Bifurcation Method Research Articles

Bifurcations and exact traveling wave solutions for the regularized Schamel equation

Abstract In the present paper, we focus on studying the bifurcations and the traveling wave solutions (TWSs) for the regularized Schamel equation. Based on the bifurcation method of a dynamical system, a complete phase portrait analysis is given in various parameter conditions and some novel TWSs with the same energy of the Hamiltonian system are discovered. Various significant results on exact expressions of TWSs, including solitary waves, periodic waves, cusp waves, weak kink waves, loop solitons, compactons in different conditions are obtained.

Bifurcation of a Mathematical Model for Tumor Growth with Angiogenesis and Gibbs–Thomson Relation

In this paper, a mathematical model for solid vascular tumor growth with Gibbs–Thomson relation is studied. On the free boundary, we consider Gibbs–Thomson relation which means energy is expended to maintain the tumor structure. Supposing that the nutrient is the source of the energy, the nutrient denoted by [Formula: see text] satisfies [Formula: see text] where [Formula: see text] is a constant representing the ability of the tumor to absorb the nutrient through its blood vessels; [Formula: see text] is concentration of the nutrient outside the tumor; [Formula: see text] is the mean curvature; [Formula: see text] denotes adhesiveness between cells and [Formula: see text] denotes the exterior normal derivative on [Formula: see text] The existence, uniqueness and nonexistence of radially symmetric solutions are discussed. By using the bifurcation method, we discuss the existence of nonradially symmetric solutions. The results show that infinitely many nonradially symmetric solutions bifurcate from the radially symmetric solutions.

New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Continuation and Bifurcation in Nonlinear PDEs \u2013 Algorithms, Applications, and Experiments

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package pde2path to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

Periodic oscillation for an eight-neuron BAM neural network model with discrete delays

This paper investigates the existence of periodic oscillations for a eight-neuron BAM neural network model with discrete delays. Two theorems are provided to guarantee the existence of periodic oscillations for this model by using mathematical analysis method, which is simpler than bifurcation method. The criteria for selecting of the parameters in this network are provided. Computer simulation examples are presented to demonstrate the correctness of this method.

Imperfection sensitivity of nonlinear primary resonance behavior in bi-directional functionally graded porous material beam

This research aims to analyze the imperfection sensitivity in the nonlinear resonance behavior of bi-directional functionally graded (BDFG) porous beam. The trigonometric and hyperbolic functions are chosen to describe the global and localized geometrical imperfections. Considering the first-order shear deformation theory and von-Kármán’s geometric nonlinearity, theoretical model is formulated to depict the coupled motion including the stretching, bending and shear deformations of BDFG porous beam. Differential quadrature based Galerkin’s procedure is proposed to formulate the reduced model. The nonlinear responses are obtained by using the numerical bifurcation method. Numerical results demonstrate that resonance response of beam under goes cyclic-fold bifurcations. Resonance behavior of beam may exhibit hardening, softening or softening-hardening type nonlinearity. Imperfection sensitivity analysis shows that the variation of imperfection modes, material gradient and porosity change the characteristic of frequency-response and force-response or even the number of bifurcation points. .

Hopf Bifurcation for UAV Path Planning in Autonomous Surveillance and Landing on an UGV

A novel path planning approach based on the bifurcation theory is proposed for a quadrotor UAV to track a ground vehicle in motion. For this purpose, a brief description of the mathematical model of the quadrotor UAV and the ground vehicle is presented, and the Hopf bifurcation method around a mobile balance point is used to design the path planning for a quadrotor UAV, and autonomously it generates a circular surveillance relative to a moving ground vehicle. The stability analysis based on Lyapunov theory is presented in order to demonstrate that the trajectory satisfies the bifurcation properties. Simulation and experimental tests are carried out to validate the proposed approach.

Study on bifurcation to chaos of surface tension gradient driven flow

As the primary heat and mass transfer mechanism in space through natural convection, surface tension gradient-driven convection is a complex nonlinear process concerning strong coupling between fluid flow and heat transfer. It is also a multiple parameter coupling process that exhibits complex spatial-temporal characteristics. Therefore, the mechanism of the surface tension gradient-driven convection becomes a hotspot in microgravity fluid physics. It also has many important applications, such as in space fluid and energy management. In this paper, recent experimental and numerical results on the transition of surface tension gradient-driven convection are reviewed, especially the nonlinear analysis on the flow bifurcations to chaos. There are several numerical methods to obtain the corresponding bifurcation diagrams. One is to integrate the model forward in time starting from different parameters and initial values, and others are to calculate the asymptotic flow states and bifurcation points directly. The direct numerical simulation method and time series analysis are widely used, but searching for bifurcation points from a large number of data is burdensome. Bifurcation points can be computed directly with the numerical bifurcation method, but such calculations are more difficult to implement than the direct numerical method.

