Generalized Hopf Bifurcation Research Articles

Three limit cycles in the Kim–Forger model of the mammalian circadian clock

The Kim–Forger model is a fundamental mathematical model for understanding circadian rhythms. The limit cycles presented in the model is the dynamic mechanism of the periodic phenomena in the mammalian circadian clock. However, the full dynamics of the Kim–Forger model remain not completely understood. In this paper, we theoretically demonstrate that this model can undergo supercritical Hopf bifurcation, subcritical Hopf bifurcation, and generalized Hopf bifurcation of codimension two. Numerically, the system can exhibit 0, 1, 2, or 3 limit cycles. Our work complements and enriches the dynamical insights in the field of circadian rhythms.

Birth of Catastrophe and Strange Attractors through Generalized Hopf Bifurcations in Covid-19 Transmission Mathematical Model

Coronavirus can be transmitted through the things that people carry or the things where it sticks to after being spread by the sufferer. Instead, various preventive measures have been carried out. We create a new mathematical model that represents Coronavirus that exists in non-living objects, susceptible, and infected subpopulations interaction by considering the Coronavirus transmission through non-living objects caused by susceptible and infected subpopulations along with its prevention to characterize the dynamics of Coronavirus transmission in the population under those conditions. One disease-free and two infection equilibrium points along with their local stability and coexistence are identified. Global stability of the disease-free equilibria and basic reproduction number are also investigated. Changes in susceptible-Coronavirus interaction rate generate Fold and Hopf bifurcations which represent the emergence of a cycle and the collision of two infection equilibrium points respectively. Catastrophe generated by the collision between an attractor and a repeller is found around a Generalized Hopf bifurcation point by changing susceptible-Coronavirus interaction rate and increasing rate of Coronavirus originating from infected subpopulation. It represents a momentary unpredictable dynamics as the effect of Coronavirus addition and infection. Non-chaotic strange attractors that represent complex but still predictