Type Of Hopf Bifurcation Research Articles

Bifurcation analysis of modified density gradient continuum traffic model

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.

Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse

The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number mathcal{R}_{0} is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

A quantitative mathematical model of neurons should not only include enough details to consider the dynamics of single neurons but also minimize the complexity of the model so that the model calculation is convenient. The two-dimensional Prescott model provides a good compromise between the authenticity and computational efficiency of a neuron. The dynamic characteristics of the Prescott model under external electrical stimulation are studied by combining analytical and numerical methods in this paper. Through the analysis of the equilibrium point distribution, the influence of model parameters and external stimulus on the dynamic characteristics is described. The occurrence conditions and the type of Hopf bifurcation in the Prescott model are analyzed, and the analytical determination formula of the Hopf bifurcation type in the neuron model is obtained. Washout filter control is used to change the Hopf bifurcation type, so that the subcritical Hopf bifurcation transforms to supercritical Hopf bifurcation, so as to realize the change of the dynamic characteristics of the model.

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

Nutations in growing plant shoots as a morphoelastic flutter instability.

Growing plant shoots exhibit spontaneous oscillations that Darwin observed, and termed 'circumnutations'. Recently, they have received renewed attention for the design and optimal actuation of bioinspired robotic devices. We discuss a possible interpretation of these spontaneous oscillations as a Hopf-type bifurcation in a growing morphoelastic rod. Using a three-dimensional model and numerical simulations, we analyse the salient features of this flutter-like phenomenon (e.g. the characteristic period of the oscillations) and their dependence on the model details (in particular, the impact of choosing different growth models) finding that, overall, these features are robust with respect to changes in the details of the growth model adopted. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Bifurcation analysis of a railway wheelset with nonlinear wheel–rail contact

Stability is a key factor for the operation safety of railway vehicles, while current work employs linearized and simplified wheel/rail contact to study the bifurcation mechanism and assess the stability. To study the stability and bifurcation characters under real nonlinear wheel/rail contact, a fully parameterized nonlinear railway vehicle wheelset model is built. In modeling, the geometry nonlinearities of wheel and rail profiles come from field measurements, including the rolling radius, contact angle, and curvatures, etc. Firstly, four flange force models and their effects on the stability bifurcations are compared. It shows that an exponent fitting is more proper than a quintic polynomial one to simulate the flange and works well without changing the Hopf bifurcation type. Then the effects of each term of the nonlinear geometry of wheel/rail contact on the Hopf bifurcation and limit circle bifurcation are discussed. Both the linear term and nonlinear term of rolling radius have a significant influence on Hopf bifurcation and limit point of circle (LPC) bifurcation. The linear critical speed (Hopf bifurcation point) and the nonlinear critical speed (LPC bifurcation point) changes times while within the calculated range of the linear term of the rolling radius. Its nonlinear term changes the bifurcation type and the nonlinear critical speed almost by half. The linear term of the contact angle and the radius of curvature of the wheel and rail profiles should be taken into consideration since they can change both the bifurcation point and type, while the cubic term can be ignored. Furtherly, the field measured wheel profiles for several running mileages are employed to examine the real geometry nonlinearities and the corresponding Hopf bifurcation behavior. The result shows that a larger suspension stiffness would increase the running stability under wheel wear.

Spatiotemporal dynamics for a diffusive mussel\u2013algae model near a Hopf bifurcation point

In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

Nonlinear dynamic analysis of supercritical and subcritical Hopf bifurcations in gas foil bearing-rotor systems

