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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

Mutual enhancement and formation of multicluster chimera states.(a) Schematic illustration of m = 3 cluster chimera states in certain range of η. The colors represent the values of the order parameter R and the regions centered with red and blue colors correspond to coherent and incoherent groups of oscillators, respectively. The reference oscillator for calculating the enhancement factor I is the bottom node at the center of the coherent group. (b) Enhancement factor I for the reference oscillator as a function of η, for different values of m. The regions of maximum I values among the different m curves are specified as bold lines, which can be regarded, approximately, as the regions in which the corresponding m-cluster chimera states emerge. (c) Regions of m-cluster chimera states predicted by our mutual-enhancement theory (the colored thick horizontal lines at different m levels), including the main regions m = 4, 3, 5, 7, and 9, and the two subregions with 3&6-cluster and 5&10-cluster chimera states. The white and gray backgrounds have the same meaning as those in Fig. 2, i.e., they denote the regions of m-cluster chimera states, which are obtained from both simulation of Eq. (1) and solution of Eq. (5). The subregions for 3&6-cluster and 5&10-cluster chimera states obtained from Eqs (1) and (5) are also specified with the thin black vertical lines and the corresponding notations.
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f4: Mutual enhancement and formation of multicluster chimera states.(a) Schematic illustration of m = 3 cluster chimera states in certain range of η. The colors represent the values of the order parameter R and the regions centered with red and blue colors correspond to coherent and incoherent groups of oscillators, respectively. The reference oscillator for calculating the enhancement factor I is the bottom node at the center of the coherent group. (b) Enhancement factor I for the reference oscillator as a function of η, for different values of m. The regions of maximum I values among the different m curves are specified as bold lines, which can be regarded, approximately, as the regions in which the corresponding m-cluster chimera states emerge. (c) Regions of m-cluster chimera states predicted by our mutual-enhancement theory (the colored thick horizontal lines at different m levels), including the main regions m = 4, 3, 5, 7, and 9, and the two subregions with 3&6-cluster and 5&10-cluster chimera states. The white and gray backgrounds have the same meaning as those in Fig. 2, i.e., they denote the regions of m-cluster chimera states, which are obtained from both simulation of Eq. (1) and solution of Eq. (5). The subregions for 3&6-cluster and 5&10-cluster chimera states obtained from Eqs (1) and (5) are also specified with the thin black vertical lines and the corresponding notations.

Mentions: To gain insight into the mechanism of mutual enhancement, we analyze the stability of the coherent (or incoherent) groups in an idealized m-cluster chimera state. For a given coherent group, the contribution to the coupling from oscillators in other coherent groups tends to stabilize the state (a positive effect), while that from oscillators in the incoherent groups plays the opposite role (a negative effect). For an incoherent group, the effects of other coexisting coherent and incoherent groups are negative and positive, respectively. That is, oscillators in the like groups (coherent versus coherent or incoherent versus incoherent) tend to enhance each other’s stability, while those in the unlike groups (coherent versus incoherent or vice versa) tend to destabilize each other. To be concrete and quantitative, we define an enhancement factor I(η) that depends on the system parameter η and assume that, for an oscillator in the coherent group, the contribution from each coherent-group oscillator is +1, while that from an incoherent-group oscillator is −1. Consider the oscillator at the center of a coherent group, e.g., the bottom oscillator in Fig. 4(a). The total contribution from other oscillators to the enhancement factor for this oscillator is


Emergence of multicluster chimera states.

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

Mutual enhancement and formation of multicluster chimera states.(a) Schematic illustration of m = 3 cluster chimera states in certain range of η. The colors represent the values of the order parameter R and the regions centered with red and blue colors correspond to coherent and incoherent groups of oscillators, respectively. The reference oscillator for calculating the enhancement factor I is the bottom node at the center of the coherent group. (b) Enhancement factor I for the reference oscillator as a function of η, for different values of m. The regions of maximum I values among the different m curves are specified as bold lines, which can be regarded, approximately, as the regions in which the corresponding m-cluster chimera states emerge. (c) Regions of m-cluster chimera states predicted by our mutual-enhancement theory (the colored thick horizontal lines at different m levels), including the main regions m = 4, 3, 5, 7, and 9, and the two subregions with 3&6-cluster and 5&10-cluster chimera states. The white and gray backgrounds have the same meaning as those in Fig. 2, i.e., they denote the regions of m-cluster chimera states, which are obtained from both simulation of Eq. (1) and solution of Eq. (5). The subregions for 3&6-cluster and 5&10-cluster chimera states obtained from Eqs (1) and (5) are also specified with the thin black vertical lines and the corresponding notations.
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f4: Mutual enhancement and formation of multicluster chimera states.(a) Schematic illustration of m = 3 cluster chimera states in certain range of η. The colors represent the values of the order parameter R and the regions centered with red and blue colors correspond to coherent and incoherent groups of oscillators, respectively. The reference oscillator for calculating the enhancement factor I is the bottom node at the center of the coherent group. (b) Enhancement factor I for the reference oscillator as a function of η, for different values of m. The regions of maximum I values among the different m curves are specified as bold lines, which can be regarded, approximately, as the regions in which the corresponding m-cluster chimera states emerge. (c) Regions of m-cluster chimera states predicted by our mutual-enhancement theory (the colored thick horizontal lines at different m levels), including the main regions m = 4, 3, 5, 7, and 9, and the two subregions with 3&6-cluster and 5&10-cluster chimera states. The white and gray backgrounds have the same meaning as those in Fig. 2, i.e., they denote the regions of m-cluster chimera states, which are obtained from both simulation of Eq. (1) and solution of Eq. (5). The subregions for 3&6-cluster and 5&10-cluster chimera states obtained from Eqs (1) and (5) are also specified with the thin black vertical lines and the corresponding notations.
Mentions: To gain insight into the mechanism of mutual enhancement, we analyze the stability of the coherent (or incoherent) groups in an idealized m-cluster chimera state. For a given coherent group, the contribution to the coupling from oscillators in other coherent groups tends to stabilize the state (a positive effect), while that from oscillators in the incoherent groups plays the opposite role (a negative effect). For an incoherent group, the effects of other coexisting coherent and incoherent groups are negative and positive, respectively. That is, oscillators in the like groups (coherent versus coherent or incoherent versus incoherent) tend to enhance each other’s stability, while those in the unlike groups (coherent versus incoherent or vice versa) tend to destabilize each other. To be concrete and quantitative, we define an enhancement factor I(η) that depends on the system parameter η and assume that, for an oscillator in the coherent group, the contribution from each coherent-group oscillator is +1, while that from an incoherent-group oscillator is −1. Consider the oscillator at the center of a coherent group, e.g., the bottom oscillator in Fig. 4(a). The total contribution from other oscillators to the enhancement factor for this oscillator is

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