We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one repeatedly solves systems of nonlinear equations F (u(s), λ(s)) = 0 for a real-valued function u and a real parameter λ, given different values of the arclength s. ...

In this paper we investigate numerically positive solutions of the equation −Δu = λu+u with Dirichlet boundary condition in a boundary domain Ω for λ > 0 and 0 < q < 1 < p < 2∗, we will compute and visualize the range of λ, this problem achieves a numerical solution. Keywords—positive solutions; concave-convex; sub-supersolution method; pseudo arclength method.

In this paper, we present a new method to rigorously compute smooth branches of zeros of nonlinear operators f : R1 × B1 → R2 × B2, where B1 and B2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

We present eigenvalue bounds for perturbations of Hermitian matrices, and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications ...

A pseudo-arclength continuation method (PACM) is proposed and employed to compute the ground state and excited state solutions of spin-1 Bose-Einstein condensates (BEC). A pseudo-arclength continuation method (PACM) is employed to compute the ground state and excited state solutions of spin-1 BoseEinstein condensates (BEC). The BEC is governed by the time-independent coupled Gross-Pitaevskii eq...

Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of interest satisfies a parametrised set of algebraic equations, and branches of...

Abstract. In this paper we describe a method for continuing periodic solution bifurcations in periodic delaydifferential equations. First, the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is proved. An alternative formulation of the characteristic matrix is given, which can efficiently be computed. Defining systems of bifurcation...

We perform a rigorous mathematical analysis of a simple membrane based model of an electrostatically ac-tuated MEMS device. Using both analytical and numerical techniques, we prove the existence of a fold in the solution space of the displacement, implying the existence of a critical voltage beyond which there are no solutions of the equation. This critical voltage corresponds to the pull-in vo...

A nonlinear Euler–Bernoulli model of slender piezoelectric beams is employed to investigate parametric resonance motions driven by a pulsating voltage with DC component. The beam based on 3D electric charge conservation and 1D reduction the momentum balance laws assuming as space coordinate arclength along base line coordinate. constitutive relationships for material are specialized according a...