Critical coupling strength of the Cucker–Smale model for flocking

2017 ◽  
Vol 27 (06) ◽  
pp. 1051-1087 ◽  
Author(s):  
Seung-Yeal Ha ◽  
Dongnam Ko ◽  
Yinglong Zhang
Keyword(s):  
Lower Bound ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Short Range ◽  
Critical Value ◽  
Emergent Behavior ◽  
Initial Configuration ◽  
Upper And Lower Bounds ◽  
Critical Coupling ◽  
Large Coupling

We present a non-trivial positive lower bound for the critical coupling strength of the Cucker–Smale model with a short-range communication weight from the viewpoint of mono-cluster (global) flocking. For a long-range communication weight, it is well known [F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007) 852–862; S.-Y. Ha and J.-G. Liu, A proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297–325] that as long as the coupling strength is positive, mono-cluster flocking occurs asymptotically for any initial configuration. Hence, the critical coupling strength is simply zero. However, for a short-range communication weight, numerical simulations indicate that for a given initial configuration, mono-cluster flocking is possible only in a large-coupling-strength regime depending on the initial configuration. This suggests the positivity of the critical coupling strength in the sense that if the coupling strength is above the critical value, mono-cluster flocking emerges, whereas if it is below the critical value, mono-cluster flocking does not occur. Thus, it is interesting to determine the exact critical value for the coupling strength depending on the initial configuration or at least to estimate the possible range of the coupling strength. In this paper, we show that the critical coupling strength exists and is positive by providing a positive lower bound. We also present the results of several numerical simulations for the upper and lower bounds of the critical coupling strength depending on the initial configurations, and we compare them with analytical results.

scholarly journals Phase Transition and Hysteresis in an Ensemble of Stochastic Spiking Neurons

2007 ◽  
Vol 19 (11) ◽  
pp. 3011-3050 ◽  
Author(s):  
Andreas Kaltenbrunner ◽  
Vicenç Gómez ◽  
Vicente López
Keyword(s):  
Phase Transition ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Stochastic Dynamics ◽  
Upper And Lower Bounds ◽  
Critical Coupling ◽  
Sustained Activity ◽  
Large Ensembles ◽  
Coupling Strengths ◽  
Pulse Coupled Oscillators

An ensemble of stochastic nonleaky integrate-and-fire neurons with global, delayed, and excitatory coupling and a small refractory period is analyzed. Simulations with adiabatic changes of the coupling strength indicate the presence of a phase transition accompanied by a hysteresis around a critical coupling strength. Below the critical coupling production of spikes in the ensemble is governed by the stochastic dynamics, whereas for coupling greater than the critical value, the stochastic dynamics loses its influence and the units organize into several clusters with self-sustained activity. All units within one cluster spike in unison, and the clusters themselves are phase-locked. Theoretical analysis leads to upper and lower bounds for the average interspike interval of the ensemble valid for all possible coupling strengths. The bounds allow calculating the limit behavior for large ensembles and characterize the phase transition analytically. These results may be extensible to pulse-coupled oscillators.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals Dipole radiation and beyond from axion stars in electromagnetic fields

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Mustafa A. Amin ◽  
Andrew J. Long ◽  
Zong-Gang Mou ◽  
Paul M. Saffin
Keyword(s):  
Electromagnetic Fields ◽  
Parametric Resonance ◽  
Coupling Strength ◽  
Coupling Parameter ◽  
Strong Dependence ◽  
Dipole Radiation ◽  
Large Coupling ◽  
External Electromagnetic Fields ◽  
Order Of Magnitude ◽  
Dimensionless Coupling

Abstract We investigate the production of photons from coherently oscillating, spatially localized clumps of axionic fields (oscillons and axion stars) in the presence of external electromagnetic fields. We delineate different qualitative behaviour of the photon luminosity in terms of an effective dimensionless coupling parameter constructed out of the axion-photon coupling, and field amplitude, oscillation frequency and radius of the axion star. For small values of this dimensionless coupling, we provide a general analytic formula for the dipole radiation field and the photon luminosity per solid angle, including a strong dependence on the radius of the configuration. For moderate to large coupling, we report on a non-monotonic behavior of the luminosity with the coupling strength in the presence of external magnetic fields. After an initial rise in luminosity with the coupling strength, we see a suppression (by an order of magnitude or more compared to the dipole radiation approximation) at moderately large coupling. At sufficiently large coupling, we find a transition to a regime of exponential growth of the luminosity due to parametric resonance. We carry out 3+1 dimensional lattice simulations of axion electrodynamics, at small and large coupling, including non-perturbative effects of parametric resonance as well as backreaction effects when necessary. We also discuss medium (plasma) effects that lead to resonant axion to photon conversion, relevance of the coherence of the soliton, and implications of our results in astrophysical and cosmological settings.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Numerical Simulations of Electric Field Driven Self-Assembly of Monolayers of Mixtures of Nanoparticles

2017 ◽  
Author(s):  
E. Amah ◽  
N. Musunuri ◽  
Ian S. Fischer ◽  
Pushpendra Singh
Keyword(s):  
Electric Field ◽  
Physical Properties ◽  
Numerical Simulations ◽  
Self Assembly ◽  
Composite Particle ◽  
Critical Value ◽  
Direct Numerical Simulations ◽  
Composite Particles ◽  
Liquid Interfaces ◽  
Induced Dipole

