Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hopf Bifurcation and Phase Patterns in Symmetric Ring Networks

Ian Stewart

TL;DR

Basic aspects of the theory are reviewed in some detail and general properties of ODEs coupled with the topology of a closed ring are derived and new results are obtained characterising the first bifurcation for long-range couplings and the direction in which discrete rotating wave states rotate.

Abstract

Systems of ODEs coupled with the topology of a closed ring are common models in biology, robotics, electrical engineering, and many other areas of science. When the component systems and couplings are identical, the system has a cyclic symmetry group for unidirectional rings and a dihedral symmetry group for bidirectional rings. Hopf bifurcation in equivariant and network dynamics predicts the generic occurrence of periodic discrete rotating waves whose phase patterns are determined by the symmetry group. We review basic aspects of the theory in some detail and derive general properties of such rings. New results are obtained characterising the first bifurcation for long-range couplings and the direction in which discrete rotating wave states rotate.

Hopf Bifurcation and Phase Patterns in Symmetric Ring Networks

TL;DR

Basic aspects of the theory are reviewed in some detail and general properties of ODEs coupled with the topology of a closed ring are derived and new results are obtained characterising the first bifurcation for long-range couplings and the direction in which discrete rotating wave states rotate.

Abstract

Systems of ODEs coupled with the topology of a closed ring are common models in biology, robotics, electrical engineering, and many other areas of science. When the component systems and couplings are identical, the system has a cyclic symmetry group for unidirectional rings and a dihedral symmetry group for bidirectional rings. Hopf bifurcation in equivariant and network dynamics predicts the generic occurrence of periodic discrete rotating waves whose phase patterns are determined by the symmetry group. We review basic aspects of the theory in some detail and derive general properties of such rings. New results are obtained characterising the first bifurcation for long-range couplings and the direction in which discrete rotating wave states rotate.
Paper Structure (32 sections, 13 theorems, 84 equations, 7 figures)

This paper contains 32 sections, 13 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Hopf Bifurcation
  3. Phase Patterns
  4. Networks and Ring Topology
  5. Symmetry Induces Phase Patterns
  6. Summary of Paper
  7. Background on Dynamics and Bifurcations
  8. Local Bifurcation
  9. Normalised Period and Phase Shifts
  10. Symmetries
  11. Equivariant and Network Dynamics
  12. Equivariant Dynamics
  13. Admissible ODEs in Network Dynamics
  14. Local Bifurcation in Equivariant Dynamics
  15. Isotropy Subgroups and Fixed-Point Subspaces
  16. ...and 17 more sections

Key Result

Proposition 3.4

Let $f:X \rightarrow X$ be a $\Gamma$-equivariant map, and let $\Sigma$ be any subgroup of $\Gamma$. Then $\hbox{\rm Fix}(\Sigma)$ is an invariant subspace for $f$, and hence for the dynamics of E:ODEapp.

Figures (7)

  • Figure 1: Walk of an elephant at $\hbox{\footnotesize $\frac{1}{4}$}$-period intervals.
  • Figure 2: Left: 5-node unidirectional ring. Right: 6-node bidirectional ring.
  • Figure 3: Periodic state created by Hopf bifurcation in a $\mathbb{Z}_3$-symmetric ring. Solid thick = $x_1$, solid thin = $x_2$, dashed = $x_3$. Parameters $a_0= -0.9, a_1 = -2$ (just after bifurcation).
  • Figure 4: Hopf bifurcation in a $\mathbb{Z}_5$ ring for the ODE \ref{['E:Z51D']}. Horizontal coordinate: $\lambda$. Vertical coordinates: $x_i$ for $0 \leq i \leq 4$. Thick solid = node 1, thick dash = node 2, thin solid = node 3, thin dash = node 4, thick dot-dash = node 5. Parameters $\lambda= -1.1,a = -2$.
  • Figure 5: A $7$-node ring with NN (solid arrows) and NNN (dashed arrows) coupling.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Example 2.2
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • proof
  • Definition 3.6
  • ...and 24 more