Supercritical Hopf Bifurcation Research Articles

Bifurcation analysis for the coexistence in a Gause-type four-species food web model with general functional responses

<p>The dynamics of an ordinary differential equations (ODEs) system modelling the interaction of four species (one prey or resource population, two mesopredator populations, and one super-predator population) was analyzed. It was assumed that the functional responses for each interaction were general. We showed parameter conditions that ensured that the differential system underwent a supercritical Hopf bifurcation or a Bogdanov-Takens bifurcation, from which the coexistence of the four species was guaranteed. In addition, the results were illustrated by several applications, where the prey had a logistic growth rate. For the interaction of the mesopredators and prey, we considered classical Holling-type functional responses, and for the rest of the interactions, we proposed certain generalized functional responses similar to the well-known "Beddington-DeAngelis" or "Crowley-Martin" functional responses. At the end, some numerical simulations were given.</p>

Acceleration sensing based on the bifurcation dynamics of parametrically excited mode-localized resonators

A parametrically excited mode-localized accelerometer is designed using the bifurcation phenomenon to improve the robustness of the fluctuation of the driving voltage and damping while maintaining high sensitivity. A dynamic multi-physics model was established while considering both mechanical and electrostatic nonlinearities. The equation was solved by method of multiple scales and verified by harmonic balanced method coupled with the asymptotic numerical method. Two types of bifurcation exist in amplitude frequency response, namely Saddle-Node bifurcation and Supercritical Hopf bifurcation. By introducing Saddle-Node bifurcation, the response amplitude and measurement range can be improved by 100% and 1000%, respectively, while the sensitivity of the amplitude ratio is about 2 orders of magnitude higher than that based on the frequency ratio. At the Supercritical Hopf bifurcation point, a small acceleration will change the topological structure from Supercritical Hopf to Saddle-Node bifurcation. The variation in the amplitude ratio of the Supercritical Hopf point with acceleration is similar to the sign function, which leads to an extremely high sensitivity of 10000%/g in a dynamic range of ±0.001 g. Moreover, the Supercritical Hopf bifurcation point is not affected by the amplitude of the excitation voltage and damping coefficient, which provides a new method for improving the sensing robustness. Ethical Compliance: All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. Conflict of Interest declaration: The authors declare that they have NO affiliations with or involvement in any organization or entity with any financial interest in the subject matter or materials discussed in this manuscript.

Dynamics of a silicone oil drop submerged in a stratified ethanol-water bath.

We analyze numerically the Marangoni flow around an immiscible droplet submerged in a stably stratified mixture of ethanol and water. The linear stability analysis shows that the base flow undergoes a supercritical Hopf bifurcation that leads to oscillations. The theoretical prediction for the critical droplet radius is consistent with previous experimental results. Ethanol diffusion in water is critical in the flow stability for both low and high droplet viscosity. Direct numerical simulations of the nonlinear oscillatory flow show that the frequency of those oscillations approximately equals that of the critical eigenmode. The nonlinear convective term of the ethanol diffusive-convective transport equationfixes the amplitude of the droplet oscillations. The viscous dissipation associated with the Marangoni flow inside the droplet considerably reduces the oscillation.

Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional de Holling tipo II

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

Self-swaying of oblique bending cantilevers under steady illumination

Self-sustained systems are able to absorb energy from steady external environment to maintain their own motion without additional energy or control. This feature has facilitated their widespread use in micromachines, actuatosr and soft robotics. To the address the relative complexity and difficulty in fabrication of the current self-sustained systems, this paper constructs a novel self-oscillating liquid crystal elastomer (LCE) fiber-beam system, which can sway continuously and periodically under steady illumination. It consists of a LCE fiber, two oblique bending cantilevers and two masses. In conjunction with the well-established LCE dynamic model and beam theory, the governing equations of the self-swaying oblique bending cantilevers system are established in this paper. The self-swaying process of the oblique bending cantilevers system under steady illumination is described and its motion mechanism is explained in detail. Numerical results show that the system can undergo supercritical Hopf bifurcation between the static regime and self-swaying regime. The effects of system parameters on the self-swaying amplitude and frequency are discussed quantitatively. The results of this paper can deepen the understanding of self-swaying and provide guidance for autonomous robots, energy harvesters, sensors and bionic instruments.

