i \mapsto F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X_i ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n, }}\varepsilon \ll {\text{1,}} $$ where the variables X _i ∈ ℝm, the parameters λ ∈ Λ,ρ ∈ R and the functions F _i and G _i have the same meaning as in previous chapters. The weakly connected mapping (7.1) can be also written in the form $$ X_i^{k + 1} = F_i \left( {X_i^k ,\lambda } \right) + \varepsilon G_i \left( {X^k ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n, }}\varepsilon \ll {\text{1,}} $$ where X _k is the kth iteration of the variable X. In this chapter we use form (7.1) unless we explicitly specify otherwise.