The Hopf bifurcation behavior is an important issue for the nonlinear dynamic analysis of gas foil bearing (GFB)-rotor systems. However, there is a lack of detailed study on different types of Hopf bifurcation and their corresponding characteristics for GFB-rotor systems. This paper is intended to provide a clear and systematic insight into the nonlinear dynamic characteristics of GFB-rotor systems with a supercritical or a subcritical Hopf bifurcation. The onset speed (OS) of instability (i.e., the bifurcation point) for the system is calculated by the linear stability analysis. The periodic solution of the system before or after the bifurcation point is obtained by the shooting method, and its stability is assessed by the Floquet multipliers. The shock stability and the unbalanced response characteristics of the GFB-rotor system with a supercritical or a subcritical Hopf bifurcation are presented. A GFB-rotor system with a supercritical Hopf bifurcation shows better dynamic characteristics than a system with a subcritical Hopf bifurcation. The parameter analysis reveals that the aspect ratio and the foil stiffness of the GFBs have obvious effects on the Hopf bifurcation type, while the loss factor has a relatively small effect. It is remarkable that although a lower foil stiffness increases the OS of instability, the actual speed limit would probably decrease as the Hopf bifurcation changes from a supercritical to a subcritical type. This can contribute to an understanding of the necessity of studies on actual available operating speed based on nonlinear analysis rather than conventional linear analysis for the bearing design.

Numerical solution of hybrid mathematical model of dengue transmission with relapse and memory via Adam–Bashforth–Moulton predictor-corrector scheme

• A mathematical model of the dengue transmission process with memory and relapse between host-to-vector and vice-versa is presented. • The dengue model contains a major parameter namely relapse and memory. • The fractional order α shows the memory behaviour. Taking α = 0.5 gives threshold value R 0 < 1 which is no longer a sufficient condition but necessary for die out of the disease. • It is found the transmission memory increases α approaches to zero, the chance of dengue fever persistence in the host will go on the increase. In this paper, a novel hybrid compartmental model of the dengue transmission process is proposed and studied with memory and relapse between host-to-vector and vice versa. The memory and correlated learning system in the dengue models by using the fractional differential operators such as Riemann–Liouville and Caputo has been a fascinating area of research. A threshold parameter which is called basic reproduction number R 0 is investigated and calculated by next-generation technique. It’s also shows that if basic reproduction number R 0 < 1 , the disease-free equilibrium(DFE) is locally asymptotically stable(LAS) and if R 0 > 1 then, the DFE is unstable. It’s also found that the fractional-order α also depends upon R 0 . Therefore, if fractional-order α = 1 and R 0 > 1 , then dengue fever model doesn’t show Hopf-type bifurcation. Further, it’s also worth mentioning that although R 0 < 1 , the DFE E 0 may not be always stable but it’s necessary and the model shows a Hopf-type bifurcation. We employed the scheme of Adams–Bashforth–Moulton predictor-corrector to find an approximate the solution of the dengue model. The numerical simulation is carried out to validate the analytic solution.

Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Journal Bearing Considering Journal Angular Motion

Abstract The effect of bearing length to diameter (L/D) ratio and large disk position on nonlinear vibration of a flexible rotor-bearing system was investigated. The rotor consisted of a shaft modeled by one-dimensional finite elements (FEs) and disks. It was supported by a self-aligning ball bearing (BB) and an axial-groove journal bearing (JB). Two JB's L/D ratios of 0.4 and 0.6, two large disk positions of 340 and 575 mm measured from the BB, and two bearing models that consider both journal's lateral and angular motion (model A) and consider only journal's lateral motion (model B) were investigated. The degrees-of-freedom (DOF) of the equation of motion (EOM) were reduced to those of the boundary DOF by real mode component mode synthesis (CMS) that retains only the first forward and backward modes of the internal DOF. Shooting method and Floquet multiplier analysis were applied to the reduced EOM to obtain limit cycles and their stability, which indicates Hopf bifurcation type. Numerical results indicated that supercritical bifurcation only occurred in the case of L/D = 0.4 and large disk position 575 mm for both bearing models. Otherwise, the subcritical bifurcation occurred except the case of L/D = 0.6 with the large disk position 575 mm that supercritical bifurcation occurred if model B was used. The experiment with the same parameters used in the calculation was conducted as verification. The experimental results showed the same bifurcation type as calculated by using model A.

