Supercritical Hopf Bifurcation Research Articles

Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution

Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and five dimensions. The purpose of this study is to investigate how the mean delay of the distribution and the total nutrients affect the stability of the equilibrium solutions. Local stability theory and bifurcation theory are used to determine the long-time dynamics of the models. It is found that both models exhibit comparable qualitative dynamics. There are a maximum of three equilibrium points in each of the two models, and at most one of them is locally asymptotically stable. The change of stability from one equilibrium to another takes place through a transcritical bifurcation. In some hypotheses on the functional response, the nutrient–phytoplankton–zooplankton equilibrium loses stability via a supercritical Hopf bifurcation, causing the apparition of a stable limit cycle. The way in which the results are consistent with prior research and how they extend them is discussed. Finally, various application-related consequences of the results of the theoretical study are deduced.

Parameter estimation of two coupled oscillator model for pure intrinsic thermo-acoustic instability

A nonlinear phenomenological model of two coupled oscillators is proposed, which is able to describe the rich nonlinear behaviour stemming from an unstable pure intrinsic thermo-acoustic (ITA) mode of a simple combustion system. In an experimental bifurcation analysis of a pure ITA mode, it was observed that for increasing mean upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterised by low frequent amplitude and frequency modulation. It is shown that a phenomenological model consisting of two coupled oscillators is able to reproduce qualitatively all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear damped oscillator, which are nonlinearly coupled to each other. Furthermore, a parameter estimation of the model parameters is conducted, which reveals a good quantitative match between the model response and the experimental data. Hence, the presented phenomenological dynamical model accurately describes the nonlinear self-excited acoustic behaviour of premixed flames and provides a promising model structure for nonlinear time-domain flame models.

Study on Hopf bifurcation types and dominant parameters of grid‐connected voltage source converter considering non‐linear saturation elements

AbstractSubsynchronous oscillation events of grid‐connected voltage source converter (VSC) pose a potential risk to the security and stability of power systems. It is particularly challenging to reveal the subsynchronous oscillation mechanism precisely because the grid‐connected VSC consists of non‐linear components such as saturation. The describing function and the generalized Nyquist criterion are used in this paper to investigate the bifurcation types and dominant parameters of grid‐connected VSC. First, the condition of a supercritical Hopf bifurcation is investigated, which generates subsynchronous oscillations. Then, the dominant parameters of supercritical Hopf bifurcation are deduced, including the grid strength, the active power reference, or the proportional coefficient of phase‐locked loop. Finally, it is examined how the saturation's limit value impacts the limit cycle's amplitude and stability. The findings demonstrate that a system's stability depends on whether its closed‐loop frequency characteristic curve intersects the describing function's negative reciprocal curve. When the two curves intersect, the supercritical Hopf bifurcation appears, and changing the limit value can affect the limit cycle's amplitude. However, if the two curves do not intersect, altering the limit value cannot cause the supercritical Hopf bifurcation, and the subsynchronous oscillation will not happen. Variations in grid strength, active power reference, or proportional coefficient of phase‐locked loop will significantly affect the Nyquist curve, determining if supercritical Hopf bifurcation appears.

Stability and Bifurcation Analysis of Precision Motion Stage With Nonlinear Friction Isolator

Abstract The application of servocontrolled mechanical-bearing-based precision motion stages (MBMS) is well-established in advanced manufacturing, semiconductor industries, and metrological applications. Nevertheless, the performance of the motion stage is plagued by self-excited friction-induced vibrations. Recently, a passive mechanical friction isolator (FI) has been introduced to reduce the adverse impact of friction in MBMS, and accordingly, the dynamics of MBMS with FI were analyzed in the previous works. However, in the previous works, the nonlinear dynamics components of FI were not considered for the dynamical analysis of MBMS. This work presents a comprehensive, thorough analysis of an MBMS with a nonlinear FI. A servocontrolled MBMS with a nonlinear FI is modeled as a two DOF spring-mass-damper lumped parameter system. The linear stability analysis in the parametric space of reference velocity signal and differential gain reveals that including nonlinearity in FI significantly increases the local stability of the system's fixed-points. This further allows the implementation of larger differential gains in the servocontrolled motion stage. Furthermore, we perform a nonlinear analysis of the system and observe the existence of sub and supercritical Hopf bifurcation with or without any nonlinearity in the friction isolator. However, the region of sub and supercritical Hopf bifurcation on stability curves depends on the nonlinearity in FI. These observations are further verified by a detailed numerical bifurcation, which reveals the existence of nonlinear attractors in the system.

