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Journal articles on the topic 'Stationary and stable solution'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

AMBROSO, ANNALISA. "STABILITY FOR SOLUTIONS OF A STATIONARY EULER–POISSON PROBLEM." Mathematical Models and Methods in Applied Sciences 16, no. 11 (November 2006): 1817–37. http://dx.doi.org/10.1142/s0218202506001728.

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We study the stability of the solutions of a stationary Euler–Poisson problem modeling a plasma diode. The model was presented in a previous paper, where we proved the existence of multiple solutions. Since, physically speaking, we expect only one solution to be present, we propose here two strategies for selecting the physical stationary regime. On the one hand, we study the energy functional associated with the stationary problem and order the different solutions in terms of their energy level and as stationary points for the functional. On the other hand, starting from the remark that a stable stationary state is the state which is reached after an evolution, we solve numerically the corresponding time-dependent Euler–Poisson problem and show that only one of the possible solutions is indeed stable in this sense. These two independent approaches select the same solution that we label therefore as stable.

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2

Dikansky, Arnold. "Asymptotically stable stationary solutions to the reaction-diffusion equations." Bulletin of the Australian Mathematical Society 47, no. 2 (April 1993): 273–86. http://dx.doi.org/10.1017/s0004972700012508.

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We assume that there exists an asymptotically stable stationary solution of a Galerkin approximation for the reaction-diffusion system. It is shown that there exists a nearby stationary solution of the full reaction-diffusion system provided the order of the Galerkin approximation is high enough. The Lyapunov second method is used to prove the asymptotic stability of the stationary solution.

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3

Bass, L., A. J. Bracken, K. Holmåker, and B. R. F. Jefferies. "Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 29, no. 2 (October 1987): 156–94. http://dx.doi.org/10.1017/s0334270000005701.

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AbstractA model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

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4

Lan, Xiangjun, Zhihua Feng, and Fan Lv. "Stochastic Principal Parametric Resonances of Composite Laminated Beams." Shock and Vibration 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/617828.

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This paper presents a detailed study on the stochastic stability, jump, and bifurcation of the motion of the composite laminated beams subject to axial load. The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved and the results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response. The stochastic jump and bifurcation of the response are numerically calculated through the stationary joint probability and the results reveal that (a) the higher the excitation frequency is, the more probable the jump from the stable stationary nontrivial solution to the stable stationary trivial one is; (b) the most probable motion is around the nontrivial solution when the bandwidth is smaller; (c) the outer flabellate peak decreases, while the central volcano peak increases as the value of the excitation load decreases; and (d) the overall tendency of the response is that the probable motion jumps from the stable stationary nontrivial branch to the stable stationary trivial one as the fiber orientation angle of the first lamina with respect to thex-axis of the beam increases from zero to a smaller angle.

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5

SUZUKI, TAKASHI. "A NOTE ON THE STABILITY OF STATIONARY SOLUTIONS TO A SYSTEM OF CHEMOTAXIS." Communications in Contemporary Mathematics 02, no. 03 (August 2000): 373–83. http://dx.doi.org/10.1142/s0219199700000189.

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In 1970 a system of parabolic equations was proposed by Keller and Segel to describe the chemotactic feature of cellular slime molds. It has L1 preserving property for the first component and in use of this the stationary problem is reduced to a single elliptic problem concerning the second component. This problem has a variational structure and several features of the solutions are derived from it. In this paper we study linearized stable solutions in this sense and show that any of them is stable as a stationary solution to the original system of parabolic equations.

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6

FELLNER, KLEMENS, and GAËL RAOUL. "STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS." Mathematical Models and Methods in Applied Sciences 20, no. 12 (December 2010): 2267–91. http://dx.doi.org/10.1142/s0218202510004921.

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In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states [Formula: see text] approximate weakly a unique stationary state [Formula: see text]. We illustrate our results with numerical examples.

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7

Finkelshtein, Dmitri, Yuri Kondratiev, Stanislav Molchanov, and Pasha Tkachov. "Global stability in a nonlocal reaction-diffusion equation." Stochastics and Dynamics 18, no. 05 (September 12, 2018): 1850037. http://dx.doi.org/10.1142/s0219493718500375.

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We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

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8

Kong, Liang. "Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal." Journal of Mathematics Research 9, no. 1 (January 2, 2017): 1. http://dx.doi.org/10.5539/jmr.v9n1p1.