Unexpected Behaviors in a Single Mesh Josephson Junction Based Self-Reproducing Autonomous System

In the literature, existing Josephson junction based oscillators are mostly driven by external sources. Knowing the different limits of the external driven systems, we propose in this work a new autonomous one that exhibits the unusual and striking multiple phenomena among which coexist the multiple hidden attractors in self-reproducing process under the effect of initial conditions. The eight-term autonomous chaotic system has a single nonlinearity of sinusoidal type acting on only one of the state variables. A priori, the simplicity of the system does not predict the richness of its dynamics. We also find that a limit cycle attractor widens to a parameter controlling coexisting multiple-scroll attractors through the splitting and the inverse splitting of periods. Multiple types of bifurcations are found including period-doubling and period-splitting (antimonotonicity) sequences to chaos, crisis and Hopf type bifurcation. To the best of our knowledge, some of these interesting phenomena have not yet been reported in similar class of autonomous Josephson junction based circuits. Moreover, analytical investigations based on the Hopf theory analysis lead to the expressions that determine the direction of appearance of the Hopf bifurcation, confirming the existence and determining the stability of bifurcating periodic solutions. To observe this latter bifurcation and to illustrate the theoretical analysis, numerical simulations are performed. Chaos can be easily controlled by the frequency of the linear oscillator, the superconducting junction current, as well as the gain of the amplifier or circuit component values. The circuit and Field Programmable Gate Arrays (FPGA)-based implementation of the system are presented as well.

Enhancing flutter stability in nanocomposite thin panels by harnessing CNT/polymer dissipation

Numerical and analytical investigations into the aeroelastic response of thin nanocomposite panels are discussed. The mechanical data of the nanocomposite plates with different weight fractions of randomly oriented carbon nanotubes (CNT) are obtained via an extensive experimental campaign based on dynamic mechanical analysis (DMA). Panel flutter induced by high supersonic flows is studied through both linear and nonlinear dynamic analyses aimed to investigate the effect of the CNT weight fraction onto the flutter and post-flutter condition. To this end, multiple scales perturbation analysis is carried out to characterize the flutter boundaries and the type of Hopf bifurcation at flutter. It is shown that a few percent CNT weight fraction, while entailing negligible weight penalty, can considerably increase the flutter speed as well as reduce the limit cycle oscillations amplitude.

К проблеме устойчивости периодического решения в условиях бифуркации Хопфа

Данная работа посвящена проблеме устойчивости малого периодического решения нормальной автономной системы обыкновенных дифференциальных уравнений. При исследовании устойчивости периодического решения автономной системы естественно анализировать локальную динамику пересечений возмущенных траекторий с ортогональными сечениями соответствующего цикла. Путем введения специальной системы координат, в которой одна из осей направлена по касательной к траектории периодического решения, задача об орбитальной устойчивости периодического решения сводится к задаче об устойчивости по Ляпунову нулевого решения вспомогательной системы с периодической по t правой частью. Для вспомогательной системы, размерность которой на единицу меньше размерности исходной системы, в линейном приближении вопрос об устойчивости нулевого решения сводится к оценке мультипликаторов матрицы монодромии. Таким образом, по теореме Андронова — Витта реализуется классический подход к исследованию орбитальной устойчивости периодического решения. При этом имеет место некритический случай орбитальной устойчивости. Такой подход традиционно используется и в условиях бифуркации типа Хопфа для систем с параметром. В данной работе для автономной системы с параметром получены условия бифуркации малого решения, период которого близок к периоду решений соответствующей линейной однородной системы. Сформулировано определение свойства орбитальной устойчивости по параметру, согласно которому возмущенные правые полутраектории сколь угодно близки к исследуемому циклу не только за счет малости возмущений начальных значений, но и за счет малости параметра. При этом использована идея ослабления требований определения устойчивости ляпуновского типа, предложенная М.М. Хапаевым. Свойство орбитальной устойчивости по параметру может иметь место и при наличии орбитальной неустойчивости исследуемого цикла в классическом смысле. Для исследования орбитальной устойчивости малого периодического решения по параметру использовано нелинейное приближение упомянутой выше вспомогательной системы возмущенных движений.