Light-powered self-oscillation in liquid crystal elastomer auxetic metamaterials with large volume change

Self-oscillation can harvest energy directly from the environment to maintain its motion and is characterized by its autonomy, with various promising applications in the fields of actuators, soft robots and medical devices. Limited to conventional materials, challenges remain in designing novel self-oscillating systems. Auxetic metamaterials can exhibit exceptional properties of negative Poisson's ratio and volume expansion, and the integration of metamaterial structures into self-oscillation can provide additional functionality and improve existing properties and its application. This paper proposes a liquid crystal elastomer (LCE) auxetic metamaterial, and theoretically investigates the self-oscillation of the LCE auxetic metamaterial-mass system under steady illumination. The system can undergo a supercritical Hopf bifurcation between the static pattern and self-oscillation pattern, which arises from the light-driven contraction of LCE fibers and is maintained by light energy input from the environment to compensate for the damping dissipation. This paper also investigates the Hopf bifurcation conditions, as well as the vital systematic parameters affecting frequency and amplitude of the self-oscillation pattern in detail. Different from the existing self-oscillating systems, this LCE metamaterial structure can provide additional functionality and potentially improve existing properties such as large volume change. More generally, the idea of integrating LCE fibers into the re-entrant honeycomb lattices can be applied to other auxetic structures such as arrows and chirality, and it is expected to offer more diverse design ideas of metamaterial and motion control for artificial hearts, hydraulic pumps, sensors, and energy harvesters.

Assessing the impact of escalating attacks on soft targets by criminal gang: A modelling viewpoint using bifurcation analysis

The emerging threat of criminal gangs in society has severe consequences on the socio-economic well-being of individuals. To examine the impact of morally upright individuals and attacks on soft targets (i.e., vulnerable members of the society) on the coexistence dynamics of the society, we employ a system of ordinary differential equations (ODE) where individuals in the community are allowed to mix and interact. In the co-dimension one bifurcation analysis, we discover the emergence of supercritical and subcritical Hopf bifurcation, transcritical bifurcation, and saddle–node bifurcation of cycle. To understand how these different attractors interact and mediate a crime-free society, we conduct a co-dimension two bifurcation analysis (i.e., by using the morally upright individuals (h1) and attacks on soft target (e) as our bifurcation parameter). The interplay between morally upright individuals and attacks on soft targets shows the presence of a global attractor (i.e., Generalized Hopf bifurcation), which serves as a synchronization centre for the model. Several studies have reported the emergence of stable limit cycles in the criminal gang model. However, we discover a transition from stable to unstable limit cycles when the attacks on soft targets are considered under distinct coexistence factors. This study revealed that morally upright individuals and attacks on soft targets positively affected the coexistence dynamics of society. Furthermore, we observed that an unstable limit cycle is a causative factor for chaotic dynamics in the non-linear system. We further noted that the positive effect of morally upright individuals plays a crucial role in mediating a stable and crime-free society. The study concluded that the recruitment rate of susceptible individuals to the criminal gang has little consequence on the peaceful coexistence of society if the society maintains high morality.

CHAOTIC DYNAMICS OF A STAGE-STRUCTURED PREY–PREDATOR SYSTEM WITH HUNTING COOPERATION AND FEAR IN PRESENCE OF TWO DISCRETE DELAYS