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The current paper investigate the persistence of positive solutions of KPP type evolution equations with random/nonlocal dispersal in locally spatially inhomogeneous habitat. By the constructions of super/sub solutions and comparison principle, we prove that such an equation has a unique globally stable positive stationary solution.

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9

Surgailis, Donatas. "A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares." Advances in Applied Probability 40, no. 04 (December 2008): 1198–222. http://dx.doi.org/10.1017/s0001867800003025.

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.

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10

Surgailis, Donatas. "A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares." Advances in Applied Probability 40, no. 4 (December 2008): 1198–222. http://dx.doi.org/10.1239/aap/1231340170.

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.

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11

NAKAMURA, TOHRU, and SHINYA NISHIBATA. "STATIONARY WAVE ASSOCIATED WITH AN INFLOW PROBLEM IN THE HALF LINE FOR VISCOUS HEAT-CONDUCTIVE GAS." Journal of Hyperbolic Differential Equations 08, no. 04 (December 2011): 651–70. http://dx.doi.org/10.1142/s0219891611002524.

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We study the large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half space. We consider an inflow problem where the gas enter into the region through the boundary, and we show that a corresponding stationary solution is time-asymptotically stable in both the subsonic and transonic cases. The proof of asymptotic stability is based on a priori estimates of the perturbation from the stationary solution, which are derived by a standard energy method, provided the boundary strength and the initial perturbation in a certain Sobolev space are sufficiently small.

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12

BARILLON, C., G. M. MAKHVILADZE, and V. VOLPERT. "Stability of stationary solutions for a degenerate parabolic system." European Journal of Applied Mathematics 12, no. 1 (February 2001): 57–75. http://dx.doi.org/10.1017/s0956792501004430.

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The paper is devoted to the stability of stationary solutions of an evolution system, describing heat explosion in a two-phase medium, where a parabolic equation is coupled with an ordinary differential equation. Spectral properties of the problem linearized about a stationary solution are analyzed and used to study stability of continuous branches of solutions. For the convex nonlinearity specific to combustion problems it is shown that solutions on the first increasing branch are stable, solutions on all other branches are unstable. These results remain valid for the scalar equation and they generalize the results obtained before for heat explosion in the radially symmetric case [1].

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13

Moore, G. "Algorithms for constructing stable manifolds of stationary solutions." IMA Journal of Numerical Analysis 19, no. 3 (July 1, 1999): 375–424. http://dx.doi.org/10.1093/imanum/19.3.375.

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14

CIELIEBAK, K., and E. VOLKOV. "A note on the stationary Euler equations of hydrodynamics." Ergodic Theory and Dynamical Systems 37, no. 2 (October 6, 2015): 454–80. http://dx.doi.org/10.1017/etds.2015.50.

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This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

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15

Medina, Rigoberto, and M. I. Gil'. "Accurate solution estimates for nonlinear nonautonomous vector difference equations." Abstract and Applied Analysis 2004, no. 7 (2004): 603–11. http://dx.doi.org/10.1155/s1085337504306184.

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The paper deals with the vector discrete dynamical systemxk+1=Akxk+fk(xk). Thewell-known result by Perron states that this system is asymptotically stable ifAk≡A=constis stable andfk(x)≡f˜(x)=o(‖x‖). Perron's result gives no information about the size of the region of asymptotic stability and norms of solutions. In this paper, accurate estimates for the norms of solutions are derived. They give us stability conditions for (1.1) and bounds for the region of attraction of the stationary solution. Our approach is based on the freezing method for difference equations and on recent estimates for the powers of a constant matrix. We also discuss applications of our main result to partial reaction-diffusion difference equations.

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16

ZHOU, FUJUN, and JUNDE WU. "Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation." European Journal of Applied Mathematics 26, no. 4 (April 10, 2015): 401–25. http://dx.doi.org/10.1017/s0956792515000108.

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Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

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17

Guo, Shangjiang, Shangzhi Li, and Bounsanong Sounvoravong. "Oscillatory and Stationary Patterns in a Diffusive Model with Delay Effect." International Journal of Bifurcation and Chaos 31, no. 03 (March 15, 2021): 2150035. http://dx.doi.org/10.1142/s0218127421500358.

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In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns.

