Supercritical Hopf Bifurcation Research Articles

Pulsing in the Ahu‘ail‘au pond–spillway system during the 2018 Klauea eruption: a dynamical systems perspective

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of ‘pulsing’ behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava.

On the aeroelastic bifurcation of a flexible panel subjected to cavity pressure and inviscid oblique shock

The aeroelasticity of a panel in the presence of a shock is a fundamental issue of great significance in the development of hypersonic vehicles. In practical engineering, cavity pressure emerges as a crucial factor that influences the nonlinear dynamical characteristics of the panel. This study focuses on the aeroelastic bifurcation of a flexible panel subjected to both cavity pressure and oblique shock. To this end, a computational method is devised, coupling a high-fidelity reduced-order model for unsteady aerodynamic loads with nonlinear structural equations. The solution is meticulously tracked by continuous calculations. The obtained results indicate that cavity pressure plays a pivotal role in determining the bifurcation and stability characteristics of the system. First, the system exhibits hysteresis behaviour in response to the ascending and descending dynamic pressures. The evolution of hysteresis behaviour originates from the phenomenon of cusp catastrophe. Second, variations in cavity pressure induce three types of bifurcation phenomena, exhibiting characteristics akin to supercritical Hopf bifurcation, subcritical Hopf bifurcation and saddle-node bifurcation of cycles. The system's response at the critical points of these bifurcations manifests as long-period asymptotic flutter or explosive flutter. Lastly, the evolution of the dynamical system among these three types of bifurcations is an important factor contributing to the discrepancies observed in certain research results. This study enhances the understanding of the nonlinear dynamical behaviour of panel aeroelasticity in complex practical environments and provides new explanations for the discrepancies observed in certain research results.

Stability analysis for the phytoplankton-zooplankton model with depletion of dissolved oxygen and strong Allee effects

The photosynthetic activity of phytoplankton in the seas is responsible for an estimated 50–80 % of the world's oxygen generation. Both phytoplankton and zooplankton require some of this synthesized oxygen for cellular respiration. This study aims to better understand how the oxygen-phytoplankton dynamics are altered due to the Allee effect in phytoplankton development, particularly when considering the time-dependent oxygen generation rate. The dynamic analysis of the model is dedicated to finding the possible equilibrium points. The analysis reveals that three equilibrium points can be obtained. The stability study demonstrates that one of the equilibrium points is always stable. The remaining equilibrium points are stable under specific conditions. We also identify bifurcations originating from these equilibrium points, including transcritical, pitchfork, and Hopf bifurcation. We derive conditions for stable limit cycles (supercritical Hopf bifurcation) and, in some cases, establish the non-existence of solutions. Numerical simulations are performed to validate our theoretical findings. Furthermore, it is noted that the Allee threshold for the phytoplankton population (k0) significantly influences the overall dynamics of the system. When k0≤0.001, the population of plankton is at risk of extinction. On the other hand, when 0.001<k0≤0.01, the population of zooplankton is at risk of extinction. When 0.01<k0≤2, the solution reaches a stable condition of coexistence. Conversely, when k0≥2.1, the solution exhibits periodic attractor behaviour.

Mathematical Modeling of the Displacement of a Light-Fuel Self-Moving Automobile with an On-Board Liquid Crystal Elastomer Propulsion Device

The achievement and control of desired motions in active machines often involves precise manipulation of artificial muscles in a distributed and sequential manner, which poses significant challenges. A novel motion control strategy based on self-oscillation in active machines offers distinctive benefits, such as direct energy harvesting from the ambient environment and the elimination of complex controllers. Drawing inspiration from automobiles, a self-moving automobile designed for operation under steady illumination is developed, comprising two wheels and a liquid crystal elastomer fiber. To explore the dynamic behavior of this self-moving automobile under steady illumination, a nonlinear theoretical model is proposed, integrating with the established dynamic liquid crystal elastomer model. Numerical simulations are conducted using the Runge-Kutta method based on MATLAB software, and it is observed that the automobile undergoes a supercritical Hopf bifurcation, transitioning from a static state to a self-moving state. The sustained periodic self-moving is facilitated by the interplay between light energy and damping dissipation. Furthermore, the conditions under which the Hopf bifurcation occurs are analyzed in detail. It is worth noting that increasing the light intensity or decreasing rolling resistance coefficient can improve the self-moving average velocity. The innovative design of the self-moving automobile offers advantages such as not requiring an independent power source, possessing a simple structure, and being sustainable. These characteristics make it highly promising for a range of applications including actuators, soft robotics, energy harvesting, and more.