Depending on behavioral differences, reproductive capability and dependency, the life span of a species is divided mainly into two classes, namely immature and mature. In this paper, we have studied the dynamics of a predator–prey system considering stage structure in prey and the effect of predator-induced fear with two discrete time delays: maturation delay and fear response delay. We consider that predators cooperate during hunting of mature prey and also include its impact in fear term. The conditions for existence of different equilibria, their stability analysis are carried out for non-delayed system and bifurcation results are presented extensively. It is observed that the fear parameter has stabilizing effect whereas the cooperative hunting factor having destabilizing effect on the system via occurrence of supercritical Hopf-bifurcation. Further, we observe that the system exhibits backward bifurcation between interior equilibrium and predator free equilibrium and hence the situation of bi-stability occurs in the system. Thereafter, we differentiate the region of stability and instability in bi-parametric space. We have also studied the system’s dynamics with respect to maturation and fear response delay and observed that they also play a vital role in the system stability and occurrence of Hopf-bifurcation is shown with respect to both time delays. The system shows stability switching phenomenon and even higher values of fear response delay leads the system to enter in chaotic regime. The role of fear factor in switching phenomenon is discussed. Comprehensive numerical simulation and graphical presentation are carried out using MATLAB and MATCONT.

Equivalent dimensionality reduction method of doubly‐fed induction generator model and discrimination of Hopf bifurcation type

AbstractThe strong coupling, multivariate, and nonlinear characteristics of the mathematical model of a doubly‐fed induction generator (DFIG) make it challenging to analyze the bifurcation category of the DFIG. Therefore, in this study, we developed a method for analyzing the DFIG model via topologically equivalent dimensionality reduction in the neighborhood of bifurcation values based on the central manifold theorem and realized the discrimination of the Hopf bifurcation type of the DFIG. We established a complex frequency domain model of the DFIG and then calculated the critical open‐loop gain point using the root trajectory method to obtain the relationship between the critical open‐loop gain point and Hopf bifurcation. Thereafter, we performed topological equivalence dimensionality reduction in the neighborhood of the DFIG bifurcation values based on the central manifold theorem. Subsequently, the Hopf bifurcation type of the system was determined using the stability index (first Lyapunov exponent) of the Hopf bifurcation. Finally, a nonlinear time‐domain simulation was used to verify that the four and two‐dimensional DFIGs generate a stable limit cycle by supercritical Hopf bifurcation under the tuning parameter. The results indicated that the proposed method was feasible and effective in the neighborhood of bifurcation values in the state parameter space of DFIG, which provides a theoretical basis for the control of Hopf bifurcation of the DFIG.

Nonlinear dynamics of pure intrinsic thermo-acoustic modes

In this paper the nonlinear dynamics of the pure Intrinsic Thermo-Acoustic (ITA) modes are experimentally investigated for burner stabilized premixed flames by decoupling them from the acoustic modes. Thereto, a setup is used with up- and downstream mufflers that provide close to anechoic boundary conditions. This enables the investigation of the self-excited oscillations of the flame, without the influences of acoustic reflections. An experimental bifurcation analysis is conducted to systematically investigate the different asymptotic oscillatory states and their transitions. By increasing the upstream velocity, the flame looses stability through a supercritical Hopf bifurcation and reveals limit cycles with a single dominant frequency. For increasing upstream velocity, a low frequent mode appears, resulting in quasi-periodic oscillations. This regime is characterized by a beating phenomenon with simultaneous amplitude and frequency modulations. When the upstream velocity is even further increased, a period doubling bifurcation is observed. To our best knowledge, the experimental investigation of the nonlinear dynamical behavior of pure ITA modes and, in particular, the low frequency oscillation, has not been reported before, and it provides valuable insights towards developing accurate time-domain models of the nonlinear acoustic behavior of premixed flames. Finally, a phenomenological model of two coupled oscillators is introduced that is able to qualitatively reproduce all observed nonlinear oscillations of the flames.

