Local Analysis of Weakly Connected Maps

  • Frank C. Hoppensteadt
  • Eugene M. Izhikevich
Part of the Applied Mathematical Sciences book series (AMS, volume 126)


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
$$ X_i \mapsto F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X_i ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n, }}\varepsilon \ll {\text{1,}} $$
where the variables X i ∈ ℝ m , 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.


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Frank C. Hoppensteadt
    • 1
  • Eugene M. Izhikevich
    • 1
  1. 1.Center for Systems Science and EngineeringArizona State UniversityTempe

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