Stability analysis of whirl flutter in rotor-nacelle systems with freeplay nonlinearity

Tiltrotor aircraft are growing in prevalence due to the usefulness of their unique flight envelope. However, aeroelastic stability—particularly whirl flutter stability—is a major design influence that demands accurate prediction. Several nonlinearities that may be present in tiltrotor systems, such as freeplay, are often neglected for simplicity, either in the modelling or the stability analysis. However, the effects of such nonlinearities can be significant, sometimes even invalidating the stability predictions from linear analysis methods. Freeplay is a nonlinearity that may arise in tiltrotor nacelle rotation actuators due to the tension–compression loading cycles they undergo. This paper investigates the effect of a freeplay structural nonlinearity in the nacelle pitch degree of freedom. Two rotor-nacelle models of contrasting complexity are studied: one represents classical whirl flutter (propellers) and the other captures the main effects of tiltrotor aeroelasticity (proprotors). The manifestation of the freeplay in the systems’ dynamical behaviour is mapped out using Continuation and Bifurcation Methods, and consequently the change in the stability boundary is quantified. Furthermore, the effects on freeplay behaviour of (a) model complexity and (b) deadband edge sharpness are studied. Ultimately, the freeplay nonlinearity is shown to have a complex effect on the dynamics of both systems, even creating the possibility of whirl flutter in parameter ranges that linear analysis methods predict to be stable. While the size of this additional whirl flutter region is finite and bounded for the basic model, it is unbounded for the higher complexity model.

Exploring the influence of microRNA miR-34 on p53 dynamics: a numerical study* *Project supported by the National Natural Science Foundation of China under Grant No. 11762011.

The tumor suppressor p53 is at the hub of the cellular DNA damage response network. P53-dependent cell fate decision is inseparable from p53 dynamics. A type of non-coding microRNA miR-34 has the function of enhancing p53 content. An intriguing question arises: How does miR-34 affect p53 kinetics? To address this question, we reconstruct a p53 signal transduction network model containing miR-34. Some experimental phenomena of p53 pulses are reproduced to explain the rationality of the model. The method of numerical bifurcation is used to investigate the effect of miR-34 on p53 kinetics. We point out that appropriate or higher miR-34 transcription rates can prevent DNA-damaged cell proliferation by causing p53 oscillation or high steady-state kinetic behavior, respectively. However, the lack of miR-34 synthesis ability will induce p53 to remain at a low level, and cells cannot respond correctly to DNA damage. These results are well in line with the anti-cancer role of miR-34. Our work sheds light on how miR-34 carries out its tumor-suppressive function from tuning p53 dynamic aspect.

Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System

In this paper, we applied some computational tools, namely the modified extended tanh method via a Riccati equation, the general Exp $$_{a}$$ -function method and the bifurcation methods to study a nonlinear (2+1)-dimensional Bogoyavlenskii coupled system in thin-film ferroelectric medium to construct exact traveling wave solutions. By applying a classical wave transformation we obtained an ordinary differential equations. As a result, some new traveling wave solutions are obtained including hyperbolic, trigonometric, exponential functions and rational forms. If the parameters take specific values, then the periodic wave, solitary waves, kink and anti-kink wave solutions are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special cases of these nonlinear equations by the help of programming language Maple.

Abundant explicit periodic wave solutions and their limit forms to space‐time fractional Drinfel'd–Sokolov–Wilson equation

In this paper, we exploit the generalized bifurcation method to study space–time fractional Drinfel'd–Sokolov–Wilson equation and derive its various new exact explicit periodic wave solutions. Especially and interestingly, we find the so‐called M/W‐shaped periodic wave solutions, which were not found in previous studies. Furthermore, we uncover their inside limit relations as well as their limit relations with other solutions under corresponding parameters conditions. The previous results are extended.

Existence of periodic, solitary and compacton travelling wave solutions of a $$(3+1)$$-dimensional time-fractional nonlinear evolution equations with applications

In this paper, we consider the bifurcation method of dynamical systems for solving time fractional nonlinear evolution equations. We adapt and modify the methodology, incorporating new ideas from the conformable fractional derivative, to investigate exact travelling wave solutions and bifurcations of phase transitions for nonlinear evolution equations. In this study, we show the existence of periodic wave solutions, kink and anti-kink wave solutions, a bright and dark solitary wave solution and parabolic solutions. Moreover, numerical simulations method is applied to show the richer dynamical behavior of the spatial and temporal fractional order of nonlinear evolutions systems and verify the theoretical results.