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18

Egger, Herbert, and Matthias Schlottbom. "Stationary radiative transfer with vanishing absorption." Mathematical Models and Methods in Applied Sciences 24, no. 05 (March 3, 2014): 973–90. http://dx.doi.org/10.1142/s0218202513500735.

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We investigate the unique solvability of radiative transfer problems without strictly positive lower bounds on the absorption and scattering parameters. The analysis is based on a reformulation of the transfer equation as a mixed variational problem with penalty term for which we establish the well-posedness. We also prove stability of the solution with respect to perturbations in the parameters. This allows to approximate stationary radiative transfer problems by even-parity formulations even in the case of vanishing absorption. The mixed variational framework used for the analysis also enables a systematic investigation of discretization obtained by Galerkin methods. We show that, in contrast to the full problem, the widely used PN-approximations, and discretizations based on these, are not stable in the case of vanishing absorption. Some consequences and possible remedies yielding stable approximations are discussed.

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19

Vlysidis, Michail, and Yiannis Kaznessis. "On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure." Entropy 20, no. 9 (September 6, 2018): 678. http://dx.doi.org/10.3390/e20090678.

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Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.

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20

Vishnevski�, M. P. "On stable stationary solutions to a quasilinear parabolic equation." Siberian Mathematical Journal 34, no. 2 (1992): 233–41. http://dx.doi.org/10.1007/bf00970948.

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21

Simal Nascimento, Arnaldo. "Stable stationary solutions induced by spatial inhomogeneity via ?-convergence." Boletim da Sociedade Brasileira de Matem�tica 29, no. 1 (March 1998): 75–97. http://dx.doi.org/10.1007/bf01245869.

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22

Hong, Woo-Pyo. "Existence Conditions for Stable Stationary Solitons of the Cubic-Quintic Complex Ginzburg-Landau Equation with a Viscosity Term." Zeitschrift für Naturforschung A 63, no. 12 (December 1, 2008): 757–62. http://dx.doi.org/10.1515/zna-2008-1203.

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We report the existence of a new family of stable stationary solitons in the one-dimensional cubicquintic complex Ginzburg-Landau equation with the viscosity term. By applying the paraxial ray approximation, we obtain the relation between the width and the peak amplitude of the stationary soliton in terms of the model parameters. We find the bistable solitons in the presence of the small viscosity term.We verify the analytical results by direct numerical simulations and show the stability of the stationary solitons. We conclude that the existence condition obtained by the paraxial method may serve as physically more reliable initial condition for stable stationary soliton propagation in the optical system modeled by the one-dimensional cubic-quintic complex Ginzburg-Landau equation with the viscosity term, of which analytical solution is not obtainable with all nonzero parameters.

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23

Nizhnik, Irene. "Stable stationary solutions for a reaction–diffusion equation with a multi-stable nonlinearity." Physics Letters A 357, no. 4-5 (September 2006): 319–22. http://dx.doi.org/10.1016/j.physleta.2006.04.054.

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24

Norwood, Joseph. "A stable stationary solution of compressible magnetohydrodynamics associated with conservation of modified kinetic helicity." Journal of Plasma Physics 41, no. 3 (June 1989): 561–71. http://dx.doi.org/10.1017/s0022377800014082.

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It is shown that, in addition to magnetic helicity and cross-helicity, a modified form of kinetic helicity is also conserved in MHD fluids if the Hall effect is taken into account. The consequence of including this modified kinetic helicity as a conservation integral in a variation of the system energy is the emergence of an unconditionally MHD-stable solution that is realized in migma plasmas and may be expected to emerge as a self-organized configuration from an initially turbulent magnetofluid.

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25

Chakrabarti, Sandip K. "On the Accretion Disk Models by Stationary and Non-Stationary Shock Waves." Symposium - International Astronomical Union 159 (1994): 477. http://dx.doi.org/10.1017/s0074180900176545.

