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Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system
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10.1063/1.4749792
/content/aip/journal/jcp/137/10/10.1063/1.4749792
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/10/10.1063/1.4749792
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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, σ1 = 0.5, and σ2 = 0.5.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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. (a1) η = 2, σ1 = 0.5, and σ2 = 0.5. (a2) η = 0.8, σ1 = 0.5, and σ2 = 0.5. (b1) η = 2, σ1 = 0.2, and σ2 = 0.8. (b2) η = 0.8, σ1 = 0.2 and σ2 = 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.

Image of FIG. 6.
FIG. 6.

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, α1 = 0.5, and α2 = 0.5.

Image of FIG. 7.
FIG. 7.

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, α1 = 0.5, and α2 = 0.5.

Image of FIG. 8.
FIG. 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) 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, α1 = 0.5, and α2 = 0.5.

Image of FIG. 9.
FIG. 9.

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, α1 = 0.5, and α2 = 0.5. This figure continues in Fig. 10.

Image of FIG. 10.
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, α1 = 0.5, and α2 = 0.5. This figure is the continuation of Fig. 9.

Image of FIG. 11.
FIG. 11.

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, α1 = 0.5, and α2 = 0.5.

Image of FIG. 12.
FIG. 12.

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, α1 = 0.2, and α2 = 0.8.

Image of FIG. 13.
FIG. 13.

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, α1 = 0.2, and α2 = 0.8.

Image of FIG. 14.
FIG. 14.

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, α1 = 0.2, and α2 = 0.8.

Image of FIG. 15.
FIG. 15.

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, α1 = 0.2, and α2 = 0.8. This figures continues in Fig. 16.

Image of FIG. 16.
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, α1 = 0.2, and α2 = 0.8. This figure is the continuation of Fig. 15.

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/content/aip/journal/jcp/137/10/10.1063/1.4749792
2012-09-14
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/10/10.1063/1.4749792
10.1063/1.4749792
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