Multiple Stationary Solutions Research Articles

Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent

In this paper, we are concerned with some properties of the global attractor of weakly damped wave equations. We get the existence of multiple stationary solutions for wave equations with weakly damping. Furthermore, we provide some approaches to verify the small neighborhood of the origin is an attracting domain which is important to obtain the multiple equilibrium points in global attractor.

Extremum seeking control of the CANON process - existence of sub-optimal stationary solutions

The paper considers extremum seeking control (ESC) for on-line optimization of the CANON process, a new and potentially highly effective process for ammonium removal from wastewater. For gradient estimation we employ the classical method based on periodic excitation. From simulations we find that the ESC scheme can lock onto sub-optimal stationary solutions, far removed from the optimal solution, and that the ESC may have multiple stationary solutions for given controller parameters. The cause of this is investigated through analysis of a general dynamic model. Based on the analysis, it is shown that for systems for which the optimum corresponds to the input-output transfer-function having a transmission zero at the origin, there will in general exist a number of stationary solutions to the ESC with periodic excitation. The solutions are characterized by the phase lag of the system, rather than a zero gradient of the objective function, and are hence in general not related to the optimality conditions. For systems that can be described by Hammerstein or Wiener models, as typically considered in ESC, the solution will in general correspond to the zero gradient condition fulfilled at the optimum. As shown, the CANON process can not be described by Hammerstein or Wiener models, and this then explains the observed existence of sub-optimal stationary solutions.

Multiple stationary solutions of euler-poisson equations for non-isentropic gaseous stars

The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.

Deformation and breakup of a non-Newtonian slender drop in an extensional flow: inertial effects and stability

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

Multistability in Recurrent Neural Networks

Stable stationary solutions correspond to memory capacity in the application of associative memory for neural networks. In this presentation, existence of multiple stable stationary solutions for Hopfield-type neural networks with delay and without delay is investigated. Basins of attraction for these stationary solutions are also estimated. Such a scenario of dynamics is established through formulating parameter conditions based on a geometrical setting. The present theory is demonstrated by two numerical simulations on the Hopfield neural networks with delays.

Analysis of the CSTR with two-phase flow under phase equilibrium conditions

A mathematical model of the flow gas–liquid continuous stirred tank reactor (CSTR) is presented under the assumption of thermodynamic equilibrium of the two-phase flow. Phase equilibrium was calculated using the Redlich–Kwong–Soave (RKS) equation of state. An efficient algorithm based on homotopy method was used to obtain a numerical solution of the system of nonlinear algebraic equations of the model. By an example of the liquid-phase hydrogenation of benzene, steady states were analyzed in isothermal and adiabatic regimes and at partial removal of heat released as a result of exothermic reaction. It was shown that under certain conditions, a multiplicity of stationary solutions is possible due to nonlinearity of the kinetic reaction, on the one hand, and deviation of the reaction mixture properties from the ideal state, on the other hand.

Multiple stationary solutions of an irradiated slab

A mathematical model describing the heat budget of an irradiated medium is introduced. The one-dimensional form of the equations and boundary conditions are presented and analysed. Heat transport at one face of the slab occurs by absorption (and reflection) of an incoming beam of short-wave radiation with a fraction of this radiation penetrating into the body of the slab, a diffusive heat flux in the slab and a prescribed incoming heat flux term. The other face of the slab is immersed in its own melt and is considered to be a free surface. Here, temperature continuity is prescribed and evolution of the surface is determined by a Stefan condition. These boundary conditions are flexible enough to describe a range of situations such as a laser shining on an opaque medium, or the natural environment of polar sea ice or lake ice. A two-stream radiation model is used which replaces the simple Beer's law of radiation attenuation frequently used for semi-infinite domains. The stationary solutions of the governing equations are sought and it is found that there exists two possible stationary solutions for a given set of boundary conditions and a range of parameter choices. It is found that the existence of two stationary solutions is a direct result of the model of radiation absorption, due to its effect on the albedo of the medium. A linear stability analysis and numerical calculations indicate that where two stationary solutions exist, the solution corresponding to a larger thickness is always stable and the solution corresponding to a smaller thickness is unstable. Numerical simulations reveal that when there are two solutions, if the slab is thinner than the smaller stationary thickness it will melt completely, whereas if the slab is thicker than the smaller stationary thickness it will evolve toward the larger stationary thickness. These results indicate that other mechanisms (e.g. wave-induced agglomeration of crystals) are necessary to grow a slab from zero initial thickness in the parameter regime that yields two stationary solutions.

Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation \(u_t=\Delta u+f(x,t,u)\) in \(\Omega \times(0,\infty)\), where $\Omega$ is a bounded domain in \(\mathbb{R}^n\) and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form \(f(x,t,u)=m(t)g(u)\) where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.

Non-Uniqueness of Almost Unidirectional Inviscid Compressible Flow

Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with the Mach number equal to one (sonic states). Analytical quasi-one-dimensional results are supplemented by quasi-one-dimensional and axisymmetric three-dimensional finite volume computations. Good agreement between quasi-one-dimensional and axisymmetric results, including the presence of multiple stationary solutions, is presented for axisymmetric nozzles with reasonably small slopes of the radius.

Generalized multivariate Fokker–Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamics

We study many particle systems in the context of mean field forces, concentration-dependent diffusion coefficients, generalized equilibrium distributions, and quantum statistics. Using kinetic transport theory and linear nonequilibrium thermodynamics we derive for these systems a generalized multivariate Fokker–Planck equation. It is shown that this Fokker–Planck equation describes relaxation processes, has stationary maximum entropy distributions, can have multiple stationary solutions and stationary solutions that differ from Boltzmann distributions.

Optical bistability in lasers induced by active molecules with a large permanent dipole moment

We study a single-mode laser system, whose active medium consists of molecules with a large difference between excited and ground electrical permanent-dipole moments. In this case, the Maxwell-Bloch equations are further coupled by nonlinear terms involving the ratio between this difference between the dipoles and the transition dipole moment. It is found that these new terms lead to multiple stationary solutions. From the linear stability analysis, we demonstrate the bistable (or multistable) character of the lasing solutions.

Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems

We show that dissipative systems can have a multiplicity of stationary solutions in the form of both stable and unstable solitons. As a model equation, we use the complex cubic–quintic Ginzburg–Landau equation. For a given set of the equation parameters, this equation has many coexisting soliton solutions. Our stability results show that although most of them are unstable, they can have stable pieces. This partial stability leads to the phenomenon of soliton explosion.

Rotational aspects of stratified gap flows and shallow föhn

AbstractObservations of föhn in the Alps and other mountainous regions suggest that the underlying dynamics is often affected by gap‐like features in elongated ridge‐like topography. To assess the dynamics of these flows, idealized numerical experiments are conducted with a hydrostatic numerical model, using f‐plane geometry and a free‐slip lower boundary condition. The topography is taken to be a two‐dimensional ridge oriented in the west/east direction with a valley transect of depth ΔH across it. The upstream flow is westerly, with a constant wind speed U and constant Brunt‐Väisälä frequency N. The control parameters defined by this setting are a dimensionless gap depth NΔH/U, the ratio between ridge height and gap depth H/ΔH, a Rossby number describing the south‐north width of the ridge, and additional parameters associated with the shape of the gap. With intermediate Rossby numbers (Ro≈︁1) the setting resembles that of shallow Alpine south‐föhn cases, which are characterized by a cross‐Alpine flow essentially confined to valley transects. For small dimensionless gap depths and large Rossby numbers, the flow follows the predictions of linear theory and takes on an approximately symmetric pattern with respect to the ridge line. For NΔH/U ≳ 1, flow separation and splitting takes place upstream and downstream of the gap, respectively. The flow within the gap decouples from the flow aloft and is driven by the geostrophic south‐north pressure gradient to yield a föhn‐like flow. It is demonstrated that the limit f → 0 is singular (i.e. the flow solution does not converge towards the symmetric f = 0 solution), and that there exist multiple stationary solutions for f = 0 (two with northerly and southerly flow across the gap, respectively, and one with north/south symmetry). The existence of these multiple steady states is related to a wake instability, yet vortex shedding is suppressed by the presence of the ridge downstream of the gap. Additional simulations are presented which demonstrate that a transient external forcing can induce transitions between the multiple flow solutions. The relationship of the idealized setting to Alpine shallow föhn is discussed, and additional experiments are conducted to assess the effects of surface friction and of an inversion present to the south of the ridge.