An Investigation of Dynamical Behavior of a Wing Model

AbstractBifurcations of equilibria of a wing model have been investigated in this paper. It is shown that the quintic nonlinear terms in the pitch and the plunge coordinate have affected the bifurcation structure of nontrivial equilibria in different degree. In contrast with the quintic stiffening parameter in plunge, the quintic parameter in pitch has a relatively significant effect, which will affect the number, position and stability of nontrivial equilibria. Therein two pairs of nontrivial equilibria with opposite stability coexist or disappear by two fold bifurcations. If the freestream velocity has been taken as a continuation parameter, it will affect the bifurcation structure of all the equilibria, including the trivial and the nontrivial. Wherein the equilibria vary from a trivial to two nontrivial ones by a pitchfork bifurcation. Then one of nontrivial equilibria experiences a supercritical Hopf bifurcation and the bifurcated limit cycles form an ellipsoidal structure with the limit cycles bifurcated from the trivial equilibrium.

Cross-stream oscillations in the granular flow through a vertical channel

The gravity flow of a granular material between two vertical walls separated by a width $2W$ is simulated using the discrete element method (DEM). Periodic boundary conditions are applied in the flow (vertical) and the other horizontal directions. The mass flow rate is controlled by specifying the average solids fraction $\bar {\phi }$ , the ratio of the volume of the particles to the volume of the channel. A steady fully developed state can be achieved for a narrow range of $\bar {\phi }$ , $\bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$ , and the material is in free fall for $\bar {\phi } < \bar {\phi }_{min}$ . For an intermediate range of $\bar {\phi }$ ( $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ ), there are oscillations in the horizontal coordinate of the centre of mass, velocity components and stress. As $\bar {\phi }$ decreases in the range $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ , the amplitude of the oscillations increases proportional to $\sqrt {\bar {\phi }_{cr} - \bar {\phi }}$ and the frequency appears to tend to a non-zero value as $\bar {\phi } \rightarrow \bar {\phi }_{cr}$ , indicating a supercritical Hopf bifurcation. The relation between the dominant frequency and the higher harmonics of the position, velocity and stress fluctuations are explained using the momentum balance. It is found that dissipation in the inter-particle and particle–wall contacts is critical for the presence of an oscillatory state.

Self-propulsion of an elliptical phoretic disk emitting solute uniformly

Self-propulsion of chemically active droplets and phoretic disks has been studied widely; however, most research overlooks the influence of disk shape on swimming dynamics. Inspired by experimentally observed prolate composite droplets and elliptical camphor disks, we employ simulations to investigate the phoretic dynamics of an elliptical disk that emits solutes uniformly in the creeping flow regime. By varying the disk's eccentricity $e$ and the Péclet number $Pe$ , we distinguish five disk behaviours: stationary, steady, orbiting, periodic and chaotic. We perform a linear stability analysis (LSA) to predict the onset of instability and the most unstable eigenmode when a stationary disk transitions spontaneously to steady self-propulsion. In addition to the LSA, we use an alternative approach to determine the perturbation growth rate, illustrating the competing roles of advection and diffusion. The steady motion features a transition from a puller-type to a neutral-type swimmer as $Pe$ increases, which occurs as a bimodal concentration profile at the disk surface shifts to a polarized solute distribution, driven by convective solute transport. An elliptical disk achieves an orbiting motion through a chiral symmetry-breaking instability, wherein it repeatedly follows a circular path while simultaneously rotating. The periodic swinging motion, emerging from a steady motion via a supercritical Hopf bifurcation, is characterized by a wave-like trajectory. We uncover a transition from normal diffusion to superdiffusion as eccentricity $e$ increases, corresponding to a random-walking circular disk and a ballistically swimming elliptical counterpart, respectively.

