Conditions For Bifurcation Research Articles

Steady state bifurcation and pattern formation of a diffusive population model

The steady state bifurcation and spatiotemporal patterns are induced by prey-taxis in a population model, in which prey, predators and scavengers are involved. Effects of prey-taxis are manifested from the obtained results. By using the prey-taxis coefficient as the bifurcation parameter, the occurrence conditions of the steady state bifurcation is established. It is found that there is no steady state bifurcation without prey-taxis or with the attractive type prey-taxis, meanwhile, the steady state bifurcation will occur when the repulsive type prey-taxis is present. In the sequel, by employing the Crandall–Rabinowitz local bifurcation theory, the existence of the bifurcating solution and its stability are further explored. It is stable if the second-order perturbation term is less than zero, and unstable if greater than zero. The resulting nonconstant steady states are displayed through numerical simulation, which are in agreement with theoretical analysis. No steady state bifurcation or patterns will appear in simulation when the prey-taxis is absent.

Investigation of space-time dynamics of perturbed and unperturbed Chen-Lee-Liu equation: Unveiling bifurcations and chaotic structures

In this paper, the fractional space-time nonlinear Chen-Lee-Liu equation has been considered using various methods. The investigation of the transition from periodic to quasi-periodic behavior has been conducted using a saddle-node bifurcation approach. The paper reports the conditions of multi-dimensional bifurcations of dynamical solutions. Additionally, a direct algebraic method has been used to calculate various 2D and 3D solitonic structures of the equation, and an analysis of their accuracy and effectiveness has been conducted. Furthermore, the Galilean transformation has been used to convert the equation into a planar dynamical system, which is further utilized to obtain bifurcations and chaotic structures. Chaotic structures of perturbed dynamical system are observed and detected through chaos detecting tools such as 2D-phase portrait, 3D-phase portrait, time series analysis, multistability and Lyapunov exponents over time. Further, sensitivity behavior for a range of initial conditions, both perturbed and unperturbed. The results suggest that the investigated equation exhibits a higher degree of multi-stability.

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

Bifurcations of a single species model with spatial memory environment

This paper reports on the bifurcations of a single-species model with spatial memory environment. The occurrence conditions of the Hopf bifurcation and Turing bifurcation are given with/without the spatial memory effect. Especially, we show that the positive equilibrium enjoys the stability switches with spatial memory effect.

Novel Hopf Bifurcation Exploration and Control Strategies in the Fractional-Order FitzHugh–Nagumo Neural Model Incorporating Delay

In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order delay-coupled FitzHugh–Nagumo neural model is acquired. By designing a proper PDp controller, we can efficaciously control the stability domain and the time of emergence of the bifurcation phenomenon of the considered fractional delay-coupled FitzHugh–Nagumo neural model. By exploiting a reasonable hybrid controller, we can successfully adjust the stability domain and the bifurcation onset time of the involved fractional delay-coupled FitzHugh–Nagumo neural model. This study shows that when the delay crosses a critical value, a Hopf bifurcation will arise. When we adjust the control parameter, we can find other critical values to enlarge or narrow the stability domain of the fractional-order delay-coupled FitzHugh–Nagumo neural model. In order to check the correctness of the acquired outcomes of this article, we present some simulation outcomes via Matlab 7.0 software. The obtained theoretical fruits in this article have momentous theoretical significance in running and constructing networks.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Replicator-mutator dynamics with evolutionary public goods game-environmental feedbacks.

Feedback loops between strategies and the environment are commonly observed in socio-ecological, evolution-ecological, and psychology-economic systems. However, the impact of mutations in these feedback processes is often overlooked. This study proposes a novel model that integrates the public goods game with environmental feedback, considering the presence of mutations. In our model, the enhancement factor of the public goods game combines positive and negative incentives from the environment. By employing replicator-mutator (RM) equations, we provide an objective understanding of the system's evolutionary state, focusing on identifying conditions that foster cooperation and prevent the tragedy of the commons. Specifically, mutations play a crucial role in the RM dynamics, leading to the emergence of an oscillatory tragedy of the commons. By verifying the Hopf bifurcation condition, we establish the existence of a stable limit cycle, providing valuable insights into sustained oscillation strategies. Moreover, the feedback mechanism inherent in the public goods game model offers a fresh perspective on effectively addressing the classic dilemma of the tragedy of the commons.

