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In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient.

In this paper, we study a generalized Schrödinger–Poisson system in a bounded domain of $\mathbb{R}^{3}$ and involving an asymptotically linear nonlinearity. We prove the existence of positive solutions using bifurcation theory.

The stochastic coupled Maccari's system (MS) is a kind of important nonlinear partial differential equations to describe fluid flow, plasma physics, nonlinear optics and so on. In this article, the dynamical behavior and some new exact traveling wave solutions of the system are investigated. By means of complex traveling wave transformation, the system is transformed into a nonlinear ordinary differential equation. The dynamical behavior of the system as well as its perturbation case are illustrated by bifurcation theory. And then, some new stochastic traveling wave solutions of the system are extracted based on the theory of polynomial complete discrimination system. To show the effect of stochastic factor on the solutions, their structures under different Brownian motion amplitudes are compared by several sets of graphs. The results obtained in this paper have supplemented the study of the system, and the technique used to exploit the traveling wave solutions are effective.

Abstract With the deep extension of coal mining in China, fault water inrush has become one of the major disasters threatening the safety production of coal mine. Based on the control equations of steady state and non-Darcy seepage in fractured rock mass, the multi-parameter nonlinear dynamic seepage equations of fractured rock mass are established in this paper. Based on the nonlinear dynamics theory, the function of the state variable in the system is derived, and the influence of the gradual change of non-Darcy flow factors on the structural stability of seepage system is studied. The research achievements show that there are three branches in the equilibrium state of the seepage system. Specifically, the stability of the equilibrium state changes abruptly near the limit parameter. The seepage dynamic system of fractured rock mass has the delayed bifurcation, and the coal mine disaster such as fault water inrush occurs easily at the bifurcation point. The research results are of great significance to enrich the theory of fault water inrush in coal mine, and to reveal the disastrous mechanism of fault water inrush and guide its prevention and control technology in coal mine, which can provide the theoretical reference for predicting the water seepage stability in fractured rock mass.

In this paper, we propose a spatiotemporal prey-predator model with fear and Allee effects. We first establish the global existence of solution in time and provide some sufficient conditions for the existence of non-negative spatially homogeneous equilibria. Then, we study the stability and bifurcation for the non-negative equilibria and explore the bifurcation diagram, which revealed that the Allee effect and fear factor can induce complex bifurcation scenario. We discuss that large Allee effect-driven Turing instability and pattern transition for the considered system with the Holling-Ⅰ type functional response, and how small Allee effect stabilizes the system in nature. Finally, numerical simulations illustrate the effectiveness of theoretical results. The main contribution of this work is to discover that the Allee effect can induce both codimension-one bifurcations (transcritical, saddle-node, Hopf, Turing) and codimension-two bifurcations (cusp, Bogdanov-Takens and Turing-Hopf) in a spatiotemporal predator-prey model with a fear factor. In addition, we observe that the circular rings pattern loses its stability, and transitions to the coldspot and stripe pattern in Hopf region or the Turing-Hopf region for a special choice of initial condition.

Since certain prey hide from predators to protect themselves within their habitats, predators are forced to change their diet due to a lack of prey for consumption, or on the contrary, subsist only with alternative food provided by the environment. Therefore, in this paper, we propose and mathematically contrast a predator-prey, where alternative food for predators is either considered or not when the prey population size is above the refuge threshold size. Since the model with no alternative food for predators has a Hopf bifurcation and a transcritical bifurcation, in addition to a stable limit cycle surrounding the unique interior equilibrium, such bifurcation cases are transferred to the model when considering alternative food for predators when the prey size is above the refuge. However, such a model has two saddle-node bifurcations and a homoclinic bifurcation, characterized by a homoclinic curve surrounding one of the three interior equilibrium points of the model.

Nonlinear partial differential equations (NLPDEs) have been of great interest in recent years due to their numerous applications. While there are several methods for finding exact solutions to various NLPDEs, more solutions are still required. This paper first proposes the Cham method, a new method for solving NLPDEs that can generate eight families of solutions. The method is then successfully employed to solve the (2+1)-dimensional Bogoyavlenskii's breaking soliton equations. The dynamic behaviour of these equations and the bifurcation of traveling waves are also discussed. Finally, we graphically depict some solutions corresponding to some discovered solutions with different coefficient values. The Cham method is general, effective, and adaptable to many NLPDEs.

This paper explores solitary wave solutions arising in the deformations of a hyperelastic compressible plate. Explicit traveling wave solution expressions with various parameters for the hyperelastic compressible plate are obtained and visualized. To analyze the perturbed equation, we employ geometric singular perturbation theory, Melnikov methods, and invariant manifold theory. The solitary wave solutions of the hyperelastic compressible plate do not persist under small perturbations for wave speed c>−βk2. Further exploration of nonlinear models that accurately depict the persistence of solitary wave solution on the significant physical processes under the K-S perturbation is recommended.

