Nonlinear stability of flock solutions in second-order swarming models

In this paper we consider interacting particle systems which are frequently used to model collective behavior in animal swarms and other applications. We study the stability of orientationally aligned formations called flock solutions, one of the typical patterns emerging from such dynamics. We provide an analysis showing that the nonlinear stability of flocks in second-order models entirely depends on the linear stability of the first-order aggregation equation. Flocks are shown to be nonlinearly stable as a family of states under reasonable assumptions on the interaction potential. Furthermore, we numerically verify that commonly used potentials satisfy these hypotheses and investigate the nonlinear stability of flocks by an extensive case-study of uniform perturbations.Comment: 22 pages, 1 figure, 1 tabl

Zoology of a non-local cross-diffusion model for two species

We study a non-local two species cross-interaction model with cross-diffusion. We propose a positivity preserving finite volume scheme based on the numerical method introduced in Ref. [15] and explore this new model numerically in terms of its long-time behaviours. Using the so gained insights, we compute analytical stationary states and travelling pulse solutions for a particular model in the case of attractive-attractive/attractive-repulsive cross-interactions. We show that, as the strength of the cross-diffusivity decreases, there is a transition from adjacent solutions to completely segregated densities, and we compute the threshold analytically for attractive-repulsive cross-interactions. Other bifurcating stationary states with various coexistence components of the support are analysed in the attractive-attractive case. We find a strong agreement between the numerically and the analytically computed steady states in these particular cases, whose main qualitative features are also present for more general potentials

Local well-posedness of the generalized Cucker-Smale model

In this paper, we study the local well-posedness of two types of generalized Cucker-Smale (in short C-S) flocking models. We consider two different communication weights, singular and regular ones, with nonlinear coupling velocities $v|v|^{\beta-2}$ for $\beta > \frac{3-d}{2}$. For the singular communication weight, we choose $\psi^1(x) = 1/|x|^{\alpha}$ with $\alpha \in (0,d-1)$ and $\beta \geq 2$ in dimension $d > 1$. For the regular case, we select $\psi^2(x) \geq 0$ belonging to (L_{loc}^\infty \cap \mbox{Lip}_{loc})(\mathbb{R}^d) and $\beta \in (\frac{3-d}{2},2)$. We also remark the various dynamics of C-S particle system for these communication weights when $\beta \in (0,3)$

Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as $t\to\infty$

Numerical Study of a Particle Method for Gradient Flows

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.Comment: 27 pages, 21 figure

Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure

We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and V\'azquez, where the pressure is obtained as a Riesz potential associated to the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states

Variable control tool in MATLAB for energy transformation processes

During the stages of transformation of energy in a process, exercise control over the variables that intervene in it, improve its performance, and identify undesirable conditions in these. Thus, this study is developed as a graphical interface to implement a methodology for controlling variables of energy conversion processes, such as internal combustion engines. The control tool developed in MATLAB variables is based on multivariate statistics. The methods for developing this tool of Graphic User Interface is based on the statistics of principal component analysis and failure statistics such as T! Hotelling and the Q statistic that allows the control of anomalies presented in the operation's behavior. About the methodology, first, the input data are normalized, achieving standardization of the observation matrix vs. variables, then the spectral decomposition of the normalized data is performed, reaching the generation of the matrix of auto-values, allowing the age of the projection space of the data. With this based and delimited, it is possible to establish the ranges of observation of the mentioned statisticians. The result obtained from this research corresponds to software that allows the constant observation and analysis of the behavior of each variable of the generation engine. It describes the upper limit, lower limit, arithmetic mean, principal components, graphics of the statistics, and detects the failures in real times

Fuzzy logic methodology to study the behavior of energy transformation processes based on statistics t2 and q

In the processes of energy transformation, to carry out an adequate follow-up of the process parameters represent an opportunity to propose strategies to improve the processes' performance. For this reason, it is essential to analyze the behavior of process variables under the quantitative and qualitative optics supported by the experts. Thus, this work proposes a methodology of fuzzy Mandani type logic that allows the analysis of energy transformation processes (such as internal combustion engines) based on T2 and Q statistics, as a way to identify whether the operation limits are kept within the normal or exceed the limits, achieving to identify the anomaly in the process. In the initial stage, MATLAB implements two diffuse systems; the first system aims to determine the impact variables have on the generation of an anomaly, without identifying the type of defect. In the second stage, it's defined as a function of the number guests, the kind of monster that occurs in the observations made from the transition range in the operation of the system analyzed, until the last measurement obtained. In the third stage, the statistics T2, Q, and its limits are determined from the operating variables of the selected system. Finally, the previously calculated statistics are graphically processed in the diffuse systems. The results obtained in this work show that the analysis of processes or phenomena based on qualitative observations, the methodology implemented, is a useful tool for decision making in the industrial sector