103 A Bifurcation Model Predicts Intravascular Catheter Herniation When Using Transradial Access in Neuroendovascular Procedures

INTRODUCTION: The speed and safety of acute ischemic stroke interventions can be improved with catheter-based revascularization performed via the transradial access versus the traditional transfemoral access. A critical source of delay and procedural failure is a phenomenon known as herniation where the catheter system suddenly buckles and deviates from the intended endovascular pathway as the surgeon attempts to track one device over or through another around a tight bend. METHODS: Recent modeling work suggests that herniation is associated with a bifurcation in strain energy. We hypothesized that herniation could be predicted by modeling the strain energy stored in a catheter based on its flexural rigidity and the vascular anatomy. To test our hypothesis, we developed a computational model using finite element analysis and an experimental benchtop model using 3D-printed patient-specific aortic arches. Our mathematical model was validated in vivo using observational data derived from patient and porcine endovascular surgeries. RESULTS: Vascular geometry and flexural rigidity were used to derive the bifurcation condition and predict catheter stability versus instability (herniation). Catheter instability criterion was predictive of herniation in our computational model (sensitivity 100.0%, PPV 93.5%, NPV 100.0%); experimental ex vivo model (sensitivity 92.3%, PPV 89.6%, NPV 90.0%); patient in vivo model (sensitivity 100.0%, PPV 80.0%, NPV 100.0%); and porcine in vivo model (sensitivity 100%, PPV 100%, NPV 100%). CONCLUSIONS: We validated a predictive model of catheter herniation based upon the hypothesis that bifurcation in strain energy drive the phenomenon. Key factors in the model are the curvature of the vascular pathway and the flexural rigidity of the coaxial devices. Results suggest that strategic selection of catheters and wires based on their known flexural rigidity can be used to prevent the bifurcation in strain energy that causes herniation.

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

This manuscript delves into the examination of the stochastic fractional derivative of Drinfel’d-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena.

Existence conditions for bifurcations of homoclinic orbits in a railway wheelset model

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov’s method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model.

Dynamically or Kinetically Controlled? Computational Study of the Mechanisms of Electrophilic Aminoalkenylation of Heteroaromatics with Keteniminium Ions.

Quantum chemical calculations and molecular dynamics simulations were applied to study the electrophilic aminoalkenylation of heteroaromatics with keniminium ions. Post-transition state bifurcation (PTSB) was found in the electrophilic addition step for the aminoalkenylation of pyrroles and indoles, and the selectivity for these reactions was dynamically controlled. However, the aminoalkenylation of furan was kinetically controlled because no apparent PTSB was found in the electrophilic addition step. The substituents on the keteniminium ions can also affect the dynamic results for the aminoalkenylations to pyrroles: the C2-aminoalkenylated product is much more favored over the C3-aminoalkenylated product for keteniminium ions with electron-donating substituents, while the product ratio (C2 product/C3 product) decreased when stronger electron-withdrawing substituents were applied.

