Instability Of Stationary Solutions Research Articles

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.

On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass

For semilinear elliptic equations −Δu = λ|u| p−2 u−|u| q−2 u, boundary value problems in bounded and unbounded domains are considered. In the plane of exponents p × q, the so-called curves of critical exponents are defined that divide this plane into domains with qualitatively different properties of the boundary value problems and the corresponding parabolic equations. New solvability conditions for boundary value problems, conditions for the stability and instability of stationary solutions, and conditions for the existence of global solutions to parabolic equations are found.

Stationary solutions in a model three-body problem

Two visco-elastic bodies (deformable spheres) are considered which interact with each other and move in quasi-circular orbits in the attractive force field of a fixed centre – a heavy point mass. Their axes of rotation are perpendicular to their orbital plane. Stationary solutions of the evolutionary equations of motion are found. In one particular case, they extend solutions of the restricted circular three-body problem corresponding to two collinear libration points. All three bodies are located along a straight line. This implies synchronization of motion of the barycentre of the two visco-elastic bodies relative to the attracting centre with their orbital motion relative to the barycentre in a 1:1 resonance. The rotation of the two bodies relative to their own centres of mass takes place in such a way that the bodies “view” the attracting centre and each other from the same side, i.e., they are synchronized in a 1:1 resonance with their orbital motion. Instability of stationary solutions is analytically proven.

Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

Streamline approach is often used as an alternative effective method to classical finite difference technique for solving large heterogeneous fluid flow models in petroleum reservoirs. In the case of complex multi-component fluid system, this approach is scarcely used because the hydrodynamic and thermodynamic flow equations are strongly coupled through nonindependent variables including the pressure, the saturation and the species concentrations. It has been shown recently (Oladyshkin and Panfilov, Two-phase flow with phase transitions in porous media: instability of stationary solutions and a semi-stationary model. Third Biot Conference on Poromechanics, Norman, Oklahoma, USA, 2005; Oladyshkin and Panfilov, Comptes Rendus Acad Sci M´ecanique, Elsevier 335(1):7–12, 2007) that assuming quasi steady-state for the pressure field, the hydrodynamic and thermodynamic parts can be split into a set of equations that is referred as HT-split compositional model. In this work, the HT-split model is combined with the streamline technique. This approach has been implemented in gOcad using the StreamLab plug-in. The pressure field and the streamlines are computed using the finite volume flow simulator. The equations that govern the equilibrium between phases are solved separately using a classical nonlinear solver. A multi-component 1D solver has been implemented using the HT-split equations along the streamlines. Tools for visualizing the time evolution of species compositions have been also developed. Finally a simple case study illustrating the technique is presented. It is shown that the HT-splitting method coupled to the streamline technology provides an effective tool to solve complex problems involving multi-compositional flow for any 3D reservoir geometry and for any gas–liquid system. The advantage of such a technology is that the number of components is not limited.

Instability of stationary solutions for equations of curvature-driven motion of curves

A study is made for equations of evolving curves on a two-dimensional square domainΩ. It is assumed that a curve moves depending on its curvature, normal vector, and position and is orthogonal to∂Ω at its end points. Under some conditions, instability of stationary solutions is proved through an eigenvalue analysis.

Instability of Stationary Bubbles

The paper is concerned with the study, by means of a linearized operator, of the instability of stationary solutions of a nonlinear Schrodinger equation with nonvanishing “boundary conditions.” We prove that this operator possesses a real positive eigenvalue, and that when they exist, the stationary “bubbles“ are always orbitally unstable.

On a discrete model of phase separation dynamics

A spatially discrete model for phase separation with conserved order parameter is proposed. This one-dimensional model is obtained as the deterministic limit of an anisotropic lattice gas. A particular choice is made for the jump rates (which still fulfill detailed balance conditions) so that the resulting model is mathematically tractable. It exhibits a phase transition of first-order type whose nonlinear dynamics is investigated using both analytical and numerical methods. All the stationary solutions with zero current are found and parametrized in terms of Jacobian elliptic functions, showing a striking similarity with the nonlinear (continuous) Cahn-Hilliard equation. In the limit of infinite wavelength, particular solutions are found which descrive isolated domains of arbitrary size embedded in an homogeneous infinite medium of the opposite phase. New results are also presented on the structure of the set of solutions. Time-dependent profiles are studied in the spinodal regime and the stability of bounded stationary solutions is also investigated in this context. A description of time-dependent profiles is proposed which considers only interactions between neighboring domains and makes use of isolated domain solutions. This approach results in an analytic expression for the exponents characteristic of the instability of stationary solutions and is validated by comparison to numerical values. Qualitative results are also discussed and the relation to the Cahn-Hilliard equation is emphasized.