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Emergence of multicluster chimera states.

Yao N, Huang ZG, Grebogi C, Lai YC - Sci Rep (2015)

Bottom Line: This phenomenon was typically studied in the setting of non-local coupling configurations.We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift.The theoretical prediction agrees well with numerics.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Physics, Xi'an University of Technology, Xi'an 710054, China.

ABSTRACT
A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.

No MeSH data available.


Related in: MedlinePlus

Spatiotemporal patterns associated with multicluster chimera states after link removal.(a,c) Typical spatiotemporal patterns of the order parameter R for different values of η, a parameter characterizing the extent of link removal. The magnitudes for different panels are indicated using the respective color bars. The results are obtained for parameters A = 0.995 and α = 1.39 and system size N = 256, and for over 2 × 105 time steps. The values of η for different panels are (a) 0.43, 0.70, 0.75, 0.80, 0.87, and (c) 0.46, 0.54, 0.59, 0.66, 0.68 from left to right, respectively. (b) Difference in the order parameter, , between the m-cluster chimera state and the corresponding hypothetical synchronous state, where the values of the average order parameter  are displayed in the inset. The order parameters calculated through direct simulation of Eq. (1) (red pluses) and from the PDE Eq. (5) (blue circles) agree well with each other. The number of clusters emerged as η is increased is m = 4, 3, 5, 7, 9, and so on, and they are distinguished using white and gray backgrounds.
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f2: Spatiotemporal patterns associated with multicluster chimera states after link removal.(a,c) Typical spatiotemporal patterns of the order parameter R for different values of η, a parameter characterizing the extent of link removal. The magnitudes for different panels are indicated using the respective color bars. The results are obtained for parameters A = 0.995 and α = 1.39 and system size N = 256, and for over 2 × 105 time steps. The values of η for different panels are (a) 0.43, 0.70, 0.75, 0.80, 0.87, and (c) 0.46, 0.54, 0.59, 0.66, 0.68 from left to right, respectively. (b) Difference in the order parameter, , between the m-cluster chimera state and the corresponding hypothetical synchronous state, where the values of the average order parameter are displayed in the inset. The order parameters calculated through direct simulation of Eq. (1) (red pluses) and from the PDE Eq. (5) (blue circles) agree well with each other. The number of clusters emerged as η is increased is m = 4, 3, 5, 7, 9, and so on, and they are distinguished using white and gray backgrounds.

Mentions: Numerically, we observe a variety of rich phenomena when links are systematically removed. In particular, using the real order parameter R(x, t), we can identify the emergence of multiple cluster chimera states, where each cluster corresponds to a coherent group of oscillators. Figure 2 shows the spatiotemporal patterns of the emergent m-cluster chimera states for different intervals of η, which indicates that the emergence of the chimera-state patterns is robust with respect to reasonable variations of these parameters. In the simulations, the system parameters are A = 0.995 and α = 1.39, and the initial condition is generated89 using the function ϕ(x) = 6r exp(−0.76x2), where r is a random variable uniformly distributed in [−1/2,1/2]. In fact, the results obtained from direct simulations of Eq. (1) for finite-size networks and from the PDE approach [Eq. (5)] in the continuum limit N → ∞ agree with each other with similar spatiotemporal patterns. As shown in the inset of Fig. 2(b), the degree of synchrony as characterized by for different η values differs by orders of magnitude. For clarity, we use different color bars to distinguish the magnitudes of the spatiotemporal patterns in different panels. As η is increased, the number m of clusters undergoes changes from 4 to 3 (or 3&6), to 5 (or 5&10), to 7, and to 9, etc. Here the 3&6 state (or the 5&10 state) is a state that switches between 3-cluster and 6-cluster (or between 5-cluster and 10-cluster) chimera behaviors.


Emergence of multicluster chimera states.

Yao N, Huang ZG, Grebogi C, Lai YC - Sci Rep (2015)

Spatiotemporal patterns associated with multicluster chimera states after link removal.(a,c) Typical spatiotemporal patterns of the order parameter R for different values of η, a parameter characterizing the extent of link removal. The magnitudes for different panels are indicated using the respective color bars. The results are obtained for parameters A = 0.995 and α = 1.39 and system size N = 256, and for over 2 × 105 time steps. The values of η for different panels are (a) 0.43, 0.70, 0.75, 0.80, 0.87, and (c) 0.46, 0.54, 0.59, 0.66, 0.68 from left to right, respectively. (b) Difference in the order parameter, , between the m-cluster chimera state and the corresponding hypothetical synchronous state, where the values of the average order parameter  are displayed in the inset. The order parameters calculated through direct simulation of Eq. (1) (red pluses) and from the PDE Eq. (5) (blue circles) agree well with each other. The number of clusters emerged as η is increased is m = 4, 3, 5, 7, 9, and so on, and they are distinguished using white and gray backgrounds.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4563257&req=5

f2: Spatiotemporal patterns associated with multicluster chimera states after link removal.(a,c) Typical spatiotemporal patterns of the order parameter R for different values of η, a parameter characterizing the extent of link removal. The magnitudes for different panels are indicated using the respective color bars. The results are obtained for parameters A = 0.995 and α = 1.39 and system size N = 256, and for over 2 × 105 time steps. The values of η for different panels are (a) 0.43, 0.70, 0.75, 0.80, 0.87, and (c) 0.46, 0.54, 0.59, 0.66, 0.68 from left to right, respectively. (b) Difference in the order parameter, , between the m-cluster chimera state and the corresponding hypothetical synchronous state, where the values of the average order parameter are displayed in the inset. The order parameters calculated through direct simulation of Eq. (1) (red pluses) and from the PDE Eq. (5) (blue circles) agree well with each other. The number of clusters emerged as η is increased is m = 4, 3, 5, 7, 9, and so on, and they are distinguished using white and gray backgrounds.
Mentions: Numerically, we observe a variety of rich phenomena when links are systematically removed. In particular, using the real order parameter R(x, t), we can identify the emergence of multiple cluster chimera states, where each cluster corresponds to a coherent group of oscillators. Figure 2 shows the spatiotemporal patterns of the emergent m-cluster chimera states for different intervals of η, which indicates that the emergence of the chimera-state patterns is robust with respect to reasonable variations of these parameters. In the simulations, the system parameters are A = 0.995 and α = 1.39, and the initial condition is generated89 using the function ϕ(x) = 6r exp(−0.76x2), where r is a random variable uniformly distributed in [−1/2,1/2]. In fact, the results obtained from direct simulations of Eq. (1) for finite-size networks and from the PDE approach [Eq. (5)] in the continuum limit N → ∞ agree with each other with similar spatiotemporal patterns. As shown in the inset of Fig. 2(b), the degree of synchrony as characterized by for different η values differs by orders of magnitude. For clarity, we use different color bars to distinguish the magnitudes of the spatiotemporal patterns in different panels. As η is increased, the number m of clusters undergoes changes from 4 to 3 (or 3&6), to 5 (or 5&10), to 7, and to 9, etc. Here the 3&6 state (or the 5&10 state) is a state that switches between 3-cluster and 6-cluster (or between 5-cluster and 10-cluster) chimera behaviors.

Bottom Line: This phenomenon was typically studied in the setting of non-local coupling configurations.We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift.The theoretical prediction agrees well with numerics.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Physics, Xi'an University of Technology, Xi'an 710054, China.

ABSTRACT
A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.

No MeSH data available.


Related in: MedlinePlus