Bifurcation Method Research Articles

Periodic traveling wave solutions and their limit forms for the (n + 1)-dimensional double sinh-Gordon equation

In this paper, the (n+1)-dimensional double sinh-Gordon equation∑j=1nuxjxj−utt−αsinh(u)−βsinh(2u)=0is studied by using the bifurcation method of dynamical systems. The periodic traveling wave solutions and their limit forms are investigated. When the first integral varies, we show the convergence of the smooth periodic wave solutions and the periodic blow-up wave solutions, such as the smooth periodic wave solutions converge to the solitary wave solution, the smooth periodic wave solutions converge to the smooth periodic wave solution, the periodic blow-up wave solutions converge to the blow-up wave solution and the periodic blow-up wave solutions converge to the periodic blow-up wave solution. All possible explicit exact parametric representations of various nonlinear traveling waves also are given.

Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition

In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration $ \sigma $ is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient $ \gamma $ takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.

PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION

In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated.

Global Structure of Positive Solutions for Some Second-Order Multipoint Boundary Value Problems

We investigate in this paper the following second-order multipoint boundary value problem:-(Lφ)(t)=λf(t,φ(t)),0≤t≤1,φ′0=0,φ1=∑i=1m-2βiφηi. Under some conditions, we obtain global structure of positive solution set of this boundary value problem and the behavior of positive solutions with respect to parameterλby using global bifurcation method. We also obtain the infinite interval of parameterλabout the existence of positive solution.

一类三次微分系统的异宿环分支

本文运用分支的方法，通过分析未扰系统的异宿环在小扰动下的稳定流形和不稳定流形之间的相对距离，研究了一类三次微分系统的异宿环分支极限环的问题，给出了系统至少产生一个极限环的条件。 In this paper, by using the bifurcation method to analyze the relative distance between the stable manifold and the unstable manifold after the heteroclinic loop of the unperturbed system was perturbed to break, the authors studied the existence problem of limit cycles of a class of cubic differential system and obtained the conditions to ensure the system has at least one limit cycle.

Numerical continuation methods for large-scale dissipative dynamical systems

A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.

Symmetry Reductions and Exact Solutions to the Kudryashov–Sinelshchikov Equation

Abstract In this article, based on the compatibility method, some nonclassical symmetries of Kudryashov–Sinelshchikov equation are derived. By solving the corresponding characteristic equations associated with symmetry equations, some new exact explicit solutions of Kudryashov–Sinelshchikov equation are obtained. For the exact explicit traveling wave solutions, the exact parametric representations are investigated by the integral bifurcation method.

On the new solutions of the generalized Burgers–Huxley equation

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Logarithm transformation together with the integral bifurcation method for investigating exact solutions of the CDF equation with mKdV type

In this paper, under a logarithm transformation, by using the integral bifurcation method, we study the Cologero–Degasperis–Fokas equation with mKdV type, which was introduced by Cologero, Degasperis and Fokas. Some exact traveling wave solutions such as smooth solitary wave solutions, smooth periodic wave solutions, kink wave solutions, singular kink wave solution, singular periodic wave solutions, blowup soliton solution and singular blowup wave solution are obtained. We show their profiles and discuss their dynamic properties aim at some typical solutions.

The existence of seven solutions for a superlinear elliptic boundary value problem without symmetries

In this paper we study the multiplicity of nontrivial solutions for a superlinear elliptic boundary value problem. By combining truncation techniques, minimax theorem, Morse theory and bifurcation method, we show that the problem has at least seven nontrivial solutions and give sign information for them. No symmetry is assumed on the domain or the nonlinearity.

Further results on the smooth and nonsmooth solitons of the Novikov equation

This paper is concerned with the smooth and nonsmooth soliton solutions of the Novikov equation based on the bifurcation method of dynamical systems. Two interesting results are highlighted. First, the new Hamiltonian function is established in the case of \(\varphi ^2 c\) is discussed in Li (Int J Bifurcat Chaos 24(3):1450037 2014). Second, we prove that the corresponding traveling wave system of the Novikov equation exists new smooth and nonsmooth soliton solutions.

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.

Qualitative properties of solutions to mixed-diffusion bistable equations

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

Some singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation

By using the bifurcation method of dynamical systems, we investigate the singular solutions and their limit forms for generalized Calogero–Bogoyavlenskii–Schiff equation. Via some special phase orbits, we obtain some new explicit singular wave solutions expressed in terms of elliptic integral and elliptic function. From their limit forms, we also obtain the hyperbolic function kink wave and trigonometric function singular solutions.

Study of a logistic equation with local and non-local reaction terms

We examine a logistic equation with local and non-local reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.

Positive steady states and dynamics for a diffusive predator-prey system with a degeneracy

In this article, we consider positive steady state solutions and dynamics for a spatially heterogeneous predator-prey system with modified Leslie-Gower and Holling-Type II schemes. The heterogeneity here is created by the degeneracy of the intra-specific pressures for the prey. By the bifurcation method, the degree theory, and a priori estimates, we discuss the existence and multiplicity of positive steady states. Moreover, by the comparison argument, we also discuss the dynamical behavior for the diffusive predator-prey system.

Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problem

In this paper, we study monotone radially symmetric solutions of semilinear equations with Allen–Cahn type nonlinearities by the bifurcation method. Under suitable conditions imposed on the nonlinearities, we show that the structure of the monotone nodal solutions consists of a continuous U-shaped curve bifurcating from the trivial solution at the third eigenvalue of the Laplacian. The upper branch consists of a decreasing solution and the lower branch consists of an increasing solution. In particular, we show the following equation Δu+λ(u−u∣u∣p−1)=0in B,∂u∂ν=0on ∂B has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here B is the unit ball in Rn, p>1 and λ>0.

Spatiotemporal Patterns of a Homogeneous Diffusive Predator–Prey System with Holling Type III Functional Response

The dynamics of a diffusive predator–prey system with Holling type-III functional response subject to Neumann boundary conditions is investigated. The parameter region for the stability and instability of the unique constant steady state solution is derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method and Leray–Schauder degree theory. The effect of various parameters on the existence and nonexistence of spatiotemporal patterns is analyzed. These results show that the impact of Holling type-III response essentially increases the system spatiotemporal complexity.

Using Numerical Bifurcation Analysis to Study Pattern Formation in Mussel Beds

Soft-bottomed mussel beds provide an important example of ecosystem-scale self-organisation. Field data from some intertidal regions shows banded patterns of mussels, running parallel to the shore. This paper demonstrates the use of numerical bifurcation methods to investigate in detail the predictions made by mathematical models concerning these patterns. The paper focusses on the “sediment accumulation model” proposed by Liu et al (Proc. R. Soc. Lond. B 14 (2012), 20120157). The author calculates the parameter region in which patterns exist, and the sub-region in which these patterns are stable as solutions of the original model. He then shows how his results can be used to explain numerical observations of history-dependent wavelength selection as parameters are varied slowly.