Weakly nonlinear bifurcation behaviour at the onset of instability in horizontal convection using an advective buoyancy approximation

Horizontal free convection is studied under an extended buoyancy approximation constructed to capture centrifugal effects in strongly rotating regions within buoyancy driven flows. Under this approximation, the Gay-Lussac parameter (Ga) arises in the momentum equation to characterise non-Boussinesq behaviour. The scaling of average Nusselt numbers (Nuavg) are determined, as is its dependence on Ga. Increasing Ga is found to decrease Nuavg when convection dominates the flow, while no influence is detected in the low Rayleigh number regime dominated by conduction. The influence of Ga on the onset of the time-dependent regime is investigated and the critical Rayleigh number corresponding to different Ga are determined. Results demonstrate that increasing Ga stabilises the flow, delaying transition to the time-dependent regime. An Orr–Sommerfeld type stability analysis is performed to determine the local stability at different horizontal stations. A zone of convective instability is first detected, growing from the hot end of the enclosure, at Rayleigh numbers three orders of magnitude lower than the predicted global stability threshold at the maximum investigated Ga value. Ga is found to play a key role in determining the preferred orientation of instability roll structures, with increasing Ga being accompanied by a bias toward transverse-roll instabilities over longitudinal rolls. The Stuart—Landau model is used to reveal that the non-linear characteristics of this unsteady transition are consistent with a supercritical Hopf bifurcation.

Nonlinear analysis of coupled neutronic-thermohydraulic stability characteristics of supercritical water-cooled reactor

The Supercritical Water-Cooled Reactor (SCWR) exhibits significant changes in thermophysical properties of the coolant, but its dynamic characteristics are different from existing reactors, due to the supercritical conditions of the coolant. This necessitates the study of the stability behaviour of SCWR, which requires a dynamic model of the reactor. In this work, a simple unsteady lumped parameter model (LPM) for SCWR has been developed. The LPM includes point reactor kinetics for neutron balance and a two-region model for fuel and coolant thermal hydraulics. The two regions are separated from each other by a boundary that depends on the pseudocritical temperature of the coolant. Regions of stable and unstable operation are identified in the parameter space by linear stability analysis. Bifurcation analysis is carried out to capture the non-linear dynamics of the system. Generalised Hopf (GH) points are located, and parameter ranges for supercritical (soft and safe) and subcritical (hard and dangerous) Hopf bifurcation are identified. The type of transient behaviour of the system for finite (though small) perturbations is predicted based on this analysis. The existence of predicted limit cycles is confirmed by numerical simulation of the nonlinear ODEs, using the shooting technique in case of unstable limit cycles. Furthermore, the effects of geometric, control and neutronics parameters on the stability of the system are studied.

Modeling the dynamic response of a light-powered self-rotating liquid crystal elastomer-based system

Self-oscillating systems possess the capability to convert ambient energy directly into mechanical work. Consequently, the development of novel self-oscillating systems holds significant potential for applications in energy harvesting, engines, and actuators, making them highly worthwhile to be designed and implemented. Inspired by the rotation of a skipping rope, we creatively developed a self-rotating liquid crystal elastomer (LCE)-based system composed of a LCE fiber and a mass, which triggers self-rotation under steady illumination. In this work, a nonlinear dynamic model is proposed to study the dynamic behaviors of optically-responsive LCE-based systems. Numerical calculations show that the self-rotating LCE-based system can undergo a supercritical Hopf bifurcation between the static state and the self-rotation state, which is maintained by the energy input of ambient illumination to compensate the damping dissipation. In addition, the Hopf bifurcation conditions, as well as the important system parameters affecting the frequency of self-rotation, are studied in detail. Especially for the cases of unilateral asymmetrical lighting, the system parameters have different effects on the clockwise and counterclockwise rotation. Different from the existing abundant self-oscillating system, the self-rotating LCE-based system has the advantages of simple structure, customizable size and multimode rotation, and is expected to provide more diversified design ideas for soft robotics, energy harvesters, micro-machines, etc.

Multifractality and scale-free network topology in a noise-perturbed laminar jet

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in a globally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for peak multifractality produce a complex network whose node degree distribution obeys an inverse power-law scaling with an exponent of $2 < \gamma < 3$ , indicating scale-free topology and (iii) the Hurst exponent and the global clustering coefficient can serve as early warning indicators of global instability under specific operating and forcing conditions. By characterising the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics commonly observed in turbulent flows can also be observed in laminar flows under certain stochastic forcing conditions.

