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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

Different final states associated with a multicluster chimera state.For the ring network system of size N = 256, spatiotemporal patterns of the order parameter R, instantaneous phase ϕ and short-term average velocity v for: (a) a stable 4-cluster chimera state, (b) a transient 3-cluster chimera state evolving into a globally synchronous state, and (c,d) transient 3-cluster and 6-cluster chimera states evolving into a 6π-twisted state, respectively. The phase ϕ at several instants and the corresponding short-term average phase velocity v (temporal average over a time window with length 104) are displayed to demonstrate the dynamical processes, e.g., global synchronization at t2 in (b), 6-cluster chimera states at t1, and the twisted states at t3 in (c,d). The results in (a–d) are obtained from Eq. (1) for η = 0.43, 0.48, 0.55, and 0.54, respectively.
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f3: Different final states associated with a multicluster chimera state.For the ring network system of size N = 256, spatiotemporal patterns of the order parameter R, instantaneous phase ϕ and short-term average velocity v for: (a) a stable 4-cluster chimera state, (b) a transient 3-cluster chimera state evolving into a globally synchronous state, and (c,d) transient 3-cluster and 6-cluster chimera states evolving into a 6π-twisted state, respectively. The phase ϕ at several instants and the corresponding short-term average phase velocity v (temporal average over a time window with length 104) are displayed to demonstrate the dynamical processes, e.g., global synchronization at t2 in (b), 6-cluster chimera states at t1, and the twisted states at t3 in (c,d). The results in (a–d) are obtained from Eq. (1) for η = 0.43, 0.48, 0.55, and 0.54, respectively.

Mentions: In the small neighborhood of zero η value, the observed states are conventional chimera states consisting of a coherent and an incoherent clusters. For η ~ 0.4, 4-cluster chimera states emerge. In the 4-cluster region [m = 4 region in Fig. 2(b)], the value of increases with η, which can be attributed to the increasing fraction of coherent groups, as demonstrated by the red color in the spatiotemporal patterns [first panel in Fig. 2(a)]. The behaviors in subsequent parameter regions are richer and more complicated. In particular, the 3-cluster chimera states for small values of η are stable and regular as the 4-cluster chimera states. As η is increased further, the 3-cluster configuration becomes unstable and evolves eventually to global synchronization. In the 3-cluster region, various other states can emerge, which include (in successive order) stable regular 3-cluster states, transient 3-cluster states toward global synchronization, 6π-twisted states and 3&6 cluster double-state switching process, cluster drift states, and so on. One remarkable phenomenon is spatial period doubling (or spatial cluster doubling) in the 3-cluster region, in which each cluster bifurcates into two clusters and a 6-cluster chimera state emerges consequently, as shown in Fig. 2(c) (2nd and 3rd panels). The 6-cluster chimera states are unstable and can evolve into 6π-twisted states, as shown in the 2nd panel in Fig. 2(c), which will be further discussed in Fig. 3. Analogous to chemical oscillating reactions50, self-organized double-state switching processes are observed, in which the 3-cluster and 6-cluster chimera states appear and disappear alternatively, leading to spatiotemporal patterns of switching between the two states. The switching process also takes place in the 5-cluster region, where the system alternates between 5-cluster and 10-cluster chimera states, as shown in the 3rd panel in Fig. 2(a). Overall, as η is increased in the 3-cluster region, the resulting state is a cluster drifting state with strong intrinsic correlation in the spatiotemporal dynamics, as characterized by harmonically temporal breathing and spatial drifting of the coherent and incoherent groups [4th panel in Fig. 2(c)].


Emergence of multicluster chimera states.

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

Different final states associated with a multicluster chimera state.For the ring network system of size N = 256, spatiotemporal patterns of the order parameter R, instantaneous phase ϕ and short-term average velocity v for: (a) a stable 4-cluster chimera state, (b) a transient 3-cluster chimera state evolving into a globally synchronous state, and (c,d) transient 3-cluster and 6-cluster chimera states evolving into a 6π-twisted state, respectively. The phase ϕ at several instants and the corresponding short-term average phase velocity v (temporal average over a time window with length 104) are displayed to demonstrate the dynamical processes, e.g., global synchronization at t2 in (b), 6-cluster chimera states at t1, and the twisted states at t3 in (c,d). The results in (a–d) are obtained from Eq. (1) for η = 0.43, 0.48, 0.55, and 0.54, respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Different final states associated with a multicluster chimera state.For the ring network system of size N = 256, spatiotemporal patterns of the order parameter R, instantaneous phase ϕ and short-term average velocity v for: (a) a stable 4-cluster chimera state, (b) a transient 3-cluster chimera state evolving into a globally synchronous state, and (c,d) transient 3-cluster and 6-cluster chimera states evolving into a 6π-twisted state, respectively. The phase ϕ at several instants and the corresponding short-term average phase velocity v (temporal average over a time window with length 104) are displayed to demonstrate the dynamical processes, e.g., global synchronization at t2 in (b), 6-cluster chimera states at t1, and the twisted states at t3 in (c,d). The results in (a–d) are obtained from Eq. (1) for η = 0.43, 0.48, 0.55, and 0.54, respectively.
Mentions: In the small neighborhood of zero η value, the observed states are conventional chimera states consisting of a coherent and an incoherent clusters. For η ~ 0.4, 4-cluster chimera states emerge. In the 4-cluster region [m = 4 region in Fig. 2(b)], the value of increases with η, which can be attributed to the increasing fraction of coherent groups, as demonstrated by the red color in the spatiotemporal patterns [first panel in Fig. 2(a)]. The behaviors in subsequent parameter regions are richer and more complicated. In particular, the 3-cluster chimera states for small values of η are stable and regular as the 4-cluster chimera states. As η is increased further, the 3-cluster configuration becomes unstable and evolves eventually to global synchronization. In the 3-cluster region, various other states can emerge, which include (in successive order) stable regular 3-cluster states, transient 3-cluster states toward global synchronization, 6π-twisted states and 3&6 cluster double-state switching process, cluster drift states, and so on. One remarkable phenomenon is spatial period doubling (or spatial cluster doubling) in the 3-cluster region, in which each cluster bifurcates into two clusters and a 6-cluster chimera state emerges consequently, as shown in Fig. 2(c) (2nd and 3rd panels). The 6-cluster chimera states are unstable and can evolve into 6π-twisted states, as shown in the 2nd panel in Fig. 2(c), which will be further discussed in Fig. 3. Analogous to chemical oscillating reactions50, self-organized double-state switching processes are observed, in which the 3-cluster and 6-cluster chimera states appear and disappear alternatively, leading to spatiotemporal patterns of switching between the two states. The switching process also takes place in the 5-cluster region, where the system alternates between 5-cluster and 10-cluster chimera states, as shown in the 3rd panel in Fig. 2(a). Overall, as η is increased in the 3-cluster region, the resulting state is a cluster drifting state with strong intrinsic correlation in the spatiotemporal dynamics, as characterized by harmonically temporal breathing and spatial drifting of the coherent and incoherent groups [4th panel in Fig. 2(c)].

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