Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Möbius group actions in the solvable chimera model

Vladimir Jaćimović, Aladin Crnkić

Abstract

We study actions of Möbius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe-Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotations after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators' phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the "hidden" variable that is not visible on macroscopic level.

Möbius group actions in the solvable chimera model

Abstract

We study actions of Möbius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe-Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotations after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators' phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the "hidden" variable that is not visible on macroscopic level.

Paper Structure

This paper contains 12 sections, 10 theorems, 24 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Low dimensional dynamics on orbits of the Möbius group
  3. Evolution on the Poisson manifold
  4. Mathematical setup and preliminaries
  5. Simulations: Möbius transformations in stable and breathing chimeras on the Poisson manifold
  6. Case I: Simulations of stable chimera at the Poisson manifold
  7. Case II: Simulations of breathing chimera at the Poisson manifold
  8. Discussion: Möbius group actions in the stable chimera state on the Poisson manifold
  9. Simulations: Möbius group actions on a generic 3-dimensional invariant submanifold
  10. Discussion: Difference between evolution on the Poisson manifold and on generic 3-dimensional invariant submanifolds
  11. Conclusion
  12. Broader significance

Key Result

Proposition 1

Consider a population of oscillators governed by eqs. (chimera). There exist two one-parametric families $g_t^A$ and $g_t^B$ of Möbius transformations from $G$, such that

Figures (9)

  • Figure 1: Real order parameters $r_A(t)$ and $r_B(t)$ for sub-populations $A$ and $B$ in model (\ref{['chimera']}) with $\mu = 0.623, \nu = 0.377, \beta = \frac{\pi}{2} - 0.1$ (stable chimera) and uniform initial distributions on time intervals (a) $t \in [0,300]$ and (b) $t \in [0,3000]$.
  • Figure 2: Möbius group actions as the system (\ref{['chimera']}) evolves towards a stable chimera state: (a) $\alpha_A(0;t)$ and (b) $\alpha_B(0;t)$ on time interval $t \in [0,1000]$. Initial distributions of oscillators are uniform.
  • Figure 3: Möbius group action for desynchronized sub-population $B$ in stable chimera state: $\alpha_B(1000;t)$ at time interval $t \in [1000,1500]$. Notice that $\alpha_B(1000;t)$ returns to zero infinitely many times.
  • Figure 4: Real order parameters $r_A(t)$ and $r_B(t)$ for sub-populations $A$ and $B$ in model (\ref{['chimera']}) with $\mu = 0.675, \nu = 0.325, \beta = \frac{\pi}{2} - 0.1$ (breathing chimera) and uniform initial distributions on time intervals (a) $t \in [0,230]$ and (b) $t \in [0,1000]$.
  • Figure 5: Möbius group actions as the system (\ref{['chimera']}) evolves towards breathing chimera: (a) $\alpha_A(0;t)$ and (b) $\alpha_B(0;t)$ on time interval $t \in [0,1000]$. Initial distributions of oscillators are uniform.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Proposition 3
  • Proposition 4
  • Definition 1
  • Proposition 5
  • Proposition 6
  • ...and 8 more