Two-Parameter Bifurcations and Hidden Attractors in a Class of 3D Linear Filippov Systems

We take into consideration two different kinds of two-parameter bifurcations in a class of 3D linear Filippov systems, namely pseudo-Bautin bifurcation and boundary equilibrium bifurcations for two scenarios. The bifurcation conditions for generating rich dynamic behaviors are established. The main objective is to investigate the effects of two parameters interacting simultaneously on a variety of dynamic phenomena. In order to analyze the pseudo-Bautin bifurcation, we build the Poincaré map and analyze the number of fixed points whose types are related to the crossing limit cycles. In order to analyze boundary equilibrium bifurcations for two scenarios, we perform an analysis on the existence and admissibility of equilibria. Besides, a comprehensive investigation on hidden attractors induced by boundary equilibrium bifurcations is conducted. The novelty resides in overcoming the constraints of previous studies that solely take into account the dynamics of individual parameter variations. We innovatively characterize the two-parameter bifurcation mechanism of a new class of Filippov systems, and qualitatively demonstrate the coexistence of hidden attractor and stable pseudo-equilibrium.

Surface wrinkling of a film coated to a graded substrate

We study the surface wrinkling of a stiff thin elastic film bonded to a compliant graded elastic substrate subject to compressive stress generated either by compression or growth of the bilayer. Our aim is to clarify the influence of the modulus gradient on the onset and surface pattern in this bilayer. Within the framework of finite elasticity, an exact bifurcation condition is obtained using the Stroh formulation and the surface impedance matrix method. Further analytical progress is made by focusing on the case of short wavelength limit for which the Wentzel–Kramers–Brillouin method can be used to resolve the eigenvalue problem of ordinary differential equations with variable coefficients. An explicit bifurcation condition is obtained from which the critical buckling load and the critical wavelength are derived asymptotically. In particular, we consider two distinct situations depending on the ratio β of the shear modulus at the substrate surface to that at infinity. If β is of O(1) or small, the parameters related to modulus gradient all appear in the higher-order terms and play an insignificant role in the bifurcation. In that case, it is the modulus ratio between the film and substrate surface that governs the onset of surface wrinkling. If, however, β≫1, the modulus gradient affects the critical condition through leading-order terms. Through our analysis we unravel the influence of different material and geometric parameters, including the modulus gradient, on the bifurcation threshold and the associated wavelength which can be of importance in many biological and technological settings.

Dynamics of neural fields with exponential temporal kernel.

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

Optimal control of red palm weevil model incorporating sterile insect technique, mechanical injection, and pheromone traps

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods.

Modeling the light-powered self-rotation of a liquid crystal elastomer fiber-based engine.

Self-oscillating systems possess the ability to convert ambient energy directly into mechanical work, and new types of self-oscillating systems are worth designing for practical applications in energy harvesters, engines and actuators. Taking inspiration from the four-stroke engine. A concept for a self-rotating engine is presented on the basis of photothermally responsive materials, consisting of a liquid crystal elastomer (LCE) fiber, a hinge and a turnplate, which can self-rotate under steady illumination. Based on the photo-thermal-mechanical model, a nonlinear theoretical model of the LCE-based engine under steady illumination is proposed to investigate its self-rotating behaviors. Numerical calculations reveal that the LCE-based engine experiences a supercritical Hopf bifurcation between the static regime and the self-rotation regime. The self-rotation of the LCE-based engine originates from the photothermally driven strain of the LCE fiber in illumination, and its continuous periodic motion is sustained by the correlation between photothermal energy and damping dissipation. The Hopf bifurcation conditions are also explored in detail, as well as the vital system parameters affecting self-rotation frequency. Compared to the abundant existing self-oscillating systems, this conceptual self-rotating LCE-based engine stands out due to its simple and lightweight structure, customizable dimensions and high speed, and it is expected to offer a broader range of design concepts applicable to soft robotics, energy harvesters, medical instruments, and so on.

