Stability of a pair of planar counter-rotating vortices in a rectangular box

Murakami, Youichi; Fukuta, Hiroaki

doi:10.1016/s0169-5983(02)00064-3

Cited by 18 publications

(7 citation statements)

References 6 publications

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“…In particular, coexisting co-rotating and counter-rotating systems characterized by positive and negative frequencies, respectively, are wide spread in nature. For instance, counter-rotating spirals are observed in protoplasm of the Physarum plasmodium [27], counter-rotating vortices are inevitable in the atmosphere and ocean [28][29][30], in magnetohydrodynamics of plasma flow [31], Bose-Einstein condensates [32,33], and in other physical systems [34][35][36]. Very recently, the counter-rotating frequency induced dynamical effects were also reported in the coupled Stuart-Landau oscillator with symmetry preserving as well as symmetry breaking couplings [37].…”

Section: Modelmentioning

confidence: 96%

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

M.¹,

Gupta²,

V.³

et al. 2023

Preprint

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We construct a nontrivial generalization of the paradigmatic Kuramoto model by using an additional coupling term that explicitly breaks its rotational symmetry resulting in a variant of the Winfree Model. Consequently, we observe the characteristic features of the phase diagrams of both the Kuramoto model and the Winfree model depending on the degree of the symmetry breaking coupling strength for unimodal frequency distribution. The phase diagrams of both the Kuramoto and the Winfree models resemble each other for symmetric bimodal frequency distribution for a range of the symmetry breaking coupling strength except for region shift and difference in the degree of spread of the macroscopic dynamical states and bistable regions. The dynamical transitions in the bistable states are characterized by an abrupt (first-order) transition in both the forward and reverse traces. For asymmetric bimodal frequency distribution, the onset of bistable regions depends on the degree of the asymmetry. Large degree of the symmetry breaking coupling strength promotes the synchronized stationary state, while a large degree of heterogeneity, proportional to the separation between the two central frequencies, facilitates the spread of the incoherent and standing wave states in the phase diagram for a low strength of the symmetry breaking coupling. We deduce the low-dimensional equations of motion for the complex order parameters using the Ott-Antonsen ansatz for both unimodal and bimodal frequency distributions. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations of the generalized Kuramoto model agree well with the analytical bifurcation curves.

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Section: Modelmentioning

confidence: 96%

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

M.¹,

Gupta²,

V.³

et al. 2023

Preprint

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show abstract

“…On the other hand, coexisting co-and counter-rotating dynamical activity has a wide range of applications in diverse fields, including physical [30,31], biological [32,33], and fluid dynamics [34][35][36] contexts. For instance, counter-rotating spirals can be found in biological media, such as Physarum plasmodium protoplasm [32,33].…”

Section: Introductionmentioning

confidence: 99%

Emerging chimera states under non-identical counter-rotating oscillators

Sathiyadevi,

Chandrasekar,

Lakshmanan

2022

Preprint

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Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting co-and counter-rotating systems. To illustrate the impact of counter-rotating frequencies, we consider a network of non-identical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical transitions in the absence of perturbation, demonstrating that the transition from desynchronized state to cluster oscillatory state occurs through an interesting partial synchronization state in the oscillatory regime. Followed by this, the system dynamics transits to amplitude death and oscillation death states. Importantly, we find that the observed dynamical states do not preserve the parity(P ) symmetry in the absence of perturbation. When the perturbation is increased one can note that the system dynamics exhibits a new kind of transition which corresponds to a change from incoherent mixed synchronization to coherent mixed synchronization through chimera state. In particular, incoherent mixed synchronization and coherent mixed synchronization states completely preserve the P -symmetry, whereas the chimera state preserves the P -symmetry only partially. To demonstrate the occurrence of such partial symmetry breaking (chimera) state, we use basin stability analysis and discover that partial symmetry breaking exists as a result of the coexistence of symmetry preserving and symmetry breaking behavior in the initial state space. Further, a measure of the strength of P -symmetry is established to quantify the P -symmetry in the observed dynamical states. Subsequently, the dynamical transitions are investigated in the parametric spaces. Finally, by increasing the network size, the robustness of the chimera state is also inspected and we find that the chimera state is robust even in networks of larger sizes. We also show the generality of the above results in the related phase reduced model as well as in other coupled models such as the globally coupled van der Pol and Rössler oscillators.

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“…On the other hand, coexisting co-and counter-rotating dynamical activity can be found in various natural systems such as fluid dynamics 32,33 , physical 34,35 as well as biological systems 36 . Hence special attention is deemed towards enriching our knowledge on counter-rotation induced collective dynamics in various systems.…”

Section: Introductionmentioning

confidence: 99%

Aging transition in the absence of inactive oscillators

Sathiyadevi

Gowthaman

Senthilkumar

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The role of counter-rotating oscillators in an ensemble of coexisting co-and counter-rotating oscillators is examined by increasing the proportion of the latter. The phenomenon of aging transition was identified at a critical value of the ratio of the counter-rotating oscillators, which was otherwise realized only by increasing the number of inactive oscillators to a large extent. The effect of the mean-field feedback strength in the symmetry preserving coupling is also explored. The parameter space of aging transition was increased abruptly even for a feeble decrease in the feedback strength and subsequently aging transition was observed at a critical value of the feedback strength surprisingly without any counter-rotating oscillators. Further, the study was extended to symmetry breaking coupling using conjugate variables and it was observed that the symmetry breaking coupling can facilitating the onset of aging transition even in the absence of counter-rotating oscillators and for the unit value of the feedback strength. In general, the parameter space of aging transition was found to increase by increasing the frequency of oscillators and by increasing the proportion of the counter-rotating oscillators in both the symmetry preserving and symmetry breaking couplings. Further, the transition from oscillatory to aging transition occurs via a Hopf bifurcation, while the transition from aging transition to oscillation death state emerges via Pitchfork bifurcation. Analytical expressions for the critical ratio of the counterrotating oscillators are deduced to find the stable boundaries of the aging transition.Aging is a kind of deterioration which occurs in diverse complex systems. It is evident from our daily life that living organisms (and its efficiency) degrade as it becomes older. For instance, Alzheimer's disease is an example of a cause of failure of neurons due to the aging process. In this context, the phenomenon of aging transition was reported in an ensemble of oscillators by increasing the proportion of inactive oscillator [25][26][27][28][29][30] . In the present work, we show that even an appropriate proportion of counter-rotating oscillators in an ensemble of coexisting co-and counterrotating oscillator is capable of inducing the phenomenon of aging transition. Further, we find that either the meanfield feedback or the symmetry breaking coupling alone can facilitate the onset of the aging transition.

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Stability of a pair of planar counter-rotating vortices in a rectangular box

Cited by 18 publications

References 6 publications

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

Generalization of the Kuramoto model to the Winfree model by a symmetry breaking coupling

Emerging chimera states under non-identical counter-rotating oscillators

Aging transition in the absence of inactive oscillators

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