Equilibrium Point In Phase Space Research Articles

Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit

We consider the Keller--Rubinow model for Liesegang rings in one spatial dimension in the fast reaction limit as introduced by Hilhorst, van der Hout, Mimura, and Ohnishi in 2007. Numerical evidence suggests that solutions to this model converge, independent of the initial concentration, to a universal profile for large times in parabolic similarity coordinates. For the concentration function, the notion of convergence appears to be similar to attraction to a stable equilibrium point in phase space. The reaction term, however, is discontinuous so that it can only convergence in a much weaker, averaged sense. This also means that most of the traditional analytical tools for studying the long-time behavior fail on this problem. In this paper, we identify the candidate limit profile as the solution of a certain one-dimensional boundary value problem which can be solved explicitly. We distinguish two nontrivial regimes. In the first, the transitional regime, precipitation is restricted to a bounded region in space. We prove that the concentration converges to a single asymptotic profile. In the second, the supercritical regime, we show that the concentration converges to one of a one-parameter family of asymptotic profiles, selected by a solvability condition for the one-dimensional boundary value problem. Here, our convergence result is only conditional: we prove that if convergence happens, either pointwise for the concentration or in an averaged sense for the precipitation function, then the other field converges likewise; convergence in concentration is uniform, and the asymptotic profile is indeed the profile selected by the solvability condition. A careful numerical study suggests that the actual behavior of the equation is indeed the one suggested by the theorem.

Solutions near the hydraulic control point in a gravity current model

A steady-state gravity current model that incorporates entrainment and friction is used to describe large-scale gravity currents and channel flows. When the model includes pressure effects from varying current thickness, critical points occur when the current velocity is equal to the phase velocity of waves on the interface. Some solutions have the possibility to pass from super to subcritical flow, or vice versa. These solutions pass through a hydraulic control point and the objective is to analyze the behavior of the solutions in the vicinity of such points. Using a phase space in which the hydraulic control points occur as equilibrium points, and performing Taylor expansion to the first order, the result is a system of autonomous differential equations with constant coefficients that can describe the behavior of the solutions for different parameter regimes near a hydraulic control point. If an equilibrium point in phase space represents a saddle point, it is distinguished between three different solution classes; solutions that approach the critical velocity but never reach it, solutions that reach the critical velocity and obtain infinitely large derivative, and the solutions (one from subcritical and one from supercritical) that reach the critical velocity exactly in the equilibrium point. The method gives a way to tell whether hydraulically controlled solutions exist and in special cases an algorithm for finding these solutions.

Asymptotic Behaviors in the Dynamics of Competing Species

We consider systems of interacting species obeying Lotka–Volterra equations. We show that periodic attractors may be generated from the equilibrium point in phase space by Hopf bifurcation. An exception is the case of three species systems with equal individual growth rates, where Hopf bifurcation is abnormal. We show that the dynamics of such systems is actually bidimensional, a fact which permits us to give an analytic description of the closed orbits observed by May and Leonard. We also consider the “explosive” dynamics of a system without self interactions and show that adding a constant source term in the equations of motion has a stabilizing effect, producing periodic solutions. In the limit of small external source terms we are able to describe analytically the associated cyclic evolutions. This effect of source terms on the solutions of Lotka–Volterra equations seems to be quite general, and is explained by the fact that they remove the attraction of the orbits by the coordinate planes.