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An important point which emerged from this meeting is that disks in AGNs are not simply thin, Keplerian type; they show more complex behaviour. Chakrabarti (1990a and references therein) has shown that in an inviscid accretion disk with significant angular momentum, the centrifugal barrier is strong enough to produce axisymmetric standing shock wave. Subsequently, this work was extended to include the non-axisymmetric and viscous disks (Chakrabarti, 1990b). Particularly important are the solutions with viscosity, as they show that as the viscosity is increased, the stable becomes weaker and weaker till it disappears completely. This solution has a unifying character that inviscid pressure driven disks have almost constant angular momentum and can have shock discontinuities, but viscous driven disks dissipate angular momentum quick enough not to have centrifugal barrier and therefore no shock waves. Chakrabarti & Molteni (1993), using Smoothed Particle Hydrodynamics have shown that shocks are produced in inviscid disks, exactly where they are predicted.Unlike a Keplerian disk, a disk with a shock has basically two temperature zones. The post shock solution is responsible for the Big Blue Bump and UV excess (Chakrabarti and Wiita, 1992). At the shock location, the disk is ‘bulged’ the hard radiation from this region is intercepted by the cooler pre-shock flow. The shock strength and location are sensitive to input specific energy of the flow. This configuration might be responsible for the ‘zero-lag’ correlated variability of, say, NGC 5548 (Chakrabarti, Haardt, Maraschi & Molendi, AA, submitted) discussed in this meeting. Spiral shocks which may be produced in disks in a binary system can also appear in disks around AGNs; the perturbation may be due to passage of massive objects (Chakrabarti & Wiita, 1993a). They also cause time variations in the double horned pattern from disk line emission (Chakrabarti & Wiita 1993b) as observed in, say ARP 102B. All these observations point that shocks are probably important ingredients in any accretion disk in AGNs

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26

Nakamura, Tohru, and Shinya Nishibata. "Existence and asymptotic stability of stationary waves for symmetric hyperbolic–parabolic systems in half-line." Mathematical Models and Methods in Applied Sciences 27, no. 11 (August 30, 2017): 2071–110. http://dx.doi.org/10.1142/s0218202517500397.

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In this paper, we consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic–parabolic system under assuming that the system has a strictly convex entropy function. We firstly prove existence of a stationary solution by assuming that a boundary strength is sufficiently small. The existence of the stationary solution is characterized by the number of negative characteristics. In the case where one characteristic speed is zero at spatial asymptotic state [Formula: see text], we assume that the characteristic field corresponding to the characteristic speed 0 is genuinely nonlinear in order to show existence of a degenerate stationary solution with the aid of a center manifold theory. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on an initial perturbation in the Sobolev space. The key to proof is to derive the uniform a priori estimates by using the energy method, where the stability condition of Shizuta–Kawashima type plays an essential role.

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27

Verdejo, Humberto, Wolfgang Kliemann, and Luis Vargas. "Performance of Power Systems under Sustained Random Perturbations." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/432548.

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This paper studies linear systems under sustained additive random perturbations. The stable operating point of an electric power system is replaced by an attracting stationary solution if the system is subjected to (small) random additive perturbations. The invariant distribution of this stationary solution gives rise to several performance indices that measure how well the system copes with the randomness. These indices are introduced, showing how they can be used for the optimal tuning of system parameters in the presence of noise. Results on a four-generator two-area system are presented and discussed.

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Daskopoulos, P., and A. M. Lenhoff. "Flow in curved ducts: bifurcation structure for stationary ducts." Journal of Fluid Mechanics 203 (June 1989): 125–48. http://dx.doi.org/10.1017/s0022112089001400.

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In developed laminar flow in stationary curved ducts there is, in addition to the two-vortex secondary flow structure first analysed by Dean (1927), another solution branch with a four-vortex secondary flow. These two branches have recently been shown to be joined by a third branch, but stability characteristics and the possible presence of additional branches have yet to be described. In this paper orthogonal collocation is used in conjunction with continuation techniques to characterize the bifurcation structure for symmetric flows. The two- and four-vortex solutions are stable to symmetric disturbances, while the recently reported branch joining them is unstable. A more systematic exploration of the parameter space than has hitherto been reported is performed by examining the morphogenesis of the bifurcation structure within the general framework of properties described by Benjamin (1978). The starting point is the ‘perfect’ problem of flow in an infinite curved slit, which bifurcates to give rise to a cellular structure. Addition of ‘stickiness’ at the cell boundaries turns each pair of cells into a curved duct of rectangular cross-section which, by a geometry change, leads to the curved circular tube. For the perfect problem a large number of solution branches are present, but the addition of stickiness turns most of them into isolae which vanish before the no-slip limit is reached. The solution branches that remain include, in addition to the three described previously, another solution family not connected to the other one. This family comprises two branches, both four-vortex in character and unstable to symmetric disturbances.

