Narrow Results

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized $\kappa$ distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced $\alpha$, as well as waste $\beta$ and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor $\delta$ of regularized $\kappa$-distribution (RKD) distributed ei, the superthermality of ei particles ${\kappa }_{e,i}$, the ratio of ion to electron temperature $\sigma$, and the ratio of dust to electron density $\rho$ illustrated.  It is noticed that the DAW shows promotion in width and amplitude as the frequency $\gamma$ increases and it becomes rarefaction as the cut-off parameter $\delta$ increases. On the contrary, it becomes compressive under the impact of superthermality ${\kappa }_e$ and ${\kappa }_i$. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized $\kappa$ dispersed ei, as well as in the interstellar medium.

Exact traveling wave solutions of the conformable fractional extended $(3+1)$-dimensional Kadomtsev–Petviashvili equation are constructed. We apply four methods including the modified extended $tanh$-function method, the improved $(G^{\prime }/G)$ method, the two variables $(G^{\prime }/G,1/G)$-expansion method and the extended generalized Riccati equation mapping method. These methods are all expanded around Riccati equation, but each has unique advantages. Then various forms of solutions, such as breaking wave solutions, periodic solutions and soliton solutions can be obtained. Furthermore, the bifurcation analysis of this equation is also conducted. Dynamical features of the solutions by taking different values for parameters are illustrated by the 3D and 2D plots. Finally, we conduct discussions and comparisons of the results.

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.

One of the most important factors influencing animal growth is non-genetics, which includes factors like nutrition, management and environmental conditions. By consuming their prey, predators can directly affect their ecology and evolution, but they can also have an indirect impact by affecting their prey’s nutrition and reproduction. Preys used to change their habitats, their foraging and vigilance habits as anti-predator responses. Cooperation during hunting by the predators develops significant fear in their prey which indirectly affects their nutrition. In this work, we propose a two-species stage-structured predator–prey system where the prey are classified into juvenile and mature prey. We assume that the conversion of juvenile prey to matured prey is affected by the fear of predation risk. Non-negativity and boundedness of the solutions are demonstrated theoretically. All the biologically feasible equilibrium states are determined, and their stabilities are analyzed. The role of various important factors, e.g. hunting cooperation rate, predation rate and rate of fear, on the system dynamics is discussed. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MatCont 7.3. Finally, the proposed model is extended into a harvesting model under quadratic harvesting strategy and the associated control problem has been analyzed for optimal harvesting.

A new four-dimensional continuous autonomous hyperchaotic system is considered. It possesses two parameters, and each equation of it has one quadratic cross product term. Some basic properties of it are studied. The dynamic behaviors of it are analyzed by the Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits, and Poincaré sections. The system has larger hyperchaotic region. When it is hyperchaotic, the two positive LE are both large and they are both larger than 1 if the system parameters are taken appropriately.

A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

A modified continuum traffic flow model is established in this paper based on an extended car-following model considering driver’s reaction time and distance. The linear stability of the model and the Korteweg–de Vries (KdV) equation describing the density wave of traffic flow in the metastable region are obtained. In the new model, the relaxation term and the dissipation term exist at the same time, thus the type and stability of equilibrium solution of the model can be analyzed on the phase plane. Based on the equilibrium point, the bifurcation analysis of the model is carried out, and the existence of Hopf bifurcation and saddle-node bifurcation is proved. Numerical simulations show that the model can describe the complex nonlinear dynamic phenomena observed in freeway traffic, such as local cluster effect. Various bifurcations of the model, such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles, Cusp bifurcation and Bogdanov–Takens bifurcation, are also obtained by numerical simulations, and the traffic behaviors of some bifurcations are studied. The results show that the numerical solution is consistent with the analytical solution. Consequently, some nonlinear traffic phenomena can be analyzed and predicted from the perspective of global stability.

The essence of traffic congestion is a kind of bifurcation behavior incurred by the loss of stability in the traffic flow from the perspective of system stability. Therefore, researching the bifurcation characteristics of traffic flow can provide some new methods for relieving the traffic congestion. In this paper, we study the bifurcation dynamic behavior and the bifurcation conditions of traffic flow based on Kühne’s continuum traffic flow model. We discussed the types and stabilities of the equilibrium solutions and proved the existence conditions of some bifurcation of the model. Then various bifurcations of the system are found by numerical simulation and the stability catastrophic behaviors initiated by some bifurcation in traffic flow are investigated. Combined with the measured data, we study the actual traffic flow bifurcation phenomena and analyze the internal reasons for the stability catastrophic behavior of traffic flow passed by the bifurcation point. The nonlinear characteristics and formation mechanism of traffic flow phenomena are revealed from a new perspective. It may help to design the corresponding control schemes for preventing and alleviating traffic congestion.