Rotational aspects of stratified gap flows and shallow föhn

Observations of föhn in the Alps and other mountainous regions suggest that the underlying dynamics is often affected by gap-like features in elongated ridge-like topography. To assess the dynamics of these flows, idealized numerical experiments are conducted with a hydrostatic numerical model, using f-plane geometry and a free-slip lower boundary condition. The topography is taken to be a two-dimensional ridge oriented in the west/east direction with a valley transect of depth ΔH across it. The upstream flow is westerly, with a constant wind speed U and constant Brunt-Väisälä frequency N. The control parameters defined by this setting are a dimensionless gap depth NΔH/U, the ratio between ridge height and gap depth H/ΔH, a Rossby number describing the south-north width of the ridge, and additional parameters associated with the shape of the gap. With intermediate Rossby numbers (Ro≈︁1) the setting resembles that of shallow Alpine south-föhn cases, which are characterized by a cross-Alpine flow essentially confined to valley transects. For small dimensionless gap depths and large Rossby numbers, the flow follows the predictions of linear theory and takes on an approximately symmetric pattern with respect to the ridge line. For NΔH/U ≳ 1, flow separation and splitting takes place upstream and downstream of the gap, respectively. The flow within the gap decouples from the flow aloft and is driven by the geostrophic south-north pressure gradient to yield a föhn-like flow. It is demonstrated that the limit f → 0 is singular (i.e. the flow solution does not converge towards the symmetric f = 0 solution), and that there exist multiple stationary solutions for f = 0 (two with northerly and southerly flow across the gap, respectively, and one with north/south symmetry). The existence of these multiple steady states is related to a wake instability, yet vortex shedding is suppressed by the presence of the ridge downstream of the gap. Additional simulations are presented which demonstrate that a transient external forcing can induce transitions between the multiple flow solutions. The relationship of the idealized setting to Alpine shallow föhn is discussed, and additional experiments are conducted to assess the effects of surface friction and of an inversion present to the south of the ridge.

Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients: application to postural sway.

We study the transient and stationary behavior of many-particle systems in terms of multivariate Ornstein-Uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields. Mean-field approximations of this kind of system are derived in terms of Fokker-Planck equations. In such systems, multiple stationary solutions as well as bifurcations of stationary solutions may occur. In addition, strictly monotonically decreasing steady-state autocorrelation functions that decay faster than exponential functions are found, which are used to describe the erratic motion of the center of pressure during quiet standing.

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time.

On the Dynamics of Planetary Flow Regimes. Part II: Results from a Hierarchy of Orographically Forced Models

The relationship between steady states of the large-scale flow regimes revealed by multimodality in phase space and quasi-resonant axes of a linearized atmospheric model (neutral vectors) is investigated by means of a hierarchy of three hemispheric quasigeostrophic(QG) models, in which the flow is relaxed toward a zonally symmetric equilibrium state and orography provides the only source of asymmetric forcing. Two of these models describe only the interaction between the mean zonal flow and planetary waves of zonal wavenumber 3, including one or two meridional wavenumbers. The third model also has a zonal wavenumber-3 symmetry, but includes waves up to total wavenumber 21, and therefore is able to properly represent the interactions between planetary waves and baroclinically unstable synoptic-scale waves. Three stationary solutions are found for the highly truncated models, two of which represent states with large wave amplitude but opposite phase. These two steady states produce two well-defined flow regimes in the higher-resolution hemispheric model. This result confirms that flow regimes found in highly truncated models are not merely an artifact of the excessive truncation and can still be found when a sufficiently large number of degrees of freedom is included. The agreement between large-scale steady states and regime centroids is improved if the average effect of high-frequency transients on the planetary-scale flow is parameterized by adjusting the sources of zonal available potential energy and kinetic energy in the highly truncated models. Using these hemispheric models, it is shown that the leading dynamical and generalized neutral vectors correctly identify the axis linking the centroids of the two regimes as the axis with the smallest linear time derivative. Although anomalies in transient eddies are important in determining the spatial pattern of these centroids, the ultimate source of the two regimes of the hemispheric QG model is the existence of multiple stationary solutions of the large-scale flow. There is no inconsistency between the energetically fundamental role played by high-frequency transients and the capacity of nonlinear large-scale dynamics to determine the multimodal structure of the model's phase space. The neutral vector analysis also confirms that tropical diabatic heating could effectively generate an extratropical response along an axis corresponding to the difference between opposite regime anomalies.