EFFECT Of DELAY IN A MUSCA DOMESTICA HOUSEFLIES MODEL: STABILITY AND GLOBAL HOPF BIFURCATION

The study focuses on the model of houseflies with discrete delay, which is examined both theoretically and numerically. The solution of the delayed system is bounded and positive. The delay is selected as the bifurcation parameter. Stability analysis, local and global Hopf bifurcation are given in theoretical aspect. Through computer simulations, various dynamic behaviors, such as supercritical Hopf bifurcation, are detected. The theoretical analysis and numerical observations in this research are significant contributions to the biomathematics research area.

A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0–1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

Supercritical Hopf bifurcations in the stage-structured model of housefly populations

Insect populations, which are diverse and widespread, provide a principal area of utilization of the stage-structured modeling approach. In this paper, housefly populations incorporating a stage-structured model are investigated theoretically and graphically. First, stability charts and rightmost characteristic roots of the positive equilibrium are elucidated analytically and numerically. Furthermore, the Hopf bifurcation at the positive equilibrium is derived employing geometric stability switch criterion. Second, the properties of Hopf bifurcation are determined using the center manifold theorem and by reducing the equation to the Poincaré normal form. Finally, the correctness of the theoretical derivation is confirmed using a numerical simulation based on specific parameter values. Our results show that with an increase in delay [Formula: see text], the unique positive equilibrium may undergo two stability switches: from stable to unstable, and from unstable to stable. Interestingly, the characteristic equation has pure imaginary roots at the second pair and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also simulate the unstable periodic solutions at the second pair of critical values through a bifurcation diagram. Therefore, a pair of supercritical Hopf bifurcations appear around the positive equilibrium of the housefly population stage-structured model.

A PINN approach for identifying governing parameters of noisy thermoacoustic systems

Identifying the governing parameters of self-sustained oscillation is crucial for the diagnosis, prediction and control of thermoacoustic instabilities. In this paper, we propose and validate a novel method for computing the parameters of thermoacoustic oscillation in a stochastic environment, which exploits a physics-informed neural network (PINN). Specifically, we introduce a negative log-likelihood loss function that integrates the stochastic samples and the solution of the Fokker–Planck equation. The proposed framework is validated using the numerically generated signal and the experimental data obtained from an annular combustor, both before and after the supercritical Hopf bifurcation. The results of PINN-based system identification show good agreement with the actual system parameters and the original stochastic signal, with improved accuracy compared to established methods. To the best of our knowledge, this study constitutes the first demonstration of the PINN-inverse approach that uses the noise-induced dynamics of thermoacoustic systems, opening up new pathways for diagnosing and predicting the thermoacoustic behaviour of various combustion systems.

Two predators, one prey model that integrates the effect of supplementary food resources due to one predator's kleptoparasitism under the possibility of retribution by the other predator

