Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Steady-state bifurcations for three-cell networks with asymmetric inputs

Manuela Aguiar, Ana Dias, Pedro Soares

Abstract

We consider homogeneous coupled cell networks with asymmetric inputs. We obtain general results concerning codimension-one steady-state bifurcations for networks with any number of cells and any number of asymmetric inputs. These results rely solely on the network adjacency matrices eigenvalue structure and the existence, or not, of network synchrony subspaces. For networks with three-cells, we describe the possible lattices of synchrony subspaces annotated with the eigenvalues on each synchrony subspace. Applying the previous results, we classify the synchrony-breaking steady-state bifurcations that can occur for three-cell minimal networks with one, two or six asymmetric inputs.

Steady-state bifurcations for three-cell networks with asymmetric inputs

Abstract

We consider homogeneous coupled cell networks with asymmetric inputs. We obtain general results concerning codimension-one steady-state bifurcations for networks with any number of cells and any number of asymmetric inputs. These results rely solely on the network adjacency matrices eigenvalue structure and the existence, or not, of network synchrony subspaces. For networks with three-cells, we describe the possible lattices of synchrony subspaces annotated with the eigenvalues on each synchrony subspace. Applying the previous results, we classify the synchrony-breaking steady-state bifurcations that can occur for three-cell minimal networks with one, two or six asymmetric inputs.

Paper Structure

This paper contains 16 sections, 12 theorems, 62 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Steady-state bifurcations for coupled cell networks
  3. Coupled cell networks
  4. Coupled cell systems
  5. Synchrony subspaces
  6. Steady-state bifurcations for networks
  7. Steady-state bifurcations for networks with asymmetric inputs: general results
  8. Simple eigenvalue and maximal synchrony space
  9. Semisimple eigenvalue and submaximal synchrony spaces
  10. Defective synchrony spaces
  11. Steady-state bifurcations for three-cell networks with asymmetric inputs
  12. Eigenvalue structure
  13. Lattice structures
  14. Bifurcation diagrams
  15. Case study: three-cell networks with one, two or six asymmetric inputs
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $\mathcal{N}$ be a network with $k$ asymmetric inputs and $\mu$ a network eigenvalue. Assume that $f \in \mathcal{V}_{\mu}({\mathcal{N}})$ is generic, $\mu$ is simple and that $\Delta$ is a $\mu$-maximal synchrony subspace of $\mathcal{N}$. Then, besides the trivial branch, there exists a bifurc

Figures (4)

  • Figure 1: Examples of coupled cell networks with asymmetric inputs.
  • Figure 2: A representative of the minimal class of the networks with three-cells and six asymmetric inputs.
  • Figure 4: Bifurcation diagrams displaying the synchrony of bifurcation branches of steady-state solutions emerging from bifurcation problems with the mention bifurcation condition. Here, $\Delta_0$ denotes the full-synchrony subspace and $\Delta_1$, $\Delta_2$, $\Delta_3$ denote two-dimensional synchrony subspaces. Also, $\upsilon$ is the valency eigenvalue, and $\mu$, $\mu_1$, $\mu_2$ are other network eigenvalues. It does not display the stability, growth-rate nor the number of branches.
  • Figure : $m_a(\upsilon)=1$, $m_a(\mu) = 2$ and $m_g(\mu) =1$

Theorems & Definitions (47)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 37 more