Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

doi:10.1007/s11467-022-1223-9

Front. Phys.

2023, Vol. 18

Issue (1) : 13309 https://doi.org/10.1007/s11467-022-1223-9

RESEARCH ARTICLE

Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Lei Yang, Yuan He, Bing-Wei Li(

)

School of Physics, Hangzhou Normal University, Hangzhou 311121, China

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Abstract

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → 0) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart−Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh−Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.

Keywords spiral wave chimeras reaction-diffusion systems oscillator−environment coupling pattern formation

Corresponding Author(s): Bing-Wei Li

Issue Date: 30 November 2022

Cite this article:

Lei Yang,Yuan He,Bing-Wei Li. Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment[J]. Front. Phys. , 2023, 18(1): 13309.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1223-9
https://academic.hep.com.cn/fop/EN/Y2023/V18/I1/13309

Fig.1 SWCs in populations of SL oscillators coupled via a slowly changing environment. (a) An SWC for the real component of

W

, and (b) the corresponding phase, i.e.,

ϕ

, of SWCs, and (c) enlarged view of the core region in (b). (d) The variation of

Δ ϕ = ϕ i + 1, N / 2 − ϕ i, N / 2

with respect to

x

along the horizontal central axis in (c). (e) A spiral wave for the real component of

S

, and (f) the corresponding phase of spiral waves and (g) enlarged view of the core region in (f). (h) The variation of

Δ ϕ

with respect to

x

along the horizontal central axis in (g). (i) Temporal profile of

R e W i, j

and

R e S i, j

inside the core region with

i = 500

and

j = 512

. Parameters are

α = − 0.2

and

K = 0.5

Fig.2 The amplitude (modules) of the component of

W

and

S

and corresponding phase portraits. (a)

| W |

near the core and (b) the variation of

Δ | W | = | W i + 1, N / 2 | − | W i, N / 2 |

with respect to

x

along the center line. (c) The phase portrait in the

R e W − I m W

plane. (d)

| S |

near the core and (e) the variation of

Δ | S | = | S i + 1, N / 2 | − | S i, N / 2 |

with respect to

x

along the center line. (f) The phase portrait in the

R e S − I m S

plane. All the parameters are the same as in Fig.1.

Fig.3 The dynamical state of spiral wave chimeras as a function of

K

. (a−c) three spiral wave chimeras for

K = 0.6

0.7

and

0.8

. (d−f) The averaged order parameter

? R ?

corresponding to (a−c). (g) Regions for different dynamical states for

K

. The circles in this panel denote the core diameter for the corresponding coupling strength

K

. SWC: Spiral wave chimera. SW: Spiral wave. The local dynamics parameter

α

− 0.2

Fig.4 Two typical states of unstable spiral wave chimeras observed in the small coupling strength

K

. (a) Spiral wave chimera state with core break and (b) enlarged view of the core region in (a) for

K = 0.35

. (c) The variation of

Δ R e W = R e W 491, j + 1 − R e W 491, j

with respect to

y

in (b). (d) A turbulent-like state and (e) enlarged view of the center region in (d) for

K = 0.2

. (f) The variation of

Δ R e W =

R e W 522, j + 1 − R e W 522, j

with respect to

y

in (e). Other parameters are the same as in Fig.3.

Fig.5 The effects of

α

on rotation direction of spiral wave chimeras for

K = 0.65

. (a) A spiral wave chimera for

R e W

and (b) spiral wave for

R e S

rotating clockwise for

α = + 0.35

. (c) The spatial distribution of

D (S / x)

. (d) Spiral wave chimeras for

R e W

and (e) spiral wave for

R e S

rotating counterclockwise for

α = − 0.35

. (f) The spatial distribution of

D (S / x)

. The arrows with the solid (dashed) line denote the curl (rotation) direction in (a) and (d). The arrows in (b) and (e) denote the direction of wave propagation.

Fig.6 The dispersion relation given by Eq. (9). (a)

α =

0.35

. (b)

α = − 0.35

. The other parameters are the same as in Fig.5.

Fig.7 Mechanism analysis of the spiral wave chimera formation. (a) Frequency profile with respect to

x

along the center line in Fig.1. The left and right light green regions mean synchronization between

W

and

S

while the centered yellow region denotes the desynchronized. (b)

| S |

with respect to

x

along the center line. (c) Frequency difference between the forcing and measured frequency, i.e.,

| Δ ω | = | ω m − ω f |

as a function of the forcing amplitude

A f

for a local system with

ω f = 0.0046

. Other parameters are the same as in Fig.1.

Fig.8 Spiral wave chimeras in populations of FHN oscillators coupled via a slowly changing environment. (a) A snapshot of spiral wave chimeras for the

u

component and (b) enlarged view of the core region in (a). (c) The variation of

Δ u = u i + 1, N / 2 − u i, N / 2

with respect to

x

along the center line. (d) Time-averaged amplitude,

A w (x)

, along central horizontal axis in (b). (e) Frequency difference between the forcing and measured frequency, i.e.,

| Δ ω | = | ω m − ω f |

as a function of forcing amplitude

A f

for a local system.

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[1]	Ben Cao, Ya-Feng Wang, Liang Wang, Yi-Zhen Yu, Xin-Gang Wang. Cluster synchronization in complex network of coupled chaotic circuits: An experimental study[J]. Front. Phys. , 2018, 13(5): 130505-.

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