In ecology, foraging requires animals to expend energy in order to obtain resources. The cost of foraging can be reduced through kleptoparasitism, the theft of a resource that another individual has expended effort to acquire. Thus, kleptoparasitism is one of the most significant feeding techniques in ecology. The phenomenon of kleptoparasitism has garnered significant attention from scholars due to its substantial impact on the food chain. However, the proportionate amount of mathematical modelling to facilitate the analysis has made limited progress in the literature. This circumstance motivated us to develop mathematical models that could explain the population dynamics of the prey-predator food chain. This study explores a scenario with two predators and one prey, where one predator is a kleptoparasite and the other is a host. The energy depletion caused by the predator's counterattack subsequent to kleptoparasitism, notwithstanding the nonlethal nature of this antagonism, is an additional component incorporated into this model. It has been suggested that biologically viable equilibria must meet certain parametric conditions in order to exist and to be stable both locally and globally. This article delves deeply into the occurrences of various one-parametric bifurcations, such as saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation, as well as two-parametric bifurcations, such as Bautin bifurcation. A subcritical Hopf bifurcation happens when the growth rate of the first predator is relatively low, while a supercritical Hopf bifurcation occurs when the growth rate of the first predator is quite large, allowing for the coexistence of all three species. Numerical simulations have been conducted to validate our theoretical findings.

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.

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.

Synchrony in Networks of Type 2 Interneurons Is More Robust to Noise with Hyperpolarizing Inhibition Compared to Shunting Inhibition in Both the Stochastic Population Oscillator and the Coupled Oscillator Regimes.

Synchronization in the gamma band (25-150 Hz) is mediated by PV+ inhibitory interneurons, and evidence is accumulating for the essential role of gamma oscillations in cognition. Oscillations can arise in inhibitory networks via synaptic interactions between individual oscillatory neurons (mean-driven) or via strong recurrent inhibition that destabilizes the stationary background firing rate in the fluctuation-driven balanced state, causing an oscillation in the population firing rate. Previous theoretical work focused on model neurons with Hodgkin's Type 1 excitability (integrators) connected by current-based synapses. Here we show that networks comprised of simple Type 2 oscillators (resonators) exhibit a supercritical Hopf bifurcation between synchrony and asynchrony and a gradual transition via cycle skipping from coupled oscillators to stochastic population oscillator (SPO), as previously shown for Type 1. We extended our analysis to homogeneous networks with conductance rather than current based synapses and found that networks with hyperpolarizing inhibitory synapses were more robust to noise than those with shunting synapses, both in the coupled oscillator and SPO regime. Assuming that reversal potentials are uniformly distributed between shunting and hyperpolarized values, as observed in one experimental study, converting synapses to purely hyperpolarizing favored synchrony in all cases, whereas conversion to purely shunting synapses made synchrony less robust except at very high conductance strengths. In mature neurons the synaptic reversal potential is controlled by chloride cotransporters that control the intracellular concentrations of chloride and bicarbonate ions, suggesting these transporters as a potential therapeutic target to enhance gamma synchrony and cognition.

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.

Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

In this study, the quadratic 3-dimensional differential system is considered, in which the origin of the coordinate becomes the Hopf equilibrium point. The existence and stability of limit cycles that emerge from the Hopf point are being investigated. The Lyapunov coefficients connected to the Hopf point are calculated using the projection method. First, four families of parameter conditions are driven by which the quadratic 3-dimensional differential system can exhibit codimension three of the Hopf bifurcation. The analytical proof of each parameter family of conditions is given by calculating the Lyapunov coefficients, the vanishing of the first, sconed Lyapunov, and the non-zero of the third Lyapunov coefficients. The explicit conditions are presented for the existence and stability of three limit cycles arising from each family of Hopf bifurcations. The output of existence displays a stable (unstable) Hopf point, accompanied by the emergence of two stable (unstable) limit cycles alongside one unstable (stable) limit cycle in the neighborhood of the unstable (stable) origin of the coordinated 3-dimensional quadratic system. In addition, the outcome is utilized for exploring the limit cycles of the n-scroll chaotic attractor system, which has many practical uses, including secure communication, encryption, random number generation, and autonomous mobile robots. The conditions are derived under which the origin point of this system becomes the Hopf point, and three limit cycles can exist around the Hopf point. Finally, the numerical demonstrations show that the system undergoes a supercritical Hopf bifurcation, resulting in two stable and one unstable limit cycle. Furthermore, all results are verified.

Noise-induced collective actuation in active solids.

