Stability Of Stationary Solutions Research Articles

Critical properties and stability of stationary solutions in multitransonic pseudo-Schwarzschild accretion

For inviscid, rotational accretion flows, both isothermal and polytropic, a simple dynamical system analysis of the critical points has given a very accurate mathematical scheme to understand the nature of these points, for any pseudo-potential by which the flow may be driven on to a Schwarzschild black hole. This allows us for a complete classification of the critical points for a wide range of flow parameters, and shows that the only possible critical points for this kind of flow are saddle points and centre-type points. A restrictive upper bound on the angular momentum of critical solutions has been established. A time-dependent perturbative study reveals that the form of the perturbation equation, for both isothermal and polytropic flows, is invariant under the choice of any particular pseudo-potential. Under generically true outer boundary conditions, the inviscid flow has been shown to be stable under an adiabatic and radially propagating perturbation. The perturbation equation has also served the dual purpose of enabling and understanding the acoustic geometry for inviscid and rotational flows.

Dynamic instabilities in the microtubule cytoskeleton: A state diagram

The dynamics of microtubule growth and disassembly is considered in the framework of the theory of nonequilibrium reaction-diffusion systems. The phase diagram contains regions corresponding to stable stationary and nonstationary solutions. Dynamic instabilities can arise from nonequilibrium kinetic transitions. Agents affecting the microtubule dynamics are classed into four types, and the interplay of their effects is analyzed.

Bifurcation and stability analysis of an anaerobic digestion model

This paper presents the dynamic behaviour of the anaerobic digestion process, based on a simplified model. The hydraulic, biological and physicochemical processes such as those which underpin anaerobic digestion have more than one stable stationary solution and they compete with each other. Further, the attractive domains of the stable solutions vary with the key parameters. Thus, some initial transient process moving toward one stable solution could suddenly move towards another solution, at which a so-call catastrophe takes places (e.g. washout). The paper systematically analyses the stationary solutions with their associated stability, which provides insight and guidance for anaerobic digestion reactor design, operation and control.

Flow rate distribution in evaporating parallel pipes—modeling and experimental

The motivation for the present work is related to the use of parabolic trough solar power technology for direct steam generation (DSG). So far, commercial plants use liquid oil to absorb heat from the collectors, and steam is produced in heat exchangers. A more efficient process of direct steam generation in the collectors is not used due to possible uneven flow rate distribution and other instabilities, related to liquid–vapor flow in parallel pipes. In this work we present an analysis and experimental results for the flow rate distribution of water flowing and evaporating in two parallel pipes with common inlet and outlet manifolds. A new simplified model that yields the pressure drop versus the flow rate in a single pipe is developed. This model is the basis for the calculation of steady-state solutions, stability analysis and transient simulations in two parallel pipes. Multiple steady-state solutions may be obtained depending on the flow rate and the heating power. Linear stability analysis and transient simulation allow to differentiate between the stable steady states and the unstable ones. It is also shown that when few stable state solutions are possible the one that takes place depends on the history of the inlet flow rate. Experimental results compare well with the theoretical solutions.

Discretization of Asymptotically Stable Stationary Solutions of Delay Differential Equations with a Random Stationary Delay

We prove the existence of a stationary random solution to a delay random ordinary differential system, which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be continuous in time. Moreover, we also prove that the split implicit Euler scheme associated to the random delay differential system generates a discrete time random dynamical system, which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the delay random differential equation pathwise as the stepsize goes to zero.

Stable stationary solutions for a reaction–diffusion equation with a multi-stable nonlinearity

In this Letter, we give a construction of stable stationary solutions for a spatially multi-dimensional discrete reaction–diffusion equation with a multi-stable nonlinearity and a small diffusion constant. The existence of spatial chaos and spatially chaotic solutions is shown. The exact value of the spatial entropy is estimated.

On non‐Newtonian incompressible fluids with phase transitions

AbstractA modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn–Hilliard equation coupled with a system describing a class of non‐Newtonian incompressible fluid with p‐structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p⩾ 2 and for d=3 with p⩾ 11/5. The existence of measure‐valued solutions is obtained for d=3 in the case 2⩽ p< 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |ψ| ⩽ 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law

We complete previous investigations on the dynamical stability of barotropic stars and collisionless stellar systems. A barotropic star that minimizes the energy functional at fixed mass is a nonlinearly dynamically stable stationary solution of the Euler-Poisson system. Formally, this minimization problem is similar to a condition of ``canonical stability'' in thermodynamics. A stellar system that maximizes an H-function at fixed mass and energy is a nonlinearly dynamically stable stationary solution of the Vlasov-Poisson system. Formally, this maximization problem is similar to a condition of ``microcanonical stability'' in thermodynamics. Using a thermodynamical analogy, we provide a derivation and an interpretation of the nonlinear Antonov first law in terms of ``ensembles inequivalence'': a spherical stellar system with f=f(epsilon) and f'(epsilon)<0 is nonlinearly dynamically stable with respect to the Vlasov-Poisson system if the corresponding barotropic star with the same equilibrium density distribution is nonlinearly dynamically stable with respect to the Euler-Poisson system. This is similar to the fact that ``canonical stability implies microcanonical stability'' in thermodynamics. The converse is wrong in case of ``ensembles inequivalence'' which is generic for systems with long-range interactions like gravity. We show that criteria of nonlinear dynamical stability can be obtained very simply from purely graphical constructions by using the method of series of equilibria and the turning point argument of Poincare, as in thermodynamics.