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Goldshtik, M. A., and N. I. Javorsky. "On the flow between a porous rotating disk and a plane." Journal of Fluid Mechanics 207 (October 1989): 1–28. http://dx.doi.org/10.1017/s002211208900248x.

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Axisymmetric flow of viscous incompressible fluid between a rotating porous disk and an impermeable fixed plane is investigated. It is shown that with injection and suction through a porous disk rotating with sufficiently large angular velocity there are many isolated steady self-similar solutions. In the case of suction through a fixed porous disk at a certain Reynolds number there exists bifurcation of the stable rotational regime of flow, implying a spontaneous break of the flow symmetry and an arbitrary rise of the fluid rotation within the framework of self-similarity. This unusual effect is discussed in detail, and the results of a relevant experiment are presented. Another unusual result is the existence of multicellular regimes consistent with suction, when the lift force acting on a rapidly rotating porous disk is anomalously large; in this case some of these regimes are stable relative to self-similar perturbations.With sufficiently strong suction and rotation the stationary solution with large lift becomes unstable and the regime of self-oscillations arises. Diagrams of the possible stationary flow regimes have been constructed, and the stable ones have been identified. At the limit of vanishing viscosity we find, in the case of the suction, non-classical boundary layers on the solid surfaces characterized by a finite jump of the normal component of the velocity and unlimited tangential components. In this limit, in the interior flow region the singular non-viscous solution with an infinite velocity of rotation arises, while all limited non-singular admissible non-viscous solutions are not stable.

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30

Caraballo, Tomás, Peter E. Kloeden, and José Real. "Discretization of Asymptotically Stable Stationary Solutions of Delay Differential Equations with a Random Stationary Delay." Journal of Dynamics and Differential Equations 18, no. 4 (July 18, 2006): 863–80. http://dx.doi.org/10.1007/s10884-006-9022-5.

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31

Tzanetis, D. E., and P. M. Vlamos. "SOME INTERESTING SPECIAL CASES OF A NON-LOCAL PROBLEM MODELLING OHMIC HEATING WITH VARIABLE THERMAL CONDUCTIVITY." Proceedings of the Edinburgh Mathematical Society 44, no. 3 (October 2001): 585–95. http://dx.doi.org/10.1017/s0013091500000109.

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AbstractThe non-local equation$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

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32

Aronson, D. G., A. Tesei, and H. Weinberger. "A density-dependent diffusion system with stable discontinuous stationary solutions." Annali di Matematica Pura ed Applicata 152, no. 1 (December 1988): 259–80. http://dx.doi.org/10.1007/bf01766153.

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33

Zhu, Wei Bing, Sheng Ren Zhou, and He Shun Wang. "Thermal Deformation in a Static Pressure Dry Gas Seal." Applied Mechanics and Materials 217-219 (November 2012): 2406–9. http://dx.doi.org/10.4028/www.scientific.net/amm.217-219.2406.

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Based on the basic principles of heat transfer, the two-dimensional stable state cylindrical coordinate solid heat conduction differential equation having no inner heat source is established. Use the ANSYS software to establish the temperature field finite element analysis model of rotating and stationary rings and carry on the solution to the temperature field. The temperature distributing rules of rotating and stationary rings are obtained at the same time. According to thermo-elastic deformation theory, numerical analysis method and separation method are applied to resolve and analyze thermal-structural coupling deformation of rotating and stationary rings.

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PANZARELLA, CHARLES H., STEPHEN H. DAVIS, and S. GEORGE BANKOFF. "Nonlinear dynamics in horizontal film boiling." Journal of Fluid Mechanics 402 (January 10, 2000): 163–94. http://dx.doi.org/10.1017/s0022112099006801.