Dynamics and control of an SITR COVID-19 model with awareness and hospital bed dependency

In this article, the complex dynamics of susceptible-infected-treated-recovered (SITR) model describing the COVID-19 transmission is explored. Two types of transmission are included here, such as bi-linear with unaware susceptible individuals and nonlinear with aware susceptible individuals. Modified saturation recovery in treated individuals and a difference self-quarantined parameter in susceptible individuals have been considered. Sensitivity indices of the parameters associated with the reproduction number (R0) have been derived using the normalized forward sensitivity index to recognize the significant parameters in changing the disease dynamics. The criteria of existence and local stability of equilibrium points are discussed. The backward bifurcation, forward bifurcation and Hopf bifurcation are demonstrated analytically and numerically, indicating rich dynamics of the proposed model. Chaotic dynamics and supercritical Hopf bifurcation also have been investigated by computing the Lyapunov exponent and the first Lyapunov coefficient respectively. It is observed that, difference self-quarantined rate and psychological or awareness effect ensure the existence of only disease-free equilibrium for R0<1. Optimal control theory and Pontryagin’s maximum principle are applied to compute the efficiency of vaccination and isolation controls in lowering infection while minimizing the associated costs. According to the simulation results, applying both controls together is the most efficient way to prevent the spread of the disease, compared to the situation where a single control is used to achieve the same.

Study of Nonlinear Aerodynamic Self-Excited Force in Flutter Bifurcation and Limit Cycle Oscillation of Long-Span Suspension Bridge

This article establishes a nonlinear flutter system for a long-span suspension bridge, aiming to analyze its supercritical flutter response under the influence of nonlinear aerodynamic self-excited force. By fitting the experimental discrete values of flutter derivatives using the least squares method, a polynomial function of flutter derivatives with respect to reduced wind speed is obtained. Flutter critical value is determined by the linear matrix eigenvalues of a state-space equation. The occurrence of a supercritical Hopf bifurcation in the nonlinear system is determined by the Jacobian matrix eigenvalues of the state-space equation and the system’s vibrational response at the critical state. The vibrational response of the supercritical state is obtained through Runge–Kutta integration, revealing the presence of stable limit cycle oscillation (LCO) and unstable limit cycle oscillation in the system, and through analyzing the relationship between the LCO amplitude and wind speed. Considering cubic nonlinear damping and stiffness, the effects of different factors on the nonlinear flutter system are analyzed.

Post critical characteristics of a side-box steel concrete composite girder: Experimental investigation and mechanism analysis

The present study is aimed to investigate the post-critical behavior of a side-box steel-concrete composite girder under varying wind angles of attack, by conducting a series of section model testing with single-degree and two-degrees of freedom. The essence of flutter, as well as the adverse impact of vortex shedding lock-in on exacerbating post-critical flutter responses, was revealed in detail. The results indicate that the coupled motion of flutter for the prototype composite girder can be characterized as a supercritical Hopf bifurcation with significant phase shifting between motions. The circular character of the limit cycle oscillation (LCO) expressed in the phase diagram is altered to a non-circular form due to the apparent aerodynamic stiffness nonlinearity within a wide range of wind speeds. Interestingly, the coupling vibration between vortex-induced vibrations (VIV) and flutter, i.e., VIV-flutter coupling vibration is observed for the prototype girder, due to the evolution of total damping ratio concave shape with wind speed. Two mitigation countermeasures, i.e., a L-shaped skirt plate and a sharpened wind fairing, were proposed to improve the aeroelastic stability, caused by a significant increase in uncoupled positive aerodynamic damping and alternation of VIV-flutter coupling. In addition, the specific effects of structural damping on the VIV-flutter coupling were discussed in detail.