On the forced flow of salty water in a loop

The buoyancy-driven flow of salty water in a loop is computed. This problem belongs to the general class of the convective behavior of solutal fluids, a specific example of which is the oceanic thermohaline circulation. The two cases of freshwater flux forcing and so-called virtual salt flux forcing are compared and contrasted. The former is an exact statement of the saline forcing of the ocean by the atmosphere, while the latter is an approximation used in many climate models. Analytical solutions appropriate to both cases are presented for broad parameter ranges and ultimately encapsulated in the form of bifurcation maps. These allow for comparisons between the behaviors predicted for the two cases. Furthermore, the solutions are supported by means of numerical experimentation. It is found that a simple loop model, forced by a steady flux, can possess multiple solutions, either stationary solutions and limit cycles or distinctly different limit cycles. This result is closely related to climate models. In addition, this study transcends climate applications and applies to the more general classical problem of convection in a loop. The novel aspect here is the application of freshwater flux to a salty fluid. The effect on density of this forcing is different from that due to the application of heat to a thermally sensitive fluid. Surprising and counter-intuitive behaviors have been found which reflect these differences. As an example, in the limit where diffusion is weak relative to freshwater flux, a δ-function-like salinity profile appears if freshwater flux conditions are used. Models using a virtual salt flux approximation, or a relaxation condition, yield a low mode solution for these parameters. In contrast, the virtual salt flux equations can be obtained from the freshwater-forced equations by a systematic expansion in one limit where diffusion dominates freshwater flux. Numerical experiments are used to examine the comparisons between virtual salt flux and freshwater-forced solutions, with the result that virtual salt flux generally yields accurate results when diffusion is strong.

STATIONARY ARMA SOLUTIONS TO LINEAR RATIONAL EXPECTATIONS MODELS WITH STATIONARY ARMA EXOGENOUS PROCESSES

This paper considers a general linear model with rational expectations and a stationary ARMA exogenous process. It discusses the conditions for existence and stationarity of completely determined ARMA solutions. When the number of stable roots of the characteristic equation is equal to the number of lagged endogenous variables, there is a unique, determined stationary ARMA solution having the property to exist and to be real in any case. When the model has multiple stationary solutions, it is possible to identify a particular solution having the same property. To avoid the case of no stationary solutions, additional restrictions are needed. Copyright 1996 by Blackwell Publishers Ltd and The Victoria University of Manchester

Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation

Abstract. Climate - the "coarse-gridded" state of the coupled ocean - atmosphere system - varies on many time and space scales. The challenge is to relate such variation to specific mechanisms and to produce verifiable quantitative explanations. In this paper, we study the oceanic component of the climate system and, in particular, the different circulation regimes of the mid-latitude win driven ocean on the interannual time scale. These circulations are dominated by two counterrotating, basis scale gyres: subtropical and subpolar. Numerical techniques of bifurcation theory are used to stud the multiplicity and stability of the steady-state solution of a wind-driven, double-gyre, reduced-gravity, shallow water model. Branches of stationary solutions and their linear stability are calculated systematically as parameter are varied. This is one of the first geophysical studies i which such techniques are applied to a dynamical system with tens of thousands of degrees of freedom. Multiple stationary solutions obtain as a result of nonlinear interactions between the two main recirculating cell (cyclonic and anticyclonic) of the large- scale double-gyre flow. These equilibria appear for realistic values of the forcing and dissipation parameters. They undergo Hop bifurcation and transition to aperiodic solutions eventually occurs. The periodic and chaotic behaviour is probably related to an increased number of vorticity cells interaction with each other. A preliminary comparison with observations of the Gulf Stream and Kuroshio Extensions suggests that the intern variability of our simulated mid-latitude ocean is a important factor in the observed interannual variability o these two current systems.