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This paper uses thin-film asymptotics to show how a thin vapour layer can support a liquid which is heated from below and cooled from above, a process known as horizontal film boiling. This approach leads to a single, strongly-nonlinear evolution equation which incorporates buoyancy, capillary and evaporative effects. The stability of the vapour layer is analysed using a variety of methods for both saturated and subcooled film boiling. In subcooled film boiling, there is a stationary solution, a constant-thickness vapour film, which is determined by a simple heat-conduction balance. This is Rayleigh–Taylor unstable because the heavier liquid is above the vapour, but the instability is completely suppressed for sufficient subcooling. A bifurcation analysis determines a supercritical branch of stable, spatially-periodic solutions when the basic state is no longer stable. Numerical branch tracing extends this into the strongly-nonlinear regime, revealing a hysteresis loop and a secondary bifurcation to a branch of travelling waves which are stable under certain conditions. There are no stationary solutions in saturated film boiling, but the initial development of vapour bubbles is determined by directly solving the time-dependent evolution equation. This yields important information about the transient heat transfer during bubble development.

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35

SHCHERBINA, MARIYA, and BRUNELLO TIROZZI. "STABILITY OF ASYNCHRONOUS STATES OF SPIKING NEURONS." International Journal of Modern Physics B 18, no. 04n05 (February 20, 2004): 759–71. http://dx.doi.org/10.1142/s0217979204024380.

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We consider a system of N spiking neurons with random synaptic connections which take values ±1 with equal probability. The resulting system of equations has a stationary solution equal to 1 for the fraction of neurons having potential x at time t. This solution describes an asynchronous state. We study the stability of such a state in a perturbative way and find a threshold for the parameters of the model such that for values larger than this threshold the stationary asynchronous state is stable otherwise it is unstable. In other terms the stability of the asynchronous state holds only for relatively small random perturbations.

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NEGRI, MATTEO, and CHRISTOPH ORTNER. "QUASI-STATIC CRACK PROPAGATION BY GRIFFITH'S CRITERION." Mathematical Models and Methods in Applied Sciences 18, no. 11 (November 2008): 1895–925. http://dx.doi.org/10.1142/s0218202508003236.

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We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasi-static evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith's criterion in a suitable form which takes into account both stable and unstable propagations, as well as an energy balance formula which accounts for dissipation in the unstable regime. If the load is monotonically increasing, this solution is explicit and almost everywhere unique. For more general loads we construct a solution via time discretization. Finally, we consider a finite element discretization of the problem and prove convergence of the discrete solutions.

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BUCKWAR, E., M. G. RIEDLER, and P. E. KLOEDEN. "THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 265–81. http://dx.doi.org/10.1142/s0219493711003279.

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An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

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Hansen, Bradley M. S., and Smadar Naoz. "The stationary points of the hierarchical three-body problem." Monthly Notices of the Royal Astronomical Society 499, no. 2 (September 3, 2020): 1682–700. http://dx.doi.org/10.1093/mnras/staa2602.

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ABSTRACT We study the stationary points of the hierarchical three body problem in the planetary limit (m1, m2 ≪ m0) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal stationary points of the quadrupole solution in the limit of small outer eccentricity e2 but that new families of stable fixed points occur in both prograde and retrograde cases. The most important new equilibria are those that branch off from the quadrupolar solutions and extend to large e2. The apsidal alignment of these families is a function of mass and inner planet eccentricity, and is determined by the relative directions of precession of ω1 and ω2 at the quadrupole level. These new equilibria are also the most resilient to the destabilizing effects of relativistic precession. We find additional equilibria that enable libration of the inner planet argument of pericentre in the limit of radial orbits and recover the non-linear analogue of the Laplace–Lagrange solutions in the coplanar limit. Finally, we show that the chaotic diffusion and orbital flips identified with the eccentric Kozai–Lidov mechanism and its variants can be understood in terms of the stationary points discussed here.

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Wang, Zhi-An, and Xin Xu. "Steady states and pattern formation of the density-suppressed motility model." IMA Journal of Applied Mathematics 86, no. 3 (May 24, 2021): 577–603. http://dx.doi.org/10.1093/imamat/hxab006.

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Abstract This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.

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Delale, Can F., Kohei Okita, and Yoichiro Matsumoto. "Steady-State Cavitating Nozzle Flows With Nucleation." Journal of Fluids Engineering 127, no. 4 (April 2, 2005): 770–77. http://dx.doi.org/10.1115/1.1949643.

