Stability Of Stationary Solutions Research Articles

Synchronising Energy Harvesting and Data Packets in a Wireless Sensor

We consider a wireless sensor node that gathers energy through harvesting and reaps data through sensing. The node has a wireless transmitter that sends out a data packet whenever there is at least one “energy packet” and one “data packet”, where an energy packet represents the amount of accumulated energy at the node that can allow the transmission of a data packet. We show that such a system is unstable when both the energy storage space and the data backlog buffer approach infinity, and we obtain the stable stationary solution when both buffers are finite. We then show that if a single energy packet is not sufficient to transmit a data packet, there are conditions under which the system is stable, and we provide the explicit expression for the joint probability distribution of the number of energy and data packets in the system. Since the two flows of energy and data can be viewed as flows that are instantaneously synchronised, this paper also provides a mathematical analysis of a fundamental problem in computer science related to the stability of the “join” synchronisation primitive.

Some recent results on a free boundary problem for microelectromechanical systems

AbstractThe mathematical modeling of idealized microelectromechanical systems leads to a nonlinear singular elliptic‐parabolic free boundary problem involving non‐smooth domains and depending on a parameter λ reflecting the applied voltage across the device. Recent results derived in [1,4] are presented which confirm physically important and expected features: For small λ there is (at least) one stable stationary solution and solutions to the evolution equation exist globally in time, while for large λ no stationary solution exists and the evolution equation has no global solution. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.

Traveling pulse solutions to FitzHugh–Nagumo equations

Reaction–diffusion systems serve as relevant models for studying complex patterns in several fields of nonlinear sciences. A localized pattern is a stable non-constant stationary solution usually located far away from neighborhoods of bifurcation induced by Turing’s instability. In the study of FitzHugh–Nagumo equations, we look for a standing pulse with a profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system, and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses found in the literature, a new technique is attempted by seeking a standing pulse solution with a sign change.

Thin-shell wormholes in Born–Infeld electrodynamics with modified Chaplygin gas

Abstract In this paper, we construct spherically symmetric thin-shell wormholes in the scenario of Born–Infeld electrodynamics theory. We take the modified Chaplygin gas for the description of exotic matter around the wormhole throat. The stability of static wormhole solutions with different values of charge and Born–Infeld parameter is investigated. We compare our results with those obtained for generalized Chaplygin gas [36] and conclude that stable static wormhole solutions also exist even for large value of Born–Infeld parameter.

Influence of the generation and recombination of carriers on the volt-ampere characteristics of semiconductor heterostructures

The influence of the generation and recombination of free charge carriers in the p-i-n structure in strong electric fields on the stationary volt-ampere characteristics is considered. The issues of the stability of stationary solutions are investigated. It is shown that a form of the stationary volt-ampere characteristics and their stability depend on ratios between free electrons and holes in the current injected into the i-layer and in the current outflowing from the layer. When the structure is forward biased, the oscillations may occur.

Synchronization in populations of sparsely connected pulse-coupled oscillators

We propose a population model for δ-pulse–coupled oscillators with sparse connectivity. The model is given as an evolution equation for the phase density which takes the form of a partial differential equation with a non-local term. We discuss the existence and stability of stationary solutions and exemplify our approach for integrate-and-fire–like oscillators. While for strong couplings, the firing rate of stationary solutions diverges and solutions disappear, small couplings allow for partially synchronous states which emerge at a supercritical Andronov-Hopf bifurcation.

Gregory-Laflamme instability of the BTZ black hole in new massive gravity

We find the Gregory-Laflamme $s$-mode instability of the non-rotating BTZ black hole in new massive gravity. This instability shows that the BTZ black hole could not exist as a stable static solution to the new massive gravity. For non-rotating BTZ black string in four dimensions, however, it is demonstrated that the BTZ black string can be stable against the metric perturbation.

The roles of diffusivity and curvature in patterns on surfaces of revolution

We address the question of finding sufficient conditions for existence as well as nonexistence of nonconstant stable stationary solution to the diffusion equation ut=div(a∇u)+f(u) on a surface of revolution with and without boundary. Conditions found relate the diffusivity function a and the geometry of the surface where diffusion takes place. In the case where f is a bistable function, necessary conditions for the development of inner transition layers are given.

The linearization principle for a free boundary problem for viscous, capillary incompressible fluids

The free boundary problem associated with a viscous incompressible surface wave subject to the capillary force on a free upper surface and the Dirichlet boundary condition on a fixed bottom surface is considered. In the spatially periodic case, a general linearization principle is proved, which gives, for sufficiently small perturbations from a linearly stable stationary solution, the existence of a global solution of the associated system and the exponential convergence of the latter to the stationary one. The convergence of the velocity, the pressure, and the free boundary is proved in anisotropic Sobolev--Slobodetskii spaces, and then a suitable change of variables is performed to state the problem in a fixed domain. This linearization principle is applied to the study of the rest state's stability in the case of general potential forces. Bibliography: 16 titles.

Symmetry breaking in binary chains with nonlinear sites

We consider a system of two or four nonlinear sites coupled with binary chain waveguides. When a monochromatic wave is injected into the first (symmetric) propagation channel, the presence of cubic nonlinearity can lead to symmetry breaking, giving rise to emission of antisymmetric wave into the second (antisymmetric) propagation channel of the waveguides. We found that in the case of nonlinear plaquette, there is a domain in the parameter space where neither symmetry-preserving nor symmetry-breaking stable stationary solutions exit. As a result, injection of a monochromatic symmetric wave gives rise to emission of nonsymmetric satellite waves with energies differing from the energy of the incident wave. Thus, the response exhibits nonmonochromatic behavior.

