Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling

El‐Nashar, Hassan F.; Muruganandam, Paulsamy; Ferreira, Fernando F.; Cerdeira, Hilda A.

doi:10.1063/1.3056047

Cited by 20 publications

(20 citation statements)

References 33 publications

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“…The ring topology is defined by the periodic conditions θ N+1 = θ 1 and θ 0 = θ N . There is a minimum value for the coupling constant K, denoted as critical synchronization coupling K s , that drives the system into a fully synchronized state [16][17][18]. In this state the oscillators' instantaneous frequencies assume a constant value = 1…”

Section: Locally Coupled Kuramoto Model In the Synchronized Regionmentioning

confidence: 99%

“…At this moment the frequency becomes constant and the phases lock, such that all phase differences are constant. The solution for full synchronization and its stability has been studied by many authors [15][16][17][18][19]. Zheng et al [12] already in 1998 pointed out that the behavior of the order parameter "indicates the coexistence of multiple attractors of phase locking states" above the synchronization critical coupling, K s .…”

Section: Introductionmentioning

confidence: 99%

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Multistable behavior above synchronization in a locally coupled Kuramoto model

2011

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A system of nearest neighbors Kuramoto-like coupled oscillators placed in a ring is studied above the critical synchronization transition. We find a richness of solutions when the coupling increases, which exists only within a solvability region (SR). We also find that the solutions possess different characteristics, depending on the section of the boundary of the SR where they appear. We study the birth of these solutions and how they evolve when the coupling strength increases, and determine the diagram of solutions in phase space.

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Section: Locally Coupled Kuramoto Model In the Synchronized Regionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multistable behavior above synchronization in a locally coupled Kuramoto model

2011

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“…The local model of nearest neighbor interactions or LCKM can be considered as a diffusive version of the Kuramoto model and it is expressed as [16][17][18][19][20] …”

Section: Oscillators In a Ringmentioning

confidence: 99%

“…19 Very recently, we identified two oscillators which are responsible for dragging the system into full synchronization, 20 and the difference in phase for this pair is Ϯ / 2. These two oscillators are among two pairs of oscillators which are formed by the four oscillators at the borders between major clusters in the vicinity of the critical coupling.…”

Section: Introductionmentioning

confidence: 99%

Determination of the critical coupling for oscillators in a ring

El‐Nashar¹,

Cerdeira²

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study a model of coupled oscillators with bidirectional first nearest neighbors coupling with periodic boundary conditions. We show that a stable phase-locked solution is decided by the oscillators at the borders between the major clusters, which merge to form a larger one of all oscillators at the stage of complete synchronization. We are able to locate these four oscillators depending only on the set of the initial frequencies. Using these results plus an educated guess ͑supported by numerical findings͒ of the functional dependence of the corrections due to periodic boundary conditions, we are able to obtain a formula for the critical coupling, at which the complete synchronization state occurs. Such formula fits well in very good accuracy with the results that come from numerical simulations. This also helps to determine the sizes of the major clusters in the vicinity of the stage of full synchronization. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3212939͔Weakly coupled oscillators in the chaotic state have been known to represent many physical systems as well as chemical, biological, neurological, and so on. These systems synchronize in frequency under the influence of coupling. Knowing beforehand the value of the coupling constant and the dynamical behavior of the individual oscillators for complete synchronization to occur is an important source of information for real applications. This paper is a continuation of previous theoretical results for these systems. Here, we derive relationships that allow us to determine the oscillators which first lock in phase and drag the whole system into the synchronized state as well as the size of the two existing clusters before the transition.

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“…A wide diversity of nonlinear dynamic phenomena such as locking [1], partial synchronization [6], full synchronization [7], antiphase synchronization [8], and clustering [9] have been reported in coupled oscillators. Many coupling schemes have also been tested: local [10], nearest [11], global [12], diffusive [13], adaptive [14], delayed [15], hierarchical [16], and so on. An interesting behavior of coupled oscillators is amplitude death and oscillation death, which are steady states where the coupled oscillators stop their oscillation in a permanent way and become frozen in time [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Oscillation Death in Coupled Self-Excited Elastic Beams

Barrón

Hilerio

Plascencia³

2012

Advances in Mechanical Engineering

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The emergence of the oscillation death phenomenon in a ring of four coupled self-excited elastic beams is numerically explored in this work. The beams are mathematically represented through partial differential equations which are solved by means of the finite differences method. A coupling scheme based on shared boundary conditions at the roots of the beams is assumed, and as initial conditions, zero velocity of the first beam and three normal vibration modes of a linear elastic beam are employed. The influence of the self-exciting constant on the ring dynamics is analyzed. It is observed that oscillation death arises as result of the singularity of the coupling matrix.

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Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling

Cited by 20 publications

References 33 publications

Multistable behavior above synchronization in a locally coupled Kuramoto model

Multistable behavior above synchronization in a locally coupled Kuramoto model

Determination of the critical coupling for oscillators in a ring

Numerical Analysis of Oscillation Death in Coupled Self-Excited Elastic Beams

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