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Quasi-one-dimensional steady-state cavitating nozzle flows with homogeneous bubble nucleation and nonlinear bubble dynamics are considered using a continuum bubbly liquid flow model. The onset of cavitation is modeled using an improved version of the classical theory of homogeneous nucleation, and the nonlinear dynamics of cavitating bubbles is described by the classical Rayleigh-Plesset equation. Using a polytropic law for the partial gas pressure within the bubble and accounting for the classical damping mechanisms, in a crude manner, by an effective viscosity, stable steady-state solutions with stationary shock waves as well as unstable flashing flow solutions were obtained, similar to the homogeneous bubbly flow solutions given by Wang and Brennen [J. Fluids Eng., 120, 166–170, 1998] and by Delale, Schnerr, and Sauer [J. Fluid Mech., 427, 167–204, 2001]. In particular, reductions in the maximum bubble radius and bubble collapse periods are observed for stable nucleating nozzle flows as compared to the nonnucleating stable solution of Wang and Brennen under similar conditions.

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Li, Yeping, and Peicheng Zhu. "Asymptotics toward a nonlinear wave for an outflow problem of a model of viscous ions motion." Mathematical Models and Methods in Applied Sciences 27, no. 11 (August 30, 2017): 2111–45. http://dx.doi.org/10.1142/s0218202517500403.

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We shall investigate the asymptotic stability, toward a nonlinear wave, of the solution to an outflow problem for the one-dimensional compressible Navier–Stokes–Poisson equations. First, we construct this nonlinear wave which, under suitable assumptions, is the superposition of a stationary solution and a rarefaction wave. Then it is shown that the nonlinear wave is asymptotically stable in the case that the initial data are a suitably small perturbation of the nonlinear wave. The main ingredient of the proof is the [Formula: see text]-energy method that takes into account both the effect of the self-consistent electrostatic potential and the spatial decay of the stationary part of the nonlinear wave.

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QILIN, LIU, LIANG FEI, and LI YUXIANG. "Asymptotic behaviour for a non-local parabolic problem." European Journal of Applied Mathematics 20, no. 3 (June 2009): 247–67. http://dx.doi.org/10.1017/s0956792509007803.

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In this paper, we consider the asymptotic behaviour for the non-local parabolic problemwith a homogeneous Dirichlet boundary condition, where λ > 0,p> 0 andfis non-increasing. It is found that (a) for 0 <p≤ 1,u(x,t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) for 1 <p< 2,u(x,t) is globally bounded for any λ > 0; (c) forp= 2, if 0 < λ < 2|∂Ω|2, thenu(x,t) is globally bounded; if λ = 2|∂Ω|2, there is no stationary solution andu(x,t) is a global solution andu(x,t) → ∞ ast→ ∞ for allx∈ Ω; if λ > 2|∂Ω|2, there is no stationary solution andu(x,t) blows up in finite time for allx∈ Ω; (d) forp> 2, there exists a λ* > 0 such that for λ > λ*, or for 0 < λ ≤ λ* andu0(x) sufficiently large,u(x,t) blows up in finite time. Moreover, some formal asymptotic estimates for the behaviour ofu(x,t) as it blows up are obtained forp≥ 2.

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43

Castellani, G. C., C. Giberti, C. Franceschi, and F. Bersani. "Stable State Analysis of an Immune Network Model." International Journal of Bifurcation and Chaos 08, no. 06 (June 1998): 1285–301. http://dx.doi.org/10.1142/s0218127498000991.

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The paper analyzes a model of immune system developed by different authors (Perelson, De Boer, Weisbuch and others). The model describes interactions among B-lymphocytes. It does not consider antibodies as interaction intermediaries, although it uses a typical activation curve. The relevant parameters are: an influx term, a threshold value, a proliferation rate, and a decay parameter. The study of the n-dimensional extension of the model and a bifurcation analysis of the stationary states with respect to the influx parameter show that the influx value for which biologically acceptable solutions exist decreases as n increases. When the influx term is neglected the stationary states are obtained analytically and their stability is discussed. Moreover, it is discussed how the stable solutions can be considered as "selective states", that is, with only one high idiotypic concentration, when we suppose a complete connectivity.

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44

Linton, Oliver, Jiazhu Pan, and Hui Wang. "ESTIMATION FOR A NONSTATIONARY SEMI-STRONG GARCH(1,1) MODEL WITH HEAVY-TAILED ERRORS." Econometric Theory 26, no. 1 (June 19, 2009): 1–28. http://dx.doi.org/10.1017/s0266466609090598.