Dynamic responses of a conceptual two-dimensional airfoil in hypersonic flows with random perturbations

This paper investigates a conceptual pitch–plunge airfoil in hypersonic flows with both structural and aerodynamic nonlinearity, meanwhile the effects of irregular fluctuations in the flow and external random loads modeled as a Gaussian white noise are considered. The dynamic responses of the airfoil model especially the bifurcation behaviors are analyzed via the harmonic balance method for the deterministic case. We found the airfoil system undergo both subcritical and supercritical Hopf bifurcations. Subsequently, the effects of stochasticity on the dynamical behaviors of the hypersonic airfoil system are explored in the regimes of subcritical and supercritical Hopf bifurcations depending on the system parameters. Several interesting phenomena are triggered under random perturbations such as intermittency and stochastic transition between the low-amplitude oscillation state and the undesirable high-amplitude oscillation state. These dynamics are further characterized via the probability density function and time history. This work will provide new insights into the safety and reliability design of hypersonic aircraft.

Analysis on abnormal low-frequency vertical vibration of medium–low-speed maglev vehicle

A newly-developed medium–low-speed maglev test vehicle was used to obtain the vertical dynamic response signals of the vehicle-line coupled system when the test vehicle was standing still on four different types of lines through field tests. It was found that abnormal low-frequency vertical vibration occurred on the car body, so that an excellent ride index of the test vehicle cannot be ensured. As the stability of the vehicle at standstill is the prerequisite to the stability of the vehicle running at a certain speed, it is necessary to investigate the causes underlying the abnormal vibration and propose corrective measures through theoretical studies in combination with the measured results. Therefore, this paper established a coupled vehicle-track system dynamics model considering active feedback control, and investigated the stability and bifurcation behavior of the coupled system through nonlinear dynamics theory and numerical simulation. Field test results show that: the vertical ride index of the test vehicle was rated as “Unqualified” when it was standing still; the vertical vibration eigenfrequency of the car body is basically the same (between 2.10 and 2.20 Hz) when the test vehicle was standing still on the four different lines, indicating that the abnormal vibration was basically independent of the line type; the vertical dynamic response of each measurement point on the electromagnet, F-rail and bridge was normal. The theoretical and simulation studies show that: the numerical simulation of the theoretical model can effectively reproduce the abnormal vibration; the malfunction in the nonlinear characteristics of the air spring vertical damping (i.e. the damping malfunction) is the main reason for this vibration, and the abnormal vibration corresponded to the supercritical Hopf bifurcation dynamics behavior in nonlinear dynamics; after the air spring was restored to normal, the equilibrium point of the coupled system was stable under the influence of different initial disturbances. This study may provide an important reference for reducing the abnormal low-frequency vertical vibration of the test vehicle at standstill.

Stability and bifurcation of annular electro-thermo-convection

Abstract We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.

Hopf Bifurcation Analysis of a Housefly Model with Time Delay

The oscillatory dynamics of a delayed housefly model is analyzed in this paper. The local and global stabilities at the non-negative equilibria are obtained via analyzing the distribution of eigenvalues and Lyapunov–LaSalle invariance principle, and the model undergoes the supercritical Hopf bifurcation and the transient oscillation. Based on Wu’s global Hopf bifurcation theory, the existence of the global bifurcation is established under certain conditions.

Stability Analysis and Nonlinear Chatter Prediction for Grinding a Slender Cylindrical Part

A cylindrical plunge grinding process was modeled to investigate nonlinear regenerative chatter vibration. The rotating workpiece was a slender Euler–Bernoulli beam, and the grinding wheel was a rigid body moving towards the workpiece at a very low feed speed. A numerical method was proposed to provide the critical boundaries for chatter-free grinding. It was demonstrated that the intersection set surrounded by these critical boundaries was the chatter-free region for the considered parameters. When these parameters were outside of the chatter-free region, the stable grinding process underwent a supercritical Hopf bifurcation, resulting in the loss of the chatter-free behavior and the emergence of periodic chatter motions. Then, the periodic motions of both the grinding wheel and the workpiece were predicted analytically using the method of multiple scales, showing the effect of the regenerative force on the grinding process. We demonstrated that the analytical prediction was valid since it agreed with the numerical simulation. The results showed that there exist three kinds of nonlinear chatter motion, with different amplitudes and mode frequencies.