Metastability for small random perturbations of a PDE with blow-up

We study random perturbations of a reaction–diffusion equation with a unique stable equilibrium and solutions that blow-up in finite time. If the strength of the perturbation $\epsilon >0$ is small and the initial data is in the domain of attraction of the stable equilibrium, the system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite time (but exponentially large in ${\epsilon }^{?2}$). Moreover, for initial data in the domain of explosion we show that the explosion times converge to the one of the deterministic solution.     

Approximation on attraction domain of Cohen-Grossberg neural networks

In this paper, approximations of attraction domains of the asymptotically stable equilibrium points of some typical Cohen-Grossberg neural networks are achieved. Most Cohen-Grossberg neural networks are highly nonlinear systems which makes it difficult to approximate their attraction domain. Under some weak assumptions, we are allowed to employ the optimal Lyapunov method to obtain a Lyapunov function for asymptotically stable equilibrium points of a given Cohen-Grossberg neural network. With the help of this Lyapunov function, we approximate the corresponding attraction domain by the iterative expansion approach. Numerical simulations also illustrate that the approximation obtained is really part of the attraction domain.     

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

Age-time continuous Galerkin methods for a model of population dynamics

Continuous Galerkin finite element methods in the age-time domain are proposed to approximate the solution to the model of population dynamics with unbounded mortality (coefficient) function. Stability of the method is established and a priori L²$L^2$-error estimates are obtained. Treatment of the nonlocal boundary condition is straightforward in this framework. The approximate solution is computed strip by strip marching in time. Some numerical examples are presented.     

On a method of evaluation of attraction domain for fixed point of nonlinear point mapping of arbitrary dimension

In this paper we describe the method of attraction domain evaluation for equilibrium states of nonlinear discrete dynamic system based on Lyapunov functions method. Attraction domain evaluation size is equilibrium state neighborhood where the first difference of Lyapunov function is negative. Lyapunov function is chosen as positive quadratic form for which the negativity of its first difference by virtue of linearized system is guaranteed with given supply. We propose the method of attraction domain extension.     

Characterization for stability in planar conductivities

We find a complete characterization for sets of uniformly strongly elliptic and isotropic conductivities with stable recovery in the $L^2$ norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn out to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the integral moduli of continuity of the coefficients involved and their ellipticity bound as conjectured by Alessandrini in his 2007 paper, giving explicit quantitative control for every pair of conductivities.     

Global exponential stability in DCNNs with distributed delays and unbounded activations

We study delayed cellular neural networks (DCNNs) whose state variables are governed by nonlinear integrodifferential differential equations with delays distributed continuously over unbounded intervals. The networks are designed in such a way that the connection weight matrices are not necessarily symmetric, and the activation functions are globally Lipschitzian and they are not necessarily bounded, differentiable and monotonically increasing. By applying the inequality pa^p-1b?(p-1)a^p+b^p${\mathit{pa}}^{p-1}b?(p-1)a^p+b^p$, where p   denotes a positive integer and a,b$a,b$ denote nonnegative real numbers, and constructing an appropriate form of Lyapunov functionals we obtain a set of delay independent and easily verifiable sufficient conditions under which the network has a unique equilibrium which is globally exponentially stable. A few examples added with computer simulations are given to support our results.     

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.     

A novel method for computing the Hilbert transform with Haar multiresolution approximation

In this paper, an algorithm for computing the Hilbert transform based on the Haar multiresolution approximation is proposed and the L²$L^2$-error is estimated. Experimental results show that it outperforms the library function ‘hilbert’ in Matlab (The MathWorks, Inc. 1994–2007). Finally it is applied to compute the instantaneous phase of signals approximately and is compared with three existing methods.     

On the estimation of asymptotic stability regions for autonomous nonlinear systems

The problem of computing regions of asymptotic stability forautonomous nolinear systems is reconsidered. A two-step procedureis proposed in which a suitable global Lyapunov function isfirst constructed to prove the system's nonoscillatory behaviour.Subsequently the Lyapunov function is used to compute the initialstates for a trajectory-reversing technique to estimate thesystem's stability boundaries. The method combines computationalefficiency and accuracy in obtaining a close estimate of theexact region of attraction of a stable equilibrium state.     

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points.To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.     

Global dynamics of an SEI model with acute and chronic stages

A model with acute and chronic stages in a population with exponentially varying size is proposed. An equivalent system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of these two equilibriums is controlled by the basic reproduction number _R0$R_0$. When _R0<1$R_0<1$, the disease-free equilibrium is globally stable. When _R0>1$R_0>1$, the disease-free equilibrium is unstable and the unique endemic equilibrium is locally stable. When _R0>1$R_0>1$ and γ=0,α=0$\gamma =0,\alpha =0$, the endemic equilibrium is globally stable in Γ⁰${\Gamma }^0$.     

Energy measures on p.c.f. self-similar sets

On connected post critically finite (p.c.f.) self-similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy, and as an application we also compute the L²$L^2$ dimensions of these measures on some p.c.f. self-similar sets.     

Decay of solutions of a limiting form of Ginzburg–Landau systems involving curl

In this note, we study a semilinear system involving the operator curl in an exterior domain in R³${\mathbb{R}}^3$, which is the limiting form of the Ginzburg–Landau model for superconductors in three dimensions for a large value of the Ginzburg–Landau parameter. We prove that this problem has a smooth solution, and it decays exponentially at infinity.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain

We investigate the instability and stability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain Ω   of class C²$C^2$. When the steady density is heavier with increasing height (i.e., the Rayleigh–Taylor steady-state), we show that the steady-state is linear unstable (i.e., the linear solution grows in time in H²$H^2$) by constructing a (standard) energy functional and exploiting the modified variational method. Then, by introducing a new energy functional and using a careful bootstrap argument, we further show that the steady-state is nonlinear unstable in the sense of Hadamard. When the steady density is lighter with increasing height, we show, with the help of a restricted condition imposed on steady density, that the steady-state is linearly globally stable and nonlinearly asymptotically stable in the sense of Hadamard.     

Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays

A virus dynamics model with Beddington–DeAngelis functional response and delays is introduced. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariance principle, we show that the infection-free equilibrium is globally asymptotically stable if _R0?1$R_0?1$ and the chronic-infection equilibrium is globally asymptotically stable if _R0>1$R_0>1$. Numerical simulations are also given to explain our results.     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.