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

The article considers the main approaches to the development and evaluation of the organization’s logistics strategy based on the application of bifurcation theory. The paper notes that the greatest competitive advantages in the process of carrying out activities can be obtained by the company when designing a unique logistics strategy based on the individual features of the organization’s logistics system. The authors reveal that the bifurcation theory allows you to determine the conditionality of a qualitative dynamic change in the behavior of a logistics system by an infinitesimal change in certain parameters of the system. As part of the assessment of the organization’s logistics management strategy, bifurcation theory forms the basis for determining shift in the qualitative parameter of the system and makes it possible to assess the effect of this parameter on the behavior of the entire system as a whole and each functional subsystem separately. The study reflects that it is possible to determine the critical values of individual variables, when overcoming which the logistics system itself goes into unstable state. The main concept defining the process of critical transition of the state of the logistic system is the bifurcation point.

This paper investigates the impact of combined axial through flow and radial mass flux on Taylor–Couette flow in a counter-rotating configuration, in which different branches of nontrivial solutions appear via Hopf bifurcations. Using direct numerical simulation, we elucidate flow structures, dynamics, and bifurcation behavior in qualitative and quantitative detail as a function of axial Reynolds numbers (Re) and radial mass flux (α) spanning a parameter space with a very rich variety of solutions. We have determined nonlinear properties such as anharmonicity, asymmetry, flow rates (axial and radial) and torque for toroidally closed Taylor vortices and helical spiral vortices. Small to moderate radial flow α initially decreases the symmetry of the different flows, before for larger values, α, the symmetry eventually increases, which appears to be congruent with the degree of anharmonicity. Enhancement in the total torque with α are elucidated whereby the strength varies for different flow structures, which allows for potential better selection and control. Further, depending on control parameters, heteroclinic connections (and cycles) of oscillatory type in between unstable and topological different flow structures are detected. The research results provide a theoretical basis for simple modification the conventional Taylor flow reactor with a combination of additional mass flux to enhance the mass transfer mechanism.

Abstract In this paper, the bifurcations of a modified Degasperis–Procesi equation are studied under different parametric conditions, which have not been investigated by the bifurcation theory of dynamical systems before. The existence of loop, periodic wave and smooth solitary wave solutions for the modified Degasperis–Procesi equation is proved, and exact expressions of corresponding traveling wave solutions are obtained.

In this paper, we investigated the Gilson-Pickering (GP) equation and many new solutions are obtained with the aid of two different approaches, namely Jacobi elliptic functions and exponential rational function approach. Different choices of the parameters in obtained results lead to the solutions of some well known models, which are Camassa-Holm equation, the Fornberg-Whitham equation and the Rosenau-Hyman equation. The methods considered here can also help to have a panoply of new wave surfaces concerning other related partial differential equations. Further more, 2D and 3D graphical presentations of these surfaces are presented for the various parameters. Moreover, bifurcation behavior of nonlinear travelling waves of GP equation is discussed. Bifurcation theory of planer dynamical system is utilized to observe that considered model contains nonlinear periodic wave, bell shaped solitary wave and shock wave.

This paper anatomizes the exact solutions of the resonant non-linear Schrödinger’s equation (R-NLSE) with the Kerr law non-linearity with the assistance of the new extended direct algebraic technique. The secured soliton erections are newfangled and unreservedly invigorating for investigators. The graphically comprehensive report of some specific solutions is embellished with the well-judged values of parameters to illustrate their propagation. Then a planer dynamical system is introduced and the bifurcation analysis has been executed to figure out the bifurcation structures of the non-linear and super non-linear traveling wave solutions of the heeded model. All possible phase portraits are exhibited with specific values of parameters. Furthermore, a precise class of non-trivial and first-order conserved quantities is enumerated by the intervention of the multiplier approach.

This paper studies the dynamical behaviors of nonlinear wave solutions of perturbed and unperturbed fractional Chen-Lee-Liu (CLL) equation in optical fibers with a newly defined beta derivative. The coupled amplitude-phase formulation is used for the derivation of a nonlinear differential equation which contains a fifth-degree nonlinear term describing the evolution of the wave amplitude in the nonlinear system. Variety of soliton solutions are found by using the new extended direct algebraic method. Then, discussed model is converted into the planer dynamical system with the help of Galilean transformation and the bifurcation behavior is reported. All possible forms of phase portraits with respect to the parameters of the considered problem are plotted. In addition, by applying an extrinsic periodic force the effect of physical parameters is investigated. Furthermore, sensitive analysis is applied for different initial value problems to analyze the quasiperiodic and quasiperiodic-chaotic behaviors.

In this paper, we study the multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

Thermally sensitive neurons represent a bursting-spiking activity that is indicated by rapid repetitive spiking trains of action potentials pursued by dormant periods. Synchronization of such behavior in a network of coupled spiking neurons such as the epileptogenic zone in the brain may cause some neurological disorders such as epileptic seizures. This paper introduces a new approach for predicting the seizure onset in a model of an epileptic neuron. The parameters which are used for simulations have been selected in such a way that they would be potentially applicable in the non-invasive brain stimulation approaches such as repetitive transcranial magnetic stimulation (rTMS). In this regard, a modified Huber-Braun model of a thermally sensitive neuron exposed to external rTMS-induced voltages is presented. Applying the chaos theory, the bifurcation diagram of a modified Huber-Braun model with a new bifurcation parameter is used to estimate the time at which the bifurcation takes place whereby allowing a more accurate prediction of the seizure onset based on the modified model.

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

Non-linear phenomena are particularly important in -flight dynamics of micro-class unmanned aerial vehicles. Susceptibility to atmospheric turbulence and high manoeuvrability of such aircraft under critical flight conditions cover non-linear aerodynamics and inertia coupling. The theory of dynamical systems provides methodology for studying systems of non-linear ordinary differential equations. The bifurcation theory forms part of this theory and deals with stability changes leading to qualitatively different system responses. These changes are called bifurcations. There is a number of papers, the authors of which applied the bifurcation theory for analysing aircraft flight dynamics. This article analyses the dynamics of critical micro aerial vehicle flight regimes. The flight dynamics under such conditions is highly non-linear, therefore the bifurcation theory can be applied in the course of the analysis. The application of the theory of dynamical systems enabled predicting the nature of micro aerial vehicle motion instability caused by bifurcations and analysing the post-bifurcation microdrone motion. This article presents the application of bifurcation analysis, complemented with time-domain simulations, to understand the open-loop dynamics of strake-wing micro aerial vehicle model by identifying the attractors of the dynamic system that manages upset behaviour. A number of factors have been identified to cause potential critical states, including non-oscillating spirals and oscillatory spins. The analysis shows that these spirals and spins are connected in a one-parameter space and that due to improper operation of the autopilot on the spiral, it is possible to enter the oscillatory spin.

Abstract This paper deals with an unstirred chemostat model with the Beddington-DeAngelis functional response. First, some prior estimates for positive solutions are proved by the maximum principle and the method of upper and lower solutions. Second, the calculation on the fixed point index of chemostat model is obtained by degree theory and the homotopy invariance theorem. Finally, some sufficient condition on the existence of positive steady-state solutions is established by fixed point index theory and bifurcation theory.

This paper is concerned with the study of some properties of stationary solutions to phase field crystal equations bifurcating from a trivial solution. It is assumed that at this trivial solution, the kernel of the underlying linearized operator has dimension two. By means of the multiparameter method, we give a second order approximation of these bifurcating solutions and analyse their stability properties. The main result states that the stability of these solutions can be described by the variation of a certain angle in a two dimensional parameter space. The behaviour of the parameter curve is also investigated.

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

The paper investigates nonlinear eigenvalue problem for a vertical homogeneous rod loaded with its own weight only. The critical length of the rod, for which the rod loses its stability, is found by use of bifurcation theory. Dependence of maximal deflections of the rods on their lengths is given.

Abstract In this paper, forming limit diagrams (FLDs) for an aluminum alloy are predicted through numerical simulations using various localized necking criteria. A comparative study is conducted for the FLDs determined by using the Lemaitre damage approach and those obtained with the modified Gurson-Tvergaard-Needleman (GTN) damage model. To this end, both damage models coupled with elasto-plasticity and accounting for plastic anisotropy have been implemented into the ABAQUS/Explicit software, through the user-defined subroutine VUMAT, within the framework of large plastic strains and a fully three-dimensional formulation. The resulting constitutive frameworks are then combined with four localized necking criteria to predict the limit strains for an AA6016-T4 aluminum alloy. Three of these necking criteria are based on finite element (FE) simulations of the Nakazima deep drawing test with various specimen geometries, while the fourth criterion is based on bifurcation theory. The simulation results reveal that the limit strains predicted by local criteria, which are based on FE simulations of the Nakazima test, are in good agreement with the experiments for a number of strain paths, while those obtained with the bifurcation analysis provide an upper bound to the experimental FLD.