Dynamics and thermodynamics of axisymmetric flows: Theory

We develop variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) function while conserving energy, helicity, circulation, and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding nonideal effects such as viscosity, small-scale forcing, and dissipation (for forced systems). Finally, we derive a parametrization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between two-dimensional (2D) and 3D turbulence.

Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

We address the issue of mobility of localized modes in two-dimensional nonlinear Schrödinger lattices with saturable nonlinearity. This describes, e.g., discrete spatial solitons in a tight-binding approximation of two-dimensional optical waveguide arrays made from photorefractive crystals. We discuss the numerically obtained exact stationary solutions and their stability, focusing on three different solution families with peaks at one, two, and four neighboring sites, respectively. When varying the power, there is a repeated exchange of stability between these three solutions, with symmetry-broken families of connecting intermediate stationary solutions appearing at the bifurcation points. When the nonlinearity parameter is not too large, we observe good mobility and a well-defined Peierls-Nabarro barrier measuring the minimum energy necessary for rendering a stable stationary solution mobile.

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.

An Asymptotic One‐Dimensional Model to Describe Fires in Tunnels III: The Transient Problem

This is the third in a series of papers devoted to a model describing tunnel fire events (see [I. Gasser and J. Struckmeier, Math. Methods Appl. Sci., 25 (2002), pp. 1231–1249] and [I. Gasser, Math. Methods Appl. Sci., 26 (2003), pp. 1327–1347]). Here the transient behavior of the model is studied. In particular the stability of stationary solutions is analyzed. A good understanding of related bifurcation phenomena is obtained (at least numerically). This accelerates progress in the study of the stability of flow conditions in tunnels during a tunnel fire.

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.

Stabilization rate and stability for viscous compressible barotropic symmetric flows with free boundary for a general mass force

We consider symmetric flows of a viscous compressible barotropic fluid with free boundary driven by a general mass force (depending on both the Eulerian and the Lagrangian coordinates) and an outer pressure , for a general monotone state function . The case of self-gravitation arising in astrophysics is covered. Studied first are the existence, the uniqueness, and the static stability of positive stationary solutions; a variational study of these solutions and their static stability in terms of potential energy is presented. In the astrophysical context it is proved that the stationary solution is unique and statically stable, provided that the first adiabatic exponent is at least 4/3. Next, in the case when the -limit set for the non-stationary density and free boundary contains a statically stable positive stationary solution a uniform stabilization to this solution is deduced and, as the main result, stabilization-rate bounds of exponential type as in and for the density and the velocity are established by constructing new non-trivial Lyapunov functionals for the problem. Moreover, it is proved that statically stable stationary solutions are exponentially asymptotically stable, and this non-linear dynamic stability is in addition stable with respect to small non-stationary perturbations of and . A variational condition for the stationary solution is also introduced, which ensures global (with respect to the data) dynamic stability. The study is accomplished in the Eulerian coordinates and in the Lagrangian mass coordinates alike.

Asymptotic behavior of the solution to the non-isothermal phase field equation

We consider the Penrose–Fife phase field model [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62] with homogeneous Neumann boundary condition to the nonlinear heat flux q = ∇ ( 1 / θ ) , i.e., q = 0 on the boundary, where θ > 0 is the temperature. There is a unique H 1 solution globally in time with the non-empty, connected, compact ω -limit set composed of stationary solutions, and the linearized stable stationary solution is dynamically stable.

Capillary/Gravity Film Flows on the Surface of a Rotating Cylinder

We derive equations describing the motion of a viscous incompressible capillary film on the surface of a rotating cylinder in the transverse gravity field. As a result, we obtain an equation for the film thickness that has fourth order in two space variables and first order in time. We study both space-periodic solutions in the axial coordinate and localized solutions of this equation in the stationary case. We also discuss the stability of stationary solutions. Analysis of the one-dimensional problem shows that its solution strongly depends on the Galileo number and that such a solution does not exist if this number is large. Bibliography: 15 titles.

Stationary states of pyramidal molecules coupled to an electromagnetic field below and above critical pressure

In this paper, we extend the analysis of stationary states of a nonlinear model of a gas of pyramidal molecules (already discussed in the literature) by including a linear coupling to an external electromagnetic field. It is seen that this results in a shift of the bifurcation point and consequently allows, for appropriate values of Rabi frequency, stable localized stationary solutions below the critical pressure required in the absence of molecule–field coupling.

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the Fokker–Planck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → + ∞ , a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1 / N , the collision term of a homogeneous system has the form of the Lenard–Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard–Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker–Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → + ∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → + ∞ , or for large times t ⪢ ξ - 1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, 2D vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker–Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

Synchronization of two flow excited pendula

Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency.