^{1,a)}and Hui Wu

^{1}

### Abstract

We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized oscillatorycluster patterns in a globally coupled Oregonator model where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the oscillation amplitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition exerted by the global feedback term. The transition from phase-locked to localized cluster patterns occurs as the intensity of global feedback increases. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models. The methods we use can be extended to the investigation of oscillatorychemical reactions with other types of non-local coupling.

The authors are grateful to David Fox for reading this manuscript. This work was supported by the National Science Foundation Grant DMS-0817241 (HGR).

I. INTRODUCTION

II. MODELS AND BACKGROUND

A. Oregonator model

B. Globally coupled Oregonator

C. The canard phenomenon

D. Two-cluster reduction

III. ANTIPHASE, OUT-OF-PHASE, AND IN-PHASE PATTERNS

IV. MECHANISMS OF GENERATION OF PHASE-LOCKED PATTERNS IN HOMOGENEOUS SYSTEMS

V. MECHANISMS OF GENERATION OF PHASE-LOCKED AND LOCALIZED PATTERNS IN HETEROGENEOUS SYSTEMS

VI. DISCUSSION

### Key Topics

- Oscillators
- 75.0
- Coupled cluster
- 28.0
- Phase separation
- 13.0
- Photochemical reactions
- 8.0
- Cluster formation reactions
- 7.0

## Figures

Dynamics of the uncoupled Oregonator for a representative set of parameter values. (a) Graphs of *x*(*t*) and *z*(*t*). (b) Nullclines. (c) Phase plane. We used the following parameter values: *q* = 0.01, η = 1, and ε = 0.025.

Dynamics of the uncoupled Oregonator for a representative set of parameter values. (a) Graphs of *x*(*t*) and *z*(*t*). (b) Nullclines. (c) Phase plane. We used the following parameter values: *q* = 0.01, η = 1, and ε = 0.025.

Supercritical canard phenomenon for the Oregonator for a representative set of parameters. (a) Small amplitude oscillations for η = 2.2358. (b) Large amplitude oscillations for η = 2.2357. (c) Large amplitude oscillations for η = 0.8. Left panels show the traces, middle panels show the phase-planes, and right panels show magnifications of the phase-planes around the lower knee. The canard critical value η_{ cr } lies in between the values of η corresponding to panels A and B. We used the following parameters: *q* = 0.01 and ε = 0.025. The curves *N* _{ x } and *N* _{ z } represent the *x*- and *z*-nullclines, respectively.

Supercritical canard phenomenon for the Oregonator for a representative set of parameters. (a) Small amplitude oscillations for η = 2.2358. (b) Large amplitude oscillations for η = 2.2357. (c) Large amplitude oscillations for η = 0.8. Left panels show the traces, middle panels show the phase-planes, and right panels show magnifications of the phase-planes around the lower knee. The canard critical value η_{ cr } lies in between the values of η corresponding to panels A and B. We used the following parameters: *q* = 0.01 and ε = 0.025. The curves *N* _{ x } and *N* _{ z } represent the *x*- and *z*-nullclines, respectively.

*x*- and *z*-traces for the globally coupled Oregonator model for a representative set of parameters. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 0.1, σ_{1} = 0.5, and σ_{2} = 0.5.

*x*- and *z*-traces for the globally coupled Oregonator model for a representative set of parameters. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 0.1, σ_{1} = 0.5, and σ_{2} = 0.5.

Effects of the global feedback parameter γ and the fraction α of oscillators in a cluster on the supercritical canard phenomenon for the Oregonator. (a) *x*-nullclines for α = 1 and various representative values of γ. (b) Amplitude diagram for α = 1 and various representative values of γ. (c) Amplitude diagram for γ = 1 and various representative values of α.

Effects of the global feedback parameter γ and the fraction α of oscillators in a cluster on the supercritical canard phenomenon for the Oregonator. (a) *x*-nullclines for α = 1 and various representative values of γ. (b) Amplitude diagram for α = 1 and various representative values of γ. (c) Amplitude diagram for γ = 1 and various representative values of α.

Phase evolution for the globally coupled Oregonator. The two oscillators were connected after a few cycles. The phase for these cycles correspond to the initial phase. (a_{1}) η = 2, σ_{1} = 0.5, and σ_{2} = 0.5. (a_{2}) η = 0.8, σ_{1} = 0.5, and σ_{2} = 0.5. (b_{1}) η = 2, σ_{1} = 0.2, and σ_{2} = 0.8. (b_{2}) η = 0.8, σ_{1} = 0.2 and σ_{2} = 0.8. (c) Phase planes for the autonomous part of the globally coupled Oregonator for γ = 1. In the top panel, the two cubic-like nullclines are identical, and therefore superimposed.

Phase evolution for the globally coupled Oregonator. The two oscillators were connected after a few cycles. The phase for these cycles correspond to the initial phase. (a_{1}) η = 2, σ_{1} = 0.5, and σ_{2} = 0.5. (a_{2}) η = 0.8, σ_{1} = 0.5, and σ_{2} = 0.5. (b_{1}) η = 2, σ_{1} = 0.2, and σ_{2} = 0.8. (b_{2}) η = 0.8, σ_{1} = 0.2 and σ_{2} = 0.8. (c) Phase planes for the autonomous part of the globally coupled Oregonator for γ = 1. In the top panel, the two cubic-like nullclines are identical, and therefore superimposed.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 0.8, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 0.8, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right) for the first cycle after global coupling is activated. The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Globally coupled Oregonator (Part I). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5. This figure continues in Fig. 10.

Globally coupled Oregonator (Part I). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5. This figure continues in Fig. 10.

Globally coupled Oregonator (Part II). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5. This figure is the continuation of Fig. 9.

Globally coupled Oregonator (Part II). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.5, and α_{2} = 0.5. This figure is the continuation of Fig. 9.

Globally coupled Oregonator. Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 0.8, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Globally coupled Oregonator. Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 0.8, ε = 0.025, *q* = 0.01, γ = 0.1, α_{1} = 0.5, and α_{2} = 0.5.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.2, and α_{2} = 0.8.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 1, α_{1} = 0.2, and α_{2} = 0.8.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 3.5, α_{1} = 0.2, and α_{2} = 0.8.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 3.5, α_{1} = 0.2, and α_{2} = 0.8.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8.

Dynamics of the globally coupled Oregonator model for a representative set of parameters. (a) *x*- and *z*-traces. The vertical dashed-lines indicate the time at which global coupling is activated. (For lower values of *t* the system is uncoupled.) (b) Trajectories in phase-space projected onto the (*x* _{1}, *z* _{1})-plane (left) (*x* _{2}, *z* _{2})-plane (right). The arrows indicate both the connection time and the direction of motion of the trajectory at the connection time. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8.

Globally coupled Oregonator (part I). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8. This figures continues in Fig. 16.

Globally coupled Oregonator (part I). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8. This figures continues in Fig. 16.

Globally coupled Oregonator (part II). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8. This figure is the continuation of Fig. 15.

Globally coupled Oregonator (part II). Snapshots of the phase-plane for the two globally coupled oscillators for representative values of *t*. We used the following parameters: η = 2, ε = 0.025, *q* = 0.01, γ = 5, α_{1} = 0.2, and α_{2} = 0.8. This figure is the continuation of Fig. 15.

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