Stability analysis of polarization attraction in optical fibers

The nonlinear cross-polarization interaction among two intense counterpropagating beams in a span of lossless randomly birefringent telecom optical fiber may lead to the attraction an initially polarization scrambled signal towards wave with a well-defined state of polarization at the fiber output. By exploiting exact analytical solutions of the nonlinear polarization coupling process we carry out a linear stability study which reveals that temporally stable stationary solutions are only obtained whenever the output signal polarization is nearly orthogonal to the input pump polarization. Moreover, we predict that polarization attraction is acting in full strength whenever equally intense signal and pump waves are used.

Excited Cr impurity states in Al2O3from constraint density functional theory

The excited states, ${}^{4}{T}_{2g}$ and ${}^{2}{E}_{g}$, of a Cr impurity in Al${}_{2}$O${}_{3}$ were treated by constraint density functional theory by imposing a density matrix constraint (constraint field) to control the electron occupation numbers of the $d$ orbitals. The calculated excitation energies, directly calculated from the self-consistent total energies of the ${}^{4}{A}_{2g}$ ground states and the various excited states, correctly reproduce the experimental ordering. In addition, we find that there is no stationary solution for the excited ${}^{4}{T}_{2g}$ state corresponding to the crystal-field transition state in the usual Kohn-Sham equation, i.e., with no constraint field. By contrast, the excited ${}^{2}{E}_{g}$ state of the spin-flip transition state is a (meta-) stable stationary solution, and may be responsible for the long radiative decay lifetime observed in experiments on ruby.

Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

This article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly dependent on temperature at infinite Prandtl number. Although we verify the proposed techniques solely for viscosities that depend exponentially on temperature, the methods are extensible to other dependence laws. The set-up is a 2D domain with periodic boundary conditions along the horizontal coordinate which introduces a symmetry in the problem. This is the O(2) symmetry, which is particularly well described by spectral methods and motivates the use of these methods in this context. We examine the scope of our techniques by exploring transitions from stationary regimes towards time dependent regimes. At a given aspect ratio, stable stationary solutions become unstable through a Hopf bifurcation, after which the time-dependent regime is solved by the spectral techniques proposed in this article.

Mechanical stability of cylindrical thin-shell wormholes

In this paper, we apply the cut and paste procedure to charged black string for the construction of thin-shell wormhole. We consider the Darmois-Israel formalism to determine the surface stresses of the shell. We take Chaplygin gas to deal with the matter distribution on shell. The radial perturbation approach (preserving the symmetry) is used to investigate the stability of static solutions. We conclude that stable static solutions exist both for uncharged and charged black string thin-shell wormholes for particular values of the parameters.

Stabilization of solution of cauchy problem for a non-divergent parabolic equation

Conditions for stabilization of solution of Cauchy problem for a linear second-order parabolic equation in nondivergent form are studied.

Fronts Between Periodic Patterns for Bistable Recursions on Lattices

Bistable space–time discrete systems commonly possess a large variety of stable stationary solutions with periodic profile. In this context, it is natural to ask about the fate of trajectories composed of interfaces between steady configurations with periodic pattern and in particular, to study their propagation as traveling fronts. Here, we investigate such fronts in piecewise affine bistable recursions on the one-dimensional lattice. By introducing a definition inspired by symbolic dynamics, we prove the existence of front solutions and uniqueness of their velocity, upon the existence of their ground patterns. Moreover, the velocity dependence on parameters and the co-existence of several fronts with distinct ground patterns are also described. Finally, robustness of the results to small $$C^1$$ -perturbations of the piecewise affine map is argued by mean continuation arguments.

A Fluid-Kinetic Model for Particulate Flows with Coagulation and Breakup: Stationary Solutions, Stability, and Hydrodynamic Regimes

We consider coupled models for particulate flows, where the disperse phase is made of particles subject to size variations. We are thus led to kinetic equations with coagulation and breakup operators, coupled to fluid mechanics equations. We discuss the existence and stability of stationary solutions. We also derive macroscopic models through asymptotic hydrodynamic regimes, once relevant scaling parameters have been identified.

Stability of Stationary Solutions and Existence of Oscillating Solutions to a Chemotaxis System in High Dimensional Spaces

Stability of Stationary Solutions and Existence of Oscillating Solutions to a Chemotaxis System in High Dimensional Spaces

Dynamics and interactions of spikes on smoothly curved boundaries for reaction–diffusion systems in 2D

It is known that for special types of reaction–diffusion Systems, such as the Gierer–Meinhardt model and the Gray-Scott model, stable stationary spike solutions exist on boundary points with maximal curvature. In this paper, we rigorously give the equation describing the motion of spike solutions along boundaries for general types of reaction–diffusion systems in R 2. We also apply the general results to the Gierer–Meinhardt model and show that a single spike solution moves toward a boundary point with locally maximal curvature. Moreover, by showing the repulsive interaction of spikes along boundaries for solutions of the Gierer–Meinhardt model, we have stable multispike stationary solutions in the neighborhood of a boundary point with locally maximal curvature.