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Research Papers

Amplitude Death Induced by Intrinsic Noise in a System of Three Coupled Stochastic Brusselators

[+] Author and Article Information
O. Díaz-Hernández

Facultad de Ciencias en Física y Matemáticas,
Universidad Autónoma de Chiapas,
Tuxtla Gutiérrez, Chiapas 29050, México
e-mail: orlando.diaz@unach.mx

Elizeth Ramírez-Álvarez

Facultad de Ciencias en Física y Matemáticas,
Universidad Autónoma de Chiapas,
Tuxtla Gutiérrez, Chiapas 29050, México
e-mail: eramirez@unach.mx

A. Flores-Rosas

Facultad de Ciencias en Física y Matemáticas,
Universidad Autónoma de Chiapas,
Tuxtla Gutiérrez, Chiapas 29050, México
e-mail: aros8151@gmail.com

C. I. Enriquez-Flores

Conacyt-Facultad de Ciencias en Física y
Matemáticas,
Universidad Autónoma de Chiapas,
Tuxtla Gutiérrez, Chiapas 29050, México
e-mail: chrienri@yahoo.com.mx

M. Santillán

Centro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional,
Monterrey, Nuevo León 66600, México
e-mail: msantillan@cinvestav.mx

Pablo Padilla-Longoria

Instituto de Investigaciones en Matemáticas
Aplicadas y en Sistemas,
Universidad Nacional Autónoma de México,
Mexico City 04510, México
e-mail: pablo@mym.iimas.unam.mx

Gerardo J. Escalera Santos

Facultad de Ciencias en Física y Matemáticas,
Universidad Autónoma de Chiapas,
Tuxtla Gutiérrez, Chiapas 29050, México
e-mail: gescalera.santos@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 11, 2018; final manuscript received December 11, 2018; published online February 15, 2019. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 14(4), 041004 (Feb 15, 2019) (7 pages) Paper No: CND-18-1212; doi: 10.1115/1.4042322 History: Received May 11, 2018; Revised December 11, 2018

In this work, we study the interplay between intrinsic biochemical noise and the diffusive coupling, in an array of three stochastic Brusselators that present a limit-cycle dynamics. The stochastic dynamics is simulated by means of the Gillespie algorithm. The intensity of the intrinsic biochemical noise is regulated by changing the value of the system volume (Ω), while keeping constant the chemical species' concentration. To characterize the system behavior, we measure the average spike amplitude (ASA), the order parameter R, the average interspike interval (ISI), and the coefficient of variation (CV) for the interspike interval. By analyzing how these measures depend on Ω and the coupling strength, we observe that when the coupling parameter is different from zero, increasing the level of intrinsic noise beyond a given threshold suddenly drives the spike amplitude, SA, to zero and makes ISI increase exponentially. These results provide numerical evidence that amplitude death (AD) takes place via a homoclinic bifurcation.

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Figures

Grahic Jump Location
Fig. 1

Plot of CV versus Ω, for three different K values and no-coupling, indicated inside this figure with different colors

Grahic Jump Location
Fig. 2

Time series of three uncoupled Brusselators. Plots of X versus time for three different oscillators (each color corresponds to one oscillator) with no coupling K =0 and various Ω values, the color mapping is indistinct for each oscillator. (a) Ω=0.3 V, (b) Ω=0.18 V, and (c) Ω=0.076 V.

Grahic Jump Location
Fig. 3

Time series of three coupled Brusselators. Plots of X versus time for three different oscillators (each color corresponds to one oscillator) in which K=4.0×10−2 T−1. We present simulations for various Ω values. (a) Ω=0.3 V, (b) Ω=0.18 V, and (c) Ω=0.076 V.

Grahic Jump Location
Fig. 4

Average spike amplitude for different intrinsic noise intensity and different coupling values. Plots of average spike amplitude versus Ω for different coupling values. The results corresponding to K =0 (i.e., no coupling) are represented with black-filled circles, while the results corresponding to K≠0 are represented with color-filled circles, the color code is indicate inside the figure.

Grahic Jump Location
Fig. 5

Average ISI for different intrinsic noise intensity and three coupling values. Plots of average interspike interval versus Ω, for three different values of K indicated inside this figure with different colors.

Grahic Jump Location
Fig. 6

(a) Average spike amplitude (filled circles) and ISI (empty circle) of stochastic spiking in the X variable as a function of the intrinsic noise Ω. The exponential increase in period and relatively constant amplitude of oscillation indicates a homoclinic bifurcation. In (b), the ln(Ω−ΩH) versus ISI curve can be fitted by a straight line indicating an underlying homoclinic bifurcation. The values of the ISI and the ASA are determined as the average over all stochastic oscillator.

Grahic Jump Location
Fig. 7

Plots of the order parameter for different intrinsic noise intensity and two coupling values. The results corresponding to K =0 (i.e., no coupling) are represented with empty circles, while the results corresponding to K=0.01 are represented with black-filled triangles.

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