Exploring static responses, mode transitions, and feasible tunability of Kagome-based flexible mechanical metamaterials

We consider the static responses of the uniaxially compressed flexible mechanical metamaterials, which integrate soft hinges and rigid bodies, constructed from the Kagome lattice. First, we experimentally find that the static responses of the regular-Kagome-based structure significantly differ from those of the twisted-Kagome-based structure with a very small twisting angle. Following this, we establish a theoretical model, which is combined with the deflated continuation algorithm for bifurcation analysis, to delve into the static responses and potential bifurcation behavior of these structures. We then experimentally and numerically investigate the mode transitions between various stable modes, and systematically study the role of the twisting angle in the occurrence of bifurcations. Our findings indicate that mode transitions can be feasibly realized according to the calculated bifurcation diagrams. They also provide direct evidence of an interesting physical mechanism that the transition between different stable deformation states can occur through multiple pathways, possibly passing through unstable deformation states. Moreover, by introducing the twisting angle or stiff defects, the response of the structure can be modulated, thereby enhancing the programmability and tunability of Kagome-based flexible mechanical metamaterials. Our research also reveals that novel phenomena such as meta-beam buckling and multi-phase dominated deformations can be triggered within these flexible structures, which offers valuable insights for future metamaterial designs and applications.

A light-powered self-rotating liquid crystal elastomer drill

Self-oscillating systems can directly convert ambient energy to mechanical work, and new type self-oscillating systems are worth designing for applications in energy harvesters, engines, and actuators. Taking inspiration from the hand drill, we have developed a novel self-rotating drill system, which is consist of a turnplate and a liquid crystal elastomer (LCE) fiber under steady illumination. To investigate the self-rotating behaviors of the LCE drill, we have proposed a nonlinear theoretical model of the LCE drill under steady illumination based on the well-established dynamic LCE model. Numerical calculation reveals that the LCE drill can undergo a supercritical Hopf bifurcation between the static regime and the self-rotation regime. The self-rotation of drill originates from the contraction of winding portion of LCE fiber in illumination at winding state, and its continuous periodic motion is sustained by the interrelation between light energy and damping dissipation. The Hopf bifurcation conditions are also investigated in detail, as well as the vital system parameters affecting its frequency and amplitude. In contrast to the abundant existing self-oscillating systems, this self-rotating drill stands out due to its simple and lightweight structure, customizable dimensions, and high speed, and thus facilitates the design of compact and integrated systems, enhancing their applicability in microdevices and systems. This bears great significance in fields like micro-robotics, micro-sensors, and medical instruments, enabling the realization of smaller and higher-performance devices.

A robust Study of the Treatment Delay Effect on the Dynamics of Epidemic Disease

A general treatment function with an epidemic model that involves the delay in the treatment period has been proposed and studied in this work. This model contains two compartments, namely susceptible denoted by and infected denoted by . The existence of all the fixed points has been determined. The system has two equilibrium points, namely the uninfected equilibrium point (UIEP) and the endemic equilibrium point (EEP). The conditions for local stability and Hopf bifurcation have been discussed. The stability of the periodic solutions and the direction of the Hopf bifurcation properties have been studied analytically and numerically.

Light-fueled self-fluttering aircraft with a liquid crystal elastomer-based engine

In response to external stimuli, active materials are able to alter their shape or motion. Nonetheless, considerable challenges remain in the effective realization and control of active machines. Conventional control methods, involving sensor-based and external device control as well as electronic control, are intricate and tough to put into practice. Conversely, self-oscillators built upon active materials exhibit remarkable properties, including steady motion and straightforward regulation. Drawing inspiration from flying organisms, this paper puts forward an innovative light-fueled self-fluttering aircraft with a liquid crystal elastomer (LCE) engine, capable of sustained flight in the presence of steady illumination. The governing equations of the self-fluttering aircraft are derived in accordance with the well-known dynamic LCE model. The system is capable of experiencing a supercritical Hopf bifurcation, transitioning from a stationary pattern to a self-fluttering pattern, and the energy compensation mechanism of the self-fluttering pattern is revealed as well. Moreover, the quantitative investigation covers how distinct system parameters influence the Hopf bifurcation conditions and the self-fluttering amplitude and frequency, as well as the average flight velocity. With the benefits of customizable size, electronics-free and dependence on ambient energy, the proposed system may generate fresh insights into the practical use of active materials-based self-oscillators in micro actuators, soft robotics, bionic devices, and energy harvesting.