Bifurcation of homoclinics

We show that homoclinic trajectories of nonautonomous vector fields parametrized by a circle bifurcate from the stationary solution when the asymptotic stable bundles of the linearization at plus and minus infinity are ``twisted' in different ways.

     

An ∞-dimensional inhomogeneous Langevin's equation

On a modified space Φ′ from the space $\text{J}$′ of tempered distributions, it is proven that a stochastic equation, $\text{X(t) = γ + W(t) + ∝}{}_0{}^{\mathrm{t}}\text{L}{}^1\text{(s) X(s) ds}$, has a unique solution, where W(t) is a Φ′-valued Brownian motion independent of a Φ′-valued Gaussian random variable γ and $\text{L}{}^1\text{(s)}$ is an integro-differential operator. As an application, a fluctuaton result (or central limit theorem) is shown for interacting diffusions.     

Perturbation analysis of the matrix equation

Consider the nonlinear matrix equation X-A^*X^-pA=Q with 0<p1. This paper shows that there exists a unique positive definite solution to the equation. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. We also obtain explicit expressions of the condition number for the unique positive definite solution. The theoretical results are illustrated by numerical examples.     

On the limiting behavior of Burger's equation

The limiting behavior as the viscosity goes to zero of the solution of the first boundary value problem for Burger's equation is considered. The method consists in identifying the solution of Burger's equation with the optimal control of an appropriate stochastic control problem.     

On a bi-stability regime and the existence of odd subharmonics in a Comb-drive MEMS model with cubic stiffness

In this paper we study two different problems. First we present a novel result about the existence of a family of odd subharmonics with prescribed nodal properties for a general nonlinear oscillator with bounded domain and symmetries. Then we apply the general existence result to the Comb-drive finger MEMS model with a cubic nonlinear stiffness term, and prove analytically that the odd positive subharmonic of order two is linearly stable whenever the AC load of the input voltage is small enough. Moreover, we demonstrate a bi-stability regime for this model because the trivial solution $x\equiv 0$ is also linearly stable. The general existence result was obtained through a generalization of the Ortega’s variational principle (Ortega, 2016), and the stability assertions were obtained by following the perturbation approach in Cen et al. (2020) for the linear stability of a nontrivial periodic solution that emanates from the autonomous problem, and the well-known Zukovskii criterion.     

Numerical solutions of the Laplace’s equation

This paper uses the sinc methods to construct a solution of the Laplace’s equation using two solutions of the heat equation. A numerical approximation is obtained with an exponential accuracy. We also present a reliable algorithm of Adomian decomposition method to construct a numerical solution of the Laplace’s equation in the form a rapidly convergence series and not at grid points. Numerical examples are given and comparisons are made to the sinc solution with the Adomian decomposition method. The comparison shows that the Adomian decomposition method is efficient and easy to use.     

The simplest equation method to study perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

The simplest equation method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations.In this paper, the simplest equation method is used to construct exact solutions of nonlinear Schrödinger’s equation and perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. It is shown that the proposed method is effective and general.     

Analytical solutions to Fisher’s equation with time-variable coefficients

This paper provides analytical solutions to the generalized Fisher equation with a class of time varying diffusion coefficients. To accomplish this we use the Painlevé property for partial differential equations as defined by Weiss in 1983 in “The Painlevé property for partial-differential equations”. This was first done for the variable coefficient Fisher’s equation by Ö?ün and Kart in 2007; we build on this work, finding additional solutions with a weaker restriction on the trial solution. We also use the same technique to find solutions to Fisher’s equation with time-dependent coefficients for both diffusion and nonlinear terms. Lastly we compute specific solutions to illustrate their behaviors.