We numerically study the process of self-assembly of particle mixtures on fluid-liquid interfaces when an electric field is applied in the direction normal to the interface. The force law for the dependence of the electric field induced dipole-dipole and capillary forces on the distance between the particles and their physical properties obtained in an earlier study by performing direct numerical simulations is used for conducting simulations. The inter-particle forces cause mixtures of nanoparticles to self-assemble into molecular-like hierarchical arrangements consisting of composite particles which are organized in a pattern. However, there is a critical electric intensity value below which particles move under the influence of Brownian forces and do not self-assemble. Above the critical value, when the particles sizes differed by a factor of two or more, the composite particle has a larger particle at its core and several smaller particles forming a ring around it. Approximately same sized particles, when their concentrations are approximately equal, form binary particles or chains (analogous to polymeric molecules) in which positively and negatively polarized particles alternate, but when their concentrations differ the particles whose concentration is larger form rings around the particles with smaller concentration.

Numerical simulations of temperature fields in the uncooled nozzle of a short-range anti-aircraft rocket engine with an insert in the critical section made of various materials

2021 ◽  
Vol 70 (1) ◽  
pp. 15-30
Author(s):  
Mateusz Zieliński ◽  
Piotr Koniorczyk ◽  
Janusz Zmywaczyk ◽  
Marek Preiskorn
Keyword(s):  
Numerical Simulations ◽  
Cross Section ◽  
Heat Flux Density ◽  
Short Range ◽  
Rocket Engine ◽  
Transient Heat Conduction ◽  
Critical Section ◽  
Temperature Fields ◽  
Critical Cross Section ◽  
Calculation Results

Abstract. The paper presents numerical simulations of transient heat conduction in the uncooled nozzle of a short-range anti-aircraft rocket engine. The calculations were made for the configuration of the nozzle with an insert in the critical section made of various materials. The inserts used were: POCO graphite, Al2O3 ceramics, ZrO2-3Y2O3 ceramics. For comparison, numerical simulations of the heat transfer in a nozzle made entirely of St 45 steel, the melting point of which is 1700K, were also carried out. The engine's working time was in the order of 3 s. Numerical simulations were performed using the COMSOL program. The calculation results are given in the form of temperature dependence and heat flux density as a function of time in the critical cross-section. Keywords: non-cooled nozzle, rocket engine, temperature field

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals Buckling of a coated elastic half-space when the coating and substrate have similar material properties

2015 ◽  
Vol 471 (2178) ◽  
pp. 20140979 ◽  
Author(s):  
Y. B. Fu ◽  
P. Ciarletta
Keyword(s):  
Numerical Simulations ◽  
Material Properties ◽  
Deformation Gradient ◽  
Critical Value ◽  
Elastic Half Space ◽  
Similar Material ◽  
Fully Nonlinear ◽  
Shear Moduli ◽  
Subcritical Regime ◽  
Post Buckling

This study investigates the buckling of a uni-axially compressed neo-Hookean thin film bonded to a neo-Hookean substrate. Previous studies have shown that the elastic bifurcation is supercritical if r ≡ μ f / μ s >1.74 and subcritical if r <1.74, where μ f and μ s are the shear moduli of the film and substrate, respectively. Moreover, existing numerical simulations of the fully nonlinear post-buckling behaviour have all been focused on the regime r >1.74. In this paper, we consider instead a subset of the regime r <1.74, namely when r is close to unity. Four near-critical regimes are considered. In particular, it is shown that, when r >1 and the stretch is greater than the critical stretch (the subcritical regime), there exists a localized solution that arises as the limit of modulated periodic solutions with increasingly longer and longer decaying tails. The evolution of each modulated periodic solution is followed as r is decreased, and it is found that there exists a critical value of r at which the deformation gradient develops a discontinuity and the solution becomes a static shock. The semi-analytical results presented could help future numerical simulations of the fully nonlinear post-buckling behaviour.

Dynamics of a stochastic three-species competitive model with Lévy jumps

2018 ◽  
Vol 11 (05) ◽  
pp. 1850075
Author(s):  
Yongxin Gao ◽  
Shiquan Tian
Keyword(s):  
Numerical Simulations ◽  
Critical Value ◽  
Persistence In The Mean ◽  
Competitive Model ◽  
Stability In Distribution ◽  
Parameters Analysis ◽  
The Mean ◽  
Lévy Jumps ◽  
White Noises ◽  
Theoretical Results

This paper is concerned with a three-species competitive model with both white noises and Lévy noises. We first carry out the almost complete parameters analysis for the model and establish the critical value between persistence in the mean and extinction for each species. The sufficient criteria for stability in distribution of solutions are obtained. Finally, numerical simulations are carried out to verify the theoretical results.

Efficient numerical algorithms for the phase field crystal equation

2021 ◽  
pp. 2150042
Author(s):  
Qingqu Zhuang ◽  
Shuying Zhai ◽  
Zhifeng Weng
Keyword(s):  
Free Energy ◽  
Lower Bound ◽  
Numerical Simulations ◽  
Lagrange Multiplier ◽  
Phase Field ◽  
Numerical Algorithms ◽  
Energy Potential ◽  
Energy Stable ◽  
2D And 3D ◽  
Phase Field Crystal

In this paper, based on the Lagrange Multiplier approach in time and the Fourier-spectral scheme for space, we propose efficient numerical algorithms to solve the phase field crystal equation. The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential. The unconditional energy stability of the three semi-discrete schemes is proven. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.

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