Bifurcations of Grid-Following Rectifiers and Routes to Voltage Instability in Weak AC Grids

As indispensable components in modern power systems, grid-following rectifiers would interact with the grid. Such interaction, however, may lead to unexpected collapse of the ac bus voltage. In this paper, three routes to voltage instability are identified, namely, (i) supercritical Hopf bifurcation and successive saddle-node bifurcation of periodic orbits (SNPO); (ii) subcritical Hopf bifurcation; and (iii) departure from a stable basin of attraction. The SNPO and subcritical Hopf bifurcation are reported for the first time in rectifier-based power systems. It is also found that the rectifier-based system may undergo a Bautin bifurcation, which changes a Hopf bifurcation from supercritical to subcritical, or vice versa. The current loop of the grid-following rectifier has been found to play a crucial role in these bifurcation behaviors. Considering the inevitable and immediate voltage breakdown after the onset of a subcritical Hopf bifurcation, it is recommended to increase the bandwidth of the current loop to trigger the Bautin bifurcation, so that only supercritical Hopf bifurcation may occur. Furthermore, the original high-order and nonlinear model of the rectifier-based system can be simplified to a reduced order system by focusing on the center manifold. Based on the reduced system, the occurrence mode of Hopf bifurcation can be confirmed by inspecting the first Lyapunov coefficient. Finally, relevant bifurcation diagrams are derived, providing practical design guidelines to avoid an eventual system collapse.

Spatio-temporal dynamics of a two-layer pressure-driven flow subjected to a wall-normal temperature gradient

The present study investigates the linear spatio-temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid–liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio ( $\mu _r$ ), thermal conductivity ratio ( $\kappa _r$ ), interface position ( $H$ ) and the sign of the imposed temperature gradient ( $\beta _1$ ). The analysis predicts a range of $H$ for given $\mu _r$ and $\kappa _r$ , which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of $\mu _r$ , $\kappa _r$ , $H$ , the sign of $\beta _1$ and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation.

Parameter identification of two coupled oscillator model for pure intrinsic thermoacoustic instability

Thermo-acoustic instabilities are a well-known problem in combustions systems. In general, the type of these modes can be divided in two categories: (i) modes with an acoustic origin and (ii) modes with an intrinsic thermo-acoustic (ITA) origin. It is an challenging task to observe the ITA modes experimentally. Therefore, a setup with a burner in an acoustic embedding with close to anechoic up- and downstream terminations is required. In previous research we introduced such a setup and conducted an experimental bifurcation analysis on the dominant pure ITA mode. In this bifurcation analysis, we observed that for increasing upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterized by low frequency amplitude and frequency modulation. In this research we present a phenomenological model consisting of two coupled oscillators that is able to reproduce all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear mass-spring-damper oscillator, which are nonlinearly coupled to each other. Furthermore, we executed a parameter identification of the model and concluded that the model is able to describe the experimental data quantitatively well.

Nonlinear dynamical features of vortex-acoustic lock-on in a backward-facing step combustor

Unstable shear layers characteristic of step combustors display strong vortex shedding and play a dominant role in combustion instability. Past non-premixed flame experiments with Reynolds number (Re) as the bifurcation parameter showed that during instability, the frequency of acoustic oscillations locks-on to the natural vortex shedding mode for a range of Re. In the present work, we study the dynamical features of this type of lock-on. We employ a vortex model to bring out the nonlinearity in the flame response to vortical perturbations. This is coupled with a Galerkin model for the acoustic field. Regimes of aperiodic oscillations, low-amplitude lock-on and high-amplitude lock-on are predicted in agreement with experimental trends. Phase portraits show a transition to instability during lock-on via a supercritical Hopf bifurcation with an increase in limit cycle amplitudes. Synchronisation between the acoustic pressure and vortex-driven heat release perturbation is studied using recurrence analysis. The system transitions from asynchronous to phase synchronisation in the low-amplitude lock-on regime, characterised by an increasing degree of phase correlation. In the high-amplitude lock-on regime, a state of generalised synchronisation exists, where in addition to the phase, the amplitudes also show strong correlation.

Hopf Bifurcation Analysis of a Two-Dimensional Simplified Hodgkin–Huxley Model

This paper presents a two-dimensional simplified Hodgkin–Huxley model under exposure to electric fields. The Hopf bifurcations of the simplified Hodgkin–Huxley model are investigated through qualitative analysis and numerical simulations. A necessary and sufficient condition for the existence of Hopf bifurcations is derived, and the conditions for supercritical and subcritical Hopf bifurcations are obtained. Finally, bifurcation diagrams are given for two parameters, and numerical examples are presented to illustrate the effectiveness of the theoretical results.