DYNAMICS OF SINGULAR TRAVELING WAVE SOLUTIONS OF A SHORT CAPILLARY-GRAVITY WAVE EQUATION

<abstract> In this paper, dynamical behaviour of traveling wave solutions to a short capillary-gravity equation is analyzed by using the method of bifurcation. When the phase orbits intersects the singular parabola $ y^2 = 2\phi/\lambda $ on the phase plane, then the trajectories create a weaker wave fronts than the regular traveling wave solutions. By using proper Euler transformations, we reformulate the model as a singular chaotic problem, which can then be analyzed using the singularity study. We prove existence of three types of physically realistic traveling wave solutions to the case of small diffusion for the first time, two-peaked solitary waves, three-peaked and multi-peaked periodic wave solutions. </abstract>

RANK-ONE CHAOS IN A DELAYED SIR EPIDEMIC MODEL WITH NONLINEAR INCIDENCE AND TREATMENT RATES

<abstract> The rank one chaos in a SIR model with two time-delays is studied in this paper. By using center manifold theorem, normal form theory and Hassard's method, the existence, direction and stability of Hopf bifurcation are discussed. Based on the rank-one chaos theory for delayed differential equations, the conditions for the existence of rank-one strange attractor in disturbed system are obtained. Finally, numerical simulations are given to verify the theoretical analysis results. </abstract>

Improvement of surgical treatment of Duane syndrome in children

To analyze the use of bifurcation of the lateral rectus muscle (LRM) in the surgical treatment of Duane syndrome (DS) in children. The analysis of surgical treatment involved 12 children with paralytic strabismus at the age of 8 to 17. The following research methods were conducted for all patients: the study of eye motility in different directions of gaze, the study of eyes position and the degree of deviation in different positions of gaze, the Bielschowsky test, the Park's three-step test, determination of the nature of torticollis, the forcing test - intraoperative forceps test, multislice computed tomography (MSCT) of the head and orbit, ultrasonography of the oculomotor muscles. All study children underwent bifurcation (Y-cleavage procedure) of LRM with recession in exo-DS in 17% of cases, without recession in eso-DS - 83% of cases, respectively. In all operated patients with Duane syndrome, the direction of LRM was compensated (downshoot and upshoot were eliminated). In a direct gaze, the correct position was achieved in 10 (83%) cases, in 2 (17%) the residual angle of up to 10 PD remained, and torticollis was eliminated in 9 (75%) patients. The method of LRM bifurcation - the procedure of dividing the muscle into two portions and changing its primary localization, accompanied by abnormalities (presence of deviation, retraction, anomalous vertical movements, limited ocular motility, etc.), - helps compensate for them, and provides cosmetic as well as functional improvement.

On the influence of cross-diffusion in pattern formation

In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software $\mathtt{pde2path}$. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems

Abstract By using the bifurcation method, we study the existence of an S-shaped connected component in the set of positive solutions for discrete second-order Neumann boundary value problem. By figuring the shape of unbounded connected component of positive solutions, we show that the Neumann boundary value problem has three positive solutions suggesting suitable conditions on the weight function and nonlinearity.

Влияние распределенных дислокаций на устойчивость полого нелинейно-упругого шара

The phenomenon of stability loss of a hollow elastic sphere containing distributed dislocations and loaded with external hydrostatic pressure is studied. The study was carried out in the framework of the nonlinear elasticity theory and the continuum theory of continuously distributed dislocations. An exact statement and solution of the stability problem for a three-dimensional elastic body with distributed dislocations are given. The static problem of nonlinear elasticity theory for a body with distributed dislocations is reduced to a system of equations consisting of equilibrium equations, incompatibility equations with a given dislocation density tensor, and constitutive equations of the material. The unperturbed state is caused by external pressure and a spherically symmet-ric distribution of dislocations. For distributed edge dislocations in the framework of a harmonic (semi-linear) mate-rial model, the unperturbed state is defined as an exact spherically symmetric solution to a nonlinear boundary value problem. This solution is valid for any function that characterizes the density of edge dislocations. The perturbed equilibrium state is described by a boundary value problem linearized in the neighborhood of the equilibrium. The analysis of the axisymmetric buckling of the sphere was performed using the bifurcation method. It consists in determining the equilibrium positions of an elastic body, which differ little from the unperturbed state. By solving the linearized problem, the value of the external pressure at which the sphere first loses stability is found. The effect of dislocations on the buckling of thin and thick spherical shells is analyzed.

Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction

A reaction-diffusion model describing water and plant interaction proposed by Klausmeier is studied. The existence of non-constant steady state solutions is shown through bifurcation methods, and the existence of large amplitude spatial patterned solutions is proved using associated shadow system. It is rigorously shown that non-homogeneous patterned vegetation states exist when the rainfall is at a lower level in which homogeneous vegetation state cannot survive. Even when the rainfall is very low, slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating on a small area. This provides an example of diffusion-induced persistence that non-constant steady states may exist in a reaction-diffusion system when there are no positive constant steady states.