This paper proposes a novel nonuniform continuous traffic flow model, which takes into account the differences in drivers’ expected headway and further improves the model by incorporating the influences of relative distance and relative velocity of vehicles through linear weighting. Additionally, the model also considers the dynamic response time of drivers for further refinement. With this model, bifurcation theory can also be applied in the stability analysis of traffic systems to study the stability mutation behavior of traffic systems at bifurcation points. Through linear and nonlinear analysis methods, the stability conditions of the model and the KdV–Burgers equation can be derived, the type of equilibrium point of the model can be judged, the conditions for the existence of Hopf bifurcations and the type of bifurcations can be proved, and the traffic flow problem can be transformed into the stability analysis problem of the system. Through numerical simulations in both high-density and low-density scenarios, it is shown that the model can well describe the actual traffic phenomena and can describe the stability abrupt behavior of the traffic system at Hopf bifurcation points and saddle-node bifurcation points through density-time and phase plane diagrams.

The stability of the transportation system refers to the structural stability of the system. When the system structure is unstable, local or global bifurcation phenomena will occur, which is one of the main reasons for nonlinear traffic phenomena such as congestion. To truly understand the internal mechanism of the formation of these phenomena, it is necessary to analyze the bifurcation of traffic flow. In this paper, the Hopf bifurcation control of a modified viscous macroscopic traffic flow model is studied by using the linear state feedback method, which changes the characteristics of the bifurcation phenomenon of the dynamic system and obtains the required dynamic behavior of the system. First, we can convert the original traffic model into the nonlinear ordinary differential form suitable for bifurcation analysis, solve the equilibrium point of the system, and carry out phase plane analysis. Then, the linear state feedback term is added and the corresponding controlled system is generated, the existence and type of Hopf bifurcation and the existence of saddle node bifurcation are proved. Numerical simulation results show that the analysis and control of Hopf bifurcation in the traffic model are well realized in this paper.

In this paper, the effects of external DC electric fields on the neuro-computational properties are investigated in the context of Morris–Lecar (ML) model with bifurcation analysis. We obtain the detailed bifurcation diagram in two-dimensional parameter space of externally applied DC current and trans-membrane potential induced by external DC electric field. The bifurcation sets partition the two-dimensional parameter space in terms of the qualitatively different behaviors of the ML model. Thus the neuron's information encodes the stimulus information, and vice versa, which is significant in neural control. Furthermore, we identify the electric field as a key parameter to control the transitions among four different excitability and spiking properties, which facilitates the design of electric fields based neuronal modulation method.

Based on a density gradient model proposed recently by Imran and Khan, a new heterogeneous traffic flow model considering time and lateral distance is proposed. The type and stability of the equilibrium solution of the model are discussed by using the differential equation theory, and the global distribution structure of the trajectory in the phase plane is analyzed. In addition, the density wave stability conditions and saddle-node bifurcation conditions of the model are studied, and the solitary wave solutions of the KdV equation in the metastable region are derived by using the reduced perturbation method. The numerical results show that the new model cannot only reproduce the spatiotemporal oscillation phenomenon when walking and stopping, but also describe the sudden change behavior of traffic near the critical point of saddle-node bifurcation. It is shown that the model can reproduce some complex traffic phenomena qualitatively.

Peristaltic motions are used in a variety of industrial and real-world problems such as pumping mixture with such a high solid content from the mining sector, biogases, sewers facilities, food flow through the gastrointestinal system, urinary tract, and fallopian tube of human females where extremely abrasive, gritty, and viscous fluids are present. The goal of this theoretical study is to examine the impact of the lubricated walls on the peristaltic transport of a viscous fluid in an asymmetric medium. The interfacial condition is derived under the assumptions of thin film lubrication and long wavelength approximation. The theory of a dynamical system is utilized to examine the lubrication effects on the position and bifurcation of stagnation points in the flow. For this, three flow distributions, backward flow, trapping, and augmented flow, are discussed. The obtained system of equations is solved numerically by the shooting method based on Newton–Raphson root-finding algorithm in Mathematica. The prime findings for the velocity profile, pressure increase, trapping, and reflux criteria are illustrated through graphs. Bifurcation occurs earlier by increasing the influence of lubrication. The trapping area diminishes and the augmented flow section expands with the lubrication parameter. This will be useful for medical engineering, industrial and physiological systems. The comparison for the particular case of no-slip condition is presented and found to be in excellent agreement.

In this paper, we have analyzed the bifurcation of stagnation points in the non-Newtonian flow of the FENE-P fluid through a channel induced by peristalsis. The stream function for the flow under consideration is developed under negligible inertia and streamline curvature effects. A nonlinear autonomous system is developed for scrutiny of the critical points. A dynamical system approach is used to examine the stability of bifurcation points. Streamline patterns, and local and global bifurcation diagrams are plotted for analysis of the backward flow, trapping, and augmented flow regions. The impacts of important embedded parameters, i.e., extensibility parameter and Deborah number on local and global bifurcation diagrams, are also examined. The whole analysis reveals that two critical conditions appear in the flow field that are actually responsible for the conversion of backward flow to trapping and trapping to augmented flow. It is further seen that bifurcations occur at lower flow rates for increasing Deborah’s number while an opposite trend prevails for increasing the extensibility parameter. A comparison of bifurcations between planar and axisymmetric flows is also performed. It is observed that a greater area of the trapping region is encountered for axisymmetric flow than that for planar flow.