Stability and Hopf bifurcations in rotor-bearing-foundation systems of turbines and generators

The paper investigates the type of Hopf bifurcations existing in slender rotor systems of turbines and generators with respect to the design of bearings, foundations, and rotors. Two bearing profiles applied mostly in such rotor systems, the lemon bore and the partial arc bearing are considered in the analysis with nonlinear impedance. Simplistic models of single degree of freedom per direction are considered for the bearing pedestals. Part of the paper investigates the bifurcation type in simple Jeffcott rotor mounted on nonlinear lemon bore and partial arc bearings with resilient pedestals, aiming to raise design concerns which influence the type of bifurcation. The results highlight the need for nonlinear stability assessment in the standard design of turbines and generators.

Mode-locking dynamics of corticothalamic system responses to periodic external stimuli

A physiologically based continuum model of the corticothalamic system is applied to explore mode-locking dynamics of ongoing cortical activity subject to periodic drive. The ongoing cortical activity is poised near a Hopf bifurcation. Time-dependent center manifold reduction shows that the nonlinear dynamics of driven corticothalamic system with delays can be described by an inhomogeneous low-dimensional equation near the critical point of the Hopf bifurcation. This driven normal form captures the main features of experimentally observed nonlinear interactions between brain activity and periodic stimuli. The effects of underlying physiological parameters on the type of Hopf bifurcation and onset dynamics are then studied, which allows the determination of phase-locked structures for various regions in the parameter space of the drive. It is shown that the system can adjust its oscillation regime to the frequency content of stimuli. The key features of entrainment of the alpha rhythm to periodic stimuli, including harmonic and subharmonic generation, are also explored. The theoretical predictions of these nonlinear responses are confirmed by numerical simulations. Further, photosensitive seizures are predicted for physiologically based model parameters arising from mode locking to the frequency of periodic stimulus, and their harmonics due to higher order synchronization states. These results indicate that the approach used here provides a powerful framework for the study of nonlinear interactions between brain activity and flicker by incorporating periodic driving into the corticothalamic system.

Hopf bifurcation of forced Chen system and its stability via adaptive control with arbitrary parameters

In this article, forced Chen system is analyzed for nonlinear dynamical behavior. The chaotic behavior of forced Chen system is verified by phase portraits and sensitivity dependence of system upon initial condition. Hopf bifurcation for the complex system is derived and theorem of first Lyapunov coefficient is used to investigate the type of Hopf bifurcation. It is further shown that Hopf bifurcation exists only on two equilibrium points for the proposed chaotic model. In addition, an adaptive control technique is used to control unpredictable behavior for the forced Chen system. Global stability is achieved by constructing an energy type function through Lyapunov theory, whereas its error dynamics is used to synchronize two identical forced Chen systems. Numerical simulation results are used to validate analytical results given in this article and also to demonstrate effectiveness of the considered chaotic system.

On nuclear-coupled thermal-hydraulic instability analysis of Super-Critical-Light-Water-cooled-reactor (SCLWR)

The nonlinear stability analysis of a supercritical light water reactor (SCLWR) is presented using a nuclear-coupled thermal-hydraulic reduced-order model. The analytical model is developed by coupling 1. the point-kinetics equations with one group of delayed neutrons, 2. the fuel heat transfer and 3. a 1-D reduced order model which represents the heat absorption phenomenon during the coolant flow. Unlike the existing studies, which are limited to linear stability analysis, the primary objective of the work is to present the detailed nonlinear dynamics of the SCLWR system. The said goal is achieved at two levels. The first level is the linear stability analysis wherein the linear stability boundaries are shown in two sets of parameter space namely the two intrinsic reactivity feedbacks (Doppler reactivity feedback and density reactivity feedback) and the pseudo-phase-change number and pseudo subcooling number. The parametric effects show the sensitivity of the linear stability boundaries with the system parameters. In the second level, to discuss the nonlinear characteristics of the system, two types of Hopf bifurcations (subcritical and supercritical) are studied with the help of first Lyapunov coefficients of the system. Multiple numerical simulations are performed to verify the resultant limit cycle behavior associated with these bifurcations. Moreover, the occurrence of the generalized Hopf bifurcation is shown which represents the bifurcation between the subcritical Hopf and the supercritical Hopf regions. Further, inside the stable region of the linear stability boundary, the saddle-node bifurcation is found which represents the location of the turning point and the threshold of the globally stable region of the system.

