Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays

We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structural conditions

Complex Dynamics and Search in A Cycle-Memory Neural Network

Numerical simulations of a single layer recurrent neural network model in which the synaptic connection matrix is formed by summing cyclic products of succesive patterns show that complex dynamics can occur with the reduction of a connectivity parameter which is the number of connection between neurons. The structure in these dynamics is discussed from the viewpoint of realizing complex function using complex dynamics

Stationary bumps in a piecewise smooth neural field model with synaptic depression

We analyze the existence and stability of stationary pulses or bumps in a one–dimensional piecewise smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, there exists a stable bump for sufficiently weak synaptic depression. However, as synaptic depression becomes stronger, the bump became unstable with respect to perturbations that shift the boundary of the bump, leading to the formation of a traveling pulse. The local stability of a bump is determined by the spectrum of a piecewise linear operator that keeps track of the sign of perturbations of the bump boundary. This results in a number of differences from previous studies of neural field models with Heaviside firing rate functions, where any discontinuities appear inside convolutions so that the resulting dynamical system is smooth. We also extend our results to the case of radially symmetric bumps in two–dimensional neural field models

Componentwise accurate fluid queue computations using doubling algorithms

Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue et al. (Numer Math 120:671–700, 2012) provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms. © 2014, Springer-Verlag Berlin Heidelberg

Stochastic Hopf bifurcations in vacuum optical tweezers

Funding: We acknowledge the support from the Engineering and Physical Sciences Research Council (Grant No. EP/P030017/1), the European Regional Development Fund (Grant No. CZ.02.1.01/0.0/0.0/15_003/0000476), the Czech Science Foundation (Grant No. GA19-17765S), and the Czech Academy of Sciences (Praemium Academiae, Grant No. RVO:68081731).The forces acting on an isotropic microsphere in optical tweezers are effectively conservative. However, reductions in the symmetry of the particle or trapping field can break this condition. Here we theoretically analyse the motion of a particle in a linearly non-conservative optical vacuum trap, concentrating on the case where symmetry is broken by optical birefringence, causing non-conservative coupling between rotational and translational degrees of freedom. Neglecting thermal fluctuations, we first show that the underlying deterministic motion can exhibit a Hopf bifurcation in which the trapping point destabilizes and limit cycles emerge whose amplitude grows with decreasing viscosity. When fluctuations are included, the bifurcation of the underlying deterministic system is expressed as a transition in the statistical description of the motion. For high viscosities, the probability distribution is normal, with a kurtosis of three, and persistent probability currents swirl around the stable trapping point. As the bifurcation is approached the distribution and currents spread out in phase space. Following the bifurcation the probability distribution function hollows out, reflecting the underlying limit cycle, and the kurtosis halves abruptly. The system is seen to be a noisy self sustained oscillator featuring a highly uneven limit cycle. A variety of applications, from autonomous stochastic resonance to synchronization, are discussed.Publisher PDFPeer reviewe