The mechanism of wheel polygonal wear based on rotor dynamics

A rotor dynamic model of wheel-rail coupling system was established taking wheel eccentricity and initial polygon into consideration in order to investigate the mechanism of wheel polygonal wear. Considering the complex wheel-rail contact relationship, the equivalent yaw angle was proposed and the wheel-rail relative speed model was established. The law of wheel-rail self-excited vibration and its effect on polygonal wear were studied. The possibility of variable speed operation was investigated together with the sensitivity of the system parameters. The results show that the periodic longitudinal and lateral creep forces are essential for the wheel polygonal wear, which have the characteristics of supercritical Hopf bifurcation and ‘fixed wavelength’ and ‘fixed frequency’. While in some circumstances, longitudinal and lateral creep forces jointly cause polygonal wear, in other cases, longitudinal creep forces have almost no effect. By adjusting the longitudinal and lateral stiffness, primary suspension stiffness, and rail damping parameters, as well as by selecting an appropriate speed for variable speed operation, the wheel polygonal wear may be effectively suppressed. The findings offer a fresh viewpoint for the study of mechanisms and the prevention of wheel polygonal wear.

Dynamic analysis of a predator–prey model of Gause type with Allee effect and non-Lipschitzian hyperbolic-type functional response

In this work, we study a predator–prey model of Gause type, in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response, which is neither differentiable nor locally Lipschitz at the predator axis. This kind of functional response is an extension of the so-called square root functional response, used to model systems in which the prey have a strong herd structure. We study the behavior of the solutions in the first quadrant and the existence of limit cycles. We prove that, for a wide choice of parameters, the solutions arrive at the predator axis in finite time. We also characterize the existence of an equilibrium point and, when it exists, we provide necessary and sufficient conditions for it to be a center-type equilibrium. In fact, we show that the set of parameters that yield a center-type equilibrium, is the graph of a function with an open domain. We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation. In particular, we guarantee the existence of a unique limit cycle, for small perturbations of the system.

Two- and three-dimensional wake transitions of a NACA0012 airfoil

Flow transitions are an important fluid-dynamic phenomena for many reasons, including the direct effect on the aerodynamic forces acting on the body. In the present study, two-dimensional (2-D) and three-dimensional (3-D) wake transitions of a NACA0012 airfoil are studied for angles of attack in the range $0^\circ \leq \alpha \leq 20^\circ$ and Reynolds numbers $500 \leq {\textit {Re}} \leq 5000$ . The study uses water-channel experiments and 2-D and 3-D numerical simulations based on the nodal spectral-element method, level-set function-based immersed-interface method and Floquet stability analysis. The different wake states are categorised based on the time-instantaneous wake structure, non-dimensional frequency and aerodynamic force coefficients. The wake states and transition boundaries are summarised in a wake regime map. The critical angle of attack and Reynolds number for the supercritical Hopf bifurcation (i.e. steady to periodic wake transition) varies as $\alpha _1 {\sim} {\textit {Re}}^{-0.65}$ , while the critical angle of attack for the onset of three dimensionality varies as $\alpha _{3D} {\sim} {\textit {Re}}^{-0.5}$ . Over the entire Reynolds number range, the transition to 3-D flow occurs through a mode C (subharmonic) transition. Beyond this initial transition, further instabilities of the 2-D periodic base flow arise and are investigated. For instance, at $ {\textit {Re}}=2000$ and $\alpha _{3D,2}=11.0^\circ$ , mode C coexists together with modes related to modes A and QP seen in a stationary circular cylinder wake. In contrast, at $ {\textit {Re}}=5000$ and $\alpha _{3D,2}=8.0^\circ$ , the dominant mode C coexists with mode QP. Three-dimensional simulations well beyond critical angles indicate that 2-D vortex-street transitions are approximately maintained in the fully saturated 3-D wakes in a spanwise-averaged sense.

Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect.

In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper [30]. Moreover, numerical simulations are given to verify the theoretical results.

Existence of Supercritical Hopf Bifurcation on a Type-Lorenz System

In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.

Bifurcation and overexploitation in Rosenzweig-MacArthur model

<p style='text-indent:20px;'>In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.</p>