Effect of throttling orifice head loss on dynamic behavior of hydro-turbine governing system with air cushion surge chamber

This paper aims to study the effect of throttling orifice head loss on the dynamic behavior of the hydro-turbine governing system with air cushion surge chamber. Firstly, a nonlinear mathematical model of hydro-turbine governing system considering the nonlinear head loss of throttling orifice is established. Then, the nonlinear dynamic behavior of the hydro-turbine governing system is investigated by using the Hopf bifurcation theory and direct numerical integration. The stability and dynamic response of the system are detailed described. Finally, the effect mechanism of throttling orifice head loss on the dynamic behavior of system is revealed based on the comparison between the cases with nonlinear head loss and no head loss. The results indicate that the mathematical model considering the nonlinear head loss of throttling orifice for the hydro-turbine governing system is a fifth-order nonlinear state equation. The system goes through Hopf bifurcation at the bifurcation points and the type of Hopf bifurcation is supercritical. The domain at the bottom of the bifurcation line is the stable domain. The throttling orifice cannot affect the system stability through its head loss. The throttling orifice can affect the system dynamic response through its head loss. The throttling orifice head loss is favorable for the damping of the water level oscillation in the surge chamber.

Nonlinear Torsional Vibration Analysis and Control of Semidirect Electromechanical Coupling Transmission System in Shearer

The nonlinear torsional vibration and instability oscillation caused by nonlinear damping in the shearer electromechanical coupling cutting transmission system in shearer driven by the permanent magnet synchronous motor (PMSM) is investigated in this paper. The electromechanical coupling transmission system in the shearer is equivalent to a concentrated mass model for the purpose of establishing the system dynamic model by the Lagrange–Maxwell equation. Then, the Routh–Hurwitz criterion is used to determine the torsional vibration critical point and stability region for the Hopf bifurcation for the cutting transmission system. According to the Routh–Hurwitz stability criterion, the Hopf bifurcation type and the effect of the supercritical Hopf bifurcation in the torsional vibration of the cutting transmission system are analyzed. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s large vibration amplitude and unstable oscillation. In addition, the influences of the linear gain and nonlinear gain on the bifurcation point and the limit cycle amplitude are discussed. Finally, the numerical simulation results indicate the effectiveness of the designed controller. The research achievements can provide a theoretical basis for design or optimize the cutting transmission system of high‐reliability shearer driven by PMSM.

Non-linear stability analysis of pressure drop oscillations in a heated channel

A study of non-linear stability analysis using co-dimension two (i.e., two free parameters being varied) bifurcations related to pressure drop oscillations (PDO) in a heated channel has been carried out in this paper. In the existing literature, in the context of PDO, mostly linear stability analysis is done. A few works on non-linear stability analysis are available; however, only co-dimension one bifurcation studies of PDO have been carried out in these works. However, in practice, since the inlet temperature of the coolant is also an independent operating parameter, both inlet temperature, and inlet mass flow rate need to be considered for stability analysis. It is also noted that the existing plethora of studies on PDO is limited to the prediction of supercritical Hopf bifurcation only. However, in the current study, two types of Hopf bifurcations have been identified namely subcritical and supercritical. A subcritical Hopf bifurcation exhibits unstable limit cycles, which is a signature of the existence of unstable solutions for slightly larger perturbations even in the linearly stable region. The existence of unstable solutions indicates that linear stability analysis is not sufficient to identify the overall stability behavior. Also, a generalized Hopf point on the stability boundary has been identified which denotes a boundary between subcritical and supercritical Hopf bifurcation. Furthermore, numerical simulations are carried out in different regions (both stable and unstable) of the parameter space to understand the non-linear phenomena of the system. Moreover, the bifurcations are explained in terms of the interaction of the external and internal characteristic curves of the system. The large and small amplitude cycles are shown in the characteristic curves of the system.