Local Analysis of Weakly Connected Maps

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

Abstract

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
$$ X_i \mapsto F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X_i ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n, }}\varepsilon \ll {\text{1,}} $$
(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.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations