Bifurcations in a Star-Like Network of Stuart–landau Oscillators

Frasca, Mattia; Bergner, André; Kurths, Jürgen; Fortuna, Luigi

doi:10.1142/s0218127412501738

Cited by 24 publications

(22 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The dynamics of the i-th node in the network will be governed by the paradigmatic Hopf oscillator 17,41,42…”

Section: Computational Resultsmentioning

confidence: 99%

“…The average is then based on the calculations over 100 periods. It should be noted that similar concepts of stability analysis have already been successfully applied, for example, to study the synchronization behavior of Hopf oscillators in star-like networks 42 and the occurrence of oscillation death in networks with repulsive coupling. 45 Despite the intrinsic symmetry of the attractor, the divergence may vary with time due to perturbations of the limit cycle provoked by the influence of neighboring oscillators.…”

Section: Computational Resultsmentioning

confidence: 99%

See 1 more Smart Citation

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells

Gosak

Stožer

Markovič

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

Self-sustained oscillatory dynamics is a motion along a stable limit cycle in the phase space, and it arises in a wide variety of mechanical, electrical, and biological systems. Typically, oscillations are due to a balance between energy dissipation and generation. Their stability depends on the properties of the attractor, in particular, its dissipative characteristics, which in turn determine the flexibility of a given dynamical system. In a network of oscillators, the coupling additionally contributes to the dissipation, and hence affects the robustness of the oscillatory solution. Here, we therefore investigate how a heterogeneous network structure affects the dissipation rate of individual oscillators. First, we show that in a network of diffusively coupled oscillators, the dissipation is a linearly decreasing function of the node degree, and we demonstrate this numerically by calculating the average divergence of coupled Hopf oscillators. Subsequently, we use recordings of intracellular calcium dynamics in pancreatic beta cells in mouse acute tissue slices and the corresponding functional connectivity networks for an experimental verification of the presented theory. We use methods of nonlinear time series analysis to reconstruct the phase space and calculate the sum of Lyapunov exponents. Our analysis reveals a clear tendency of cells with a higher degree, that is, more interconnected cells, having more negative values of divergence, thus confirming our theoretical predictions. We discuss these findings in the context of energetic aspects of signaling in beta cells and potential risks for pathological changes in the tissue.

show abstract

“…The dynamics of the i-th node in the network will be governed by the paradigmatic Hopf oscillator 17,41,42…”

Section: Computational Resultsmentioning

confidence: 99%

Section: Computational Resultsmentioning

confidence: 99%

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells

Gosak

Stožer

Markovič

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

“…Given that, in this example, t and pd are so close we would expect to see the network state that is most close to synchrony in the parameter window ∈ ( t , pd ), and this is confirmed in simulations. It is interesting to note that although remote synchronisation, as illustrated in Figure 6 (right), has previously been seen in star networks of Stuart-Landau oscillators [39], this has required a frequency mismatch between the hub oscillator and the leaf ones. For the homoclinic node model considered here, no such frequency mismatch is required.…”

Section: Where G(a; T) Is Given In (26) Andmentioning

confidence: 75%

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

View full text Add to dashboard Cite

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

show abstract

“…Stuart-Landau limit cycle oscillator is a general, paradigmatic model, which can be used to model a variety of nonlinear dynamical systems near the Hopf bifurcation point [41,42,43,44]. We consider a one-dimensional ring network of coupled Stuart-Landau oscillators and investigate its dynamical behavior by including a repulsive link along with a symmetry breaking coupling.…”

Section: The Modelmentioning

confidence: 99%

Long-range interaction induced collective dynamical behaviors

Sathiyadevi¹,

Chandrasekar²,

Senthilkumar³

et al. 2019

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Long-range interacting systems are omnipresent in nature. We investigate here the collective dynamical behavior in a long-range interacting system consisting of coupled Stuart-Landau limit cycle oscillators. In particular, we analyze the impact of a repulsive coupling along with a symmetry breaking coupling. We report that the addition of repulsive coupling of sufficient strength can induce a swing of the synchronized state which will start disappearing with increasing disorder as a function of the repulsive coupling. We also deduce analytical stability conditions for the oscillatory states including synchronized state, solitary state, two-cluster state as well as oscillation death state. Finally, we have also analyzed the effect of power-law exponent on the observed dynamical states.

show abstract

Bifurcations in a Star-Like Network of Stuart–landau Oscillators

Cited by 24 publications

References 39 publications

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells

The relationship between node degree and dissipation rate in networks of diffusively coupled oscillators and its significance for pancreatic beta cells

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Long-range interaction induced collective dynamical behaviors

Contact Info

Product

Resources

About