Collective actuation describes the spontaneous synchronized oscillations taking place in active solids when the elasto-active feedback, which generically couples the reorientation of the active forces and the elastic stress, is large enough. In the absence of noise, collective actuation takes the form of a strong condensation of the dynamics on a specific pair of modes and their generalized harmonics. Here we report experiments conducted with centimetric active elastic structures, where collective oscillation takes place along the single lowest energy mode of the system, gapped from the other modes because of the system's geometry. Combining the numerical and theoretical analysis of an agent-based model, we demonstrate that this form of collective actuation is noise-induced. The effect of the noise is first analyzed in a single-particle toy model that reveals the interplay between the noise and the specific structure of the phase space. We then show that in the continuous limit, any finite amount of noise turns this new form of transition to collective actuation into a bona fide supercritical Hopf bifurcation.

Bifurcations and global dynamics of a predator–prey mite model of Leslie type

AbstractIn this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Bifurcation Analysis of a Predator–Prey Model with Alternative Prey and Prey Refuges

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [ 2023 ]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two.

The Hopf whole-brain model and its linear approximation

Whole-brain models have proven to be useful to understand the emergence of collective activity among neural populations or brain regions. These models combine connectivity matrices, or connectomes, with local node dynamics, noise, and, eventually, transmission delays. Multiple choices for the local dynamics have been proposed. Among them, nonlinear oscillators corresponding to a supercritical Hopf bifurcation have been used to link brain connectivity and collective phase and amplitude dynamics in different brain states. Here, we studied the linear fluctuations of this model to estimate its stationary statistics, i.e., the instantaneous and lagged covariances and the power spectral densities. This linear approximation—that holds in the case of heterogeneous parameters and time-delays—allows analytical estimation of the statistics and it can be used for fast parameter explorations to study changes in brain state, changes in brain activity due to alterations in structural connectivity, and modulations of parameter due to non-equilibrium dynamics.

Bifurcation mechanisms underlying the nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Inaba et al, 2022) the bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered; these structures can be generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator. It is known that this oscillator can generate canards in the absence of perturbations. In this work, we investigate nested MMOs in a canard-generating driven BVP oscillator near a supercritical Hopf bifurcation point subjected to weak periodic perturbations. The analysis of this system is undertaken using the techniques presented by Kawakami, (1984). These techniques include the use of a shooting algorithm to derive the parameters of the forced oscillators at which bifurcations occur by solving the simultaneous equations that identify the conditions in which the solution of the forced oscillators is periodic and the characteristic multiplier(s) with the largest absolute value of the periodic solution on the Poincaré section crosses the unit circle. We also obtain several two-parameter bifurcation diagrams that describe the system investigated here. One motivation for this study is to demonstrate that the bifurcation structures present in nested MMOs are likely to be a widespread phenomenon. We hypothesize here that nested MMOs represent a stable phenomenon; this is likely to be the case because they appear and disappear consistently and are bound by sequential saddle–node and period-doubling bifurcation curves; this hypothesis is supported by the observation that codimension-two bifurcation points or cusps do not appear in the region close to these bifurcation boundaries.

Research on the Stability and Bifurcation Characteristics of a Landing Gear Shimming Dynamics System

A dynamic model is established to investigate the shimmy instability of a landing gear system, considering the influence of nonlinear damping. The stability criterion is utilized to determine the critical speed at which the landing gear system becomes unstable. The central manifold theorem and canonical method are employed to simplify the dynamic model of the landing gear. The first Lyapunov coefficient of the system is theoretically derived and verified using numerical simulation. Further investigation on the Hopf bifurcation characteristics and stability of the shimmy in the landing gear system is conducted. The results indicate that above a certain threshold speed, with a tire stability distance greater than half the tire length in contact with the ground plus the slack length, the aircraft remains stable during taxiing. At critical speeds, a shimmy system with higher-order nonlinear damping will undergo supercritical Hopf bifurcation. Quantitative analysis suggests an increase in the linear damping coefficient within a range that ensures a stability margin to mitigate undesired oscillation, while the nonlinear damping coefficient should be designed within a reasonable range to decrease the amplitude of the limit cycle.