In this work, a heterogeneous traffic flow model coupled with the periodic boundary condition is proposed. Based on the previous models, a heterogeneous system composed of more than one kind of vehicles is considered. By bifurcation analysis, bifurcation patterns of the heterogeneous system are discussed in three situations in detail and illustrated by diagrams of bifurcation patterns. Besides, the stability analysis of the heterogeneous system is performed to test its anti-interference ability. The relationship between the number of vehicles and the stability is obtained. Furthermore, the attractor analysis is applied to investigate the nature of the heterogeneous system near its steady-state neighborhood. Phase diagrams of the process of the heterogeneous system from initial state to equilibrium state are intuitively presented.

This paper proposes a new density gradient continuous traffic flow model, and analyzes the linear stability of the model, as well as the bifurcation type of the model. Numerical simulation of the new model verifies the usability of the model. From the perspective of system stability, the bifurcation analysis method is used to analyze the nonlinear traffic phenomena on the expressway. The equilibrium solution of the model is discussed. On this basis, Hopf bifurcation, saddle bifurcation and Bogdanov–Takens bifurcation are obtained, and the existence conditions and fractional types of Hopf bifurcation and saddle bifurcation are obtained. The traffic flow characteristics of Hopf bifurcation and saddle node bifurcation are analyzed.

In this work, the dynamics of a simplified model of three-neurons-based Hopfield neural networks (HNNs) is investigated. The simplified model is obtained by removing the synaptic weight connection of the third and second neuron in the original Hopfield networks introduced in Ref. 11. The investigations have shown that the simplified model possesses three equilibrium points among which origin of the systems coordinates. It is found that the origin is always unstable while the symmetric pair of fixed points with conditional stability has values depending on synaptic weight between the second and the first neuron that is used as bifurcation control parameter. Numerical simulations, carried out in terms of bifurcation diagrams, graph of Lyapunov exponents, phase portraits, Poincaré section, time series and frequency spectra are employed to highlight the complex dynamical behaviors exhibited by the model. The results indicate that the modified model of HNNs exhibits rich nonlinear dynamical behaviors including symmetry breaking, chaos, periodic window, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits) and coexisting self-excited attractors (e.g., coexistence of two, four and six disconnected periodic and chaotic attractors) which have not been reported in previous works focused on the dynamics of HNNs. Finally, PSpice simulations verify the results of theoretical analyses of the simplified model of three-neurons-based HNNs.

In this paper, we report a new third-order chaotic jerk system with double-hump (bimodal) nonlinearity. The bimodal nonlinearity is of basic interest in biology, physics, etc. The proposed jerk system is able to exhibit chaotic response with proper choice of parameters. Importantly, the chaotic response is also obtained from the system by tuning the nonlinearity preserving its bimodal form. We analytically obtain the symmetry, dissipativity and stability of the system and find the Hopf bifurcation condition for the emergence of oscillation. Numerical investigations are carried out and different dynamics emerging from the system are identified through the calculation of eigenvalue spectrum, two-parameter and single parameter bifurcation diagrams, Lyapunov exponent spectrum and Kaplan–Yorke dimension. We identify that the form of the nonlinearity may bring the system to chaotic regime. Effect of variation of parameters that controls the form of the nonlinearity is studied. Finally, we design the proposed system in an electronic hardware level experiment and study its behavior in the presence of noise, fluctuations, parameter mismatch, etc. The experimental results are in good analogy with that of the analytical and numerical ones.

This paper introduces a charge-controlled memristor based on the classical Chuas circuit. It also designs a novel four-dimensional chaotic system and investigates its complex dynamics, including phase portrait, Lyapunov exponent spectrum, bifurcation diagram, equilibrium point, dissipation and stability. The system appears as single-wing, double-wings chaotic attractors and the Lyapunov exponent spectrum of the system is symmetric with respect to the initial value. In addition, symmetric and asymmetric coexisting attractors are generated by changing the initial value and parameters. The findings indicate that the circuit system is equipped with excellent multi-stability. Finally, the circuit is implemented in Field Programmable Gate Array (FPGA) and analog circuits.

We investigate bifurcations in responses of two Hodgkin–Huxley neurons coupled to each other and forced by a periodic pulse train, through the characteristics of synaptic transmissions with an α-function and a time delay. Based on a computational method we previously presented, we show a mechanism of transitions among various kinds of nonperiodic as well as harmonic and subharmonic frequency-locked oscillations. The bifurcation analysis clarifies dependence of sub- and supra-threshold dynamics on coupling and forcing effects.