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This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate $\root \of n $ to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

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45

Redheffer, Ray. "The Behavior of Solutions near a Stable or Semistable Stationary Point." American Mathematical Monthly 112, no. 9 (November 1, 2005): 817. http://dx.doi.org/10.2307/30037603.

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46

Redheffer, Ray. "The Behavior of Solutions near a Stable or Semistable Stationary Point." American Mathematical Monthly 112, no. 9 (November 2005): 817–22. http://dx.doi.org/10.1080/00029890.2005.11920255.

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47

Caraballo, Tom�s, Peter E. Kloeden, and Bj�rn Schmalfu�. "Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation." Applied Mathematics and Optimization 50, no. 3 (August 18, 2004): 183–207. http://dx.doi.org/10.1007/s00245-004-0802-1.

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48

Jiang, Yu, Liquan Mei, Huiming Wei, Weijun Tian, and Jiatai Ge. "A Finite Element Variational Multiscale Method Based on Two Local Gauss Integrations for Stationary Conduction-Convection Problems." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/747391.

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A new finite element variational multiscale (VMS) method based on two local Gauss integrations is proposed and analyzed for the stationary conduction-convection problems. The valuable feature of our method is that the action of stabilization operators can be performed locally at the element level with minimal additional cost. The theory analysis shows that our method is stable and has a good precision. Finally, the numerical test agrees completely with the theoretical expectations and the “ exact solution,” which show that our method is highly efficient for the stationary conduction-convection problems.

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Tettamanti, Manuele, and Alberto Parola. "Formation Dynamics of Black- and White-Hole Horizons in an Analogue Gravity Model." Universe 6, no. 8 (July 31, 2020): 105. http://dx.doi.org/10.3390/universe6080105.

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We investigate the formation dynamics of sonic horizons in a Bose gas confined in a (quasi) one-dimensional trap. This system is one of the most promising realizations of the analogue gravity paradigm and has already been successfully studied experimentally. Taking advantage of the exact solution of the one-dimensional, hard-core, Bose model (Tonks–Girardeau gas), we show that by switching on a step potential, either a sonic, black-hole-like horizon or a black/white hole pair may form, according to the initial velocity of the fluid. Our simulations never suggest the formation of an isolated white-hole horizon, although a stable stationary solution of the dynamical equations with those properties is analytically found. Moreover, we show that the semiclassical dynamics, based on the Gross–Pitaevskii equation, conforms to the exact solution only in the case of fully subsonic flows while a stationary solution exhibiting a supersonic transition is never reached dynamically.

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Chai, Xintao, Shangxu Wang, Sanyi Yuan, Jianguo Zhao, Langqiu Sun, and Xian Wei. "Sparse reflectivity inversion for nonstationary seismic data." GEOPHYSICS 79, no. 3 (May 1, 2014): V93—V105. http://dx.doi.org/10.1190/geo2013-0313.1.

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Conventional reflectivity inversion methods are based on a stationary convolution model and theoretically require stationary seismic traces as input (i.e., those free of attenuation and dispersion effects). Reflectivity inversion for nonstationary data, which is typical for field surveys, requires us to first compensate for the earth’s [Formula: see text]-filtering effects by inverse [Formula: see text] filtering. However, the attenuation compensation for inverse [Formula: see text] filtering is inherently unstable, and offers no perfect solution. Thus, we presented a sparse reflectivity inversion method for nonstationary seismic data. We referred to this method as nonstationary sparse reflectivity inversion (NSRI); it makes the novel contribution of avoiding intrinsic instability associated with inverse [Formula: see text] filtering by integrating the earth’s [Formula: see text]-filtering operator into the stationary convolution model. NSRI also avoids time-variant wavelets that are typically required in time-variant deconvolution. Although NSRI is initially designed for nonstationary signals, it is suitable for stationary signals (i.e., using an infinite [Formula: see text]). The equations for NSRI only use reliable frequencies within the seismic bandwidth, and the basis pursuit optimizes a cost function of mixed [Formula: see text] norms to derive a stable and sparse solution. Synthetic examples show that NSRI can directly retrieve reflectivity from nonstationary data without advance inverse [Formula: see text] filtering. NSRI is satisfactorily stable in the presence of severe noise, and a slight error in the [Formula: see text] value does not greatly disturb the sensitivity of NSRI. A field data example confirmed the effectiveness of NSRI.

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