Bifurcation analysis for a modified quasilinear equation with negative exponent

In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

Exclusion regions for parameter-dependent systems of equations

This paper presents a new algorithm based on interval methods for rigorously constructing inner estimates of feasible parameter regions together with enclosures of the solution set for parameter-dependent systems of nonlinear equations in low (parameter) dimensions. The proposed method allows to explicitly construct feasible parameter sets around a regular parameter value, and to rigorously enclose a particular solution curve (resp. manifold) by a union of inclusion regions, simultaneously. The method is based on the calculation of inclusion and exclusion regions for zeros of square nonlinear systems of equations. Starting from an approximate solution at a fixed set p of parameters, the new method provides an algorithmic concept on how to construct a box $${\mathbf {s}}$$ s around p such that for each element $$s\in {\mathbf {s}}$$ s ∈ s in the box the existence of a solution can be proved within certain error bounds.

Solution region synthesis methodology of planar eight-bar linkage for four specified positions of a 4R chain

The solution region methodology for solving the problem of four-bar linkage synthesis with four specified positions was extended to solve the problem of eight-bar linkage synthesis. The processes to build solution regions for synthesizing different types of eight-bar linkages are described, and the methods of building solution regions are divided into five types. First, the synthesis equation is derived, and the curve expressed by the synthesis equation is called the solution curve. Second, the process to build the spatial solution regions from the solution curves is detailed, and a new defect identification method is developed for building the spatial feasible solution region, which is a set of linkage solutions meeting four positions and excluding defects. Finally, linkage solutions that do not meet practical engineering requirements are eliminated from the spatial feasible solution region to obtain the useful spatial solution region. The examples demonstrate the feasibility of the proposed method. The proposed synthesis methodology is simple and easy to program, and provides reference for four specified position synthesis of other multi-bar linkages.

Bifurcation Behaviors of Steady-State Solution to a Discrete General Brusselator Model

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear u in the standard Brusselator model to the nonlinear fu. Assume that f∈C0,∞,0,∞ is a strictly increasing function, and f′f−1a∈0,∞. Taking b as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point b0j,f−1a,ab/f−1a2. Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that fs/s2 is nonincreasing in 0,∞. In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem.

The Evolutionary Properties on Solitary Solutions of Nonlinear Evolution Equations

The evolution process of four class soliton solutions is investigated by basic calculus theory. For any given x, we describe the special curvature evolution following time t for the curve of soliton solution and also study the fluctuation of solution curve.

Pseudo-Arclength Continuation Algorithms for Binary Rydberg-Dressed Bose-Einstein Condensates

We study pseudo-arclength continuation methods for both Rydberg-dressed Bose-Einstein condensates (BEC), and binary Rydberg-dressed BEC which are governed by the Gross-Pitaevskii equations (GPEs). A divide-and-conquer technique is proposed for rescaling the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the affect of the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.

Minimal Energy Solutions and Infinitely Many Bifurcating Branches for a Class of Saturated Nonlinear Schrödinger Systems

We prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system$\left\{\begin{aligned} \displaystyle-\Delta u+\lambda_{1}u&\displaystyle=\frac% {\alpha u(\alpha u^{2}+\beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&% \displaystyle\text{in }\mathbb{R}^{n},\\ \displaystyle-\Delta v+\lambda_{2}v&\displaystyle=\frac{\beta v(\alpha u^{2}+% \beta v^{2})}{1+s(\alpha u^{2}+\beta v^{2})}&&\displaystyle\text{in }\mathbb{R% }^{n}\end{aligned}\right.$are necessarily semitrivial whenever ${\alpha,\hskip 0.5pt\beta,\hskip 0.5pt\lambda_{1},\hskip 0.5pt\lambda_{2}>0}$ and ${0<s<\max\{\alpha/\lambda_{1},\hskip 0.5pt\beta/\lambda_{2}\}}$ except for the symmetric case ${\lambda_{1}=\lambda_{2}}$, ${\alpha=\beta}$. Moreover, it is shown that for most parameter samples ${\alpha,\beta,\lambda_{1},\lambda_{2}}$, there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s.

The logistic modeling population: Having harvesting factor

The present paper deals with the logistic equation having harvesting factor, which is studied in two cases, constant and non-constant. In fact, the nature of equilibrium points and solutions behavior has been analyzed for both of the above cases by finding the first integral, solution curve and phase diagram. Finally, a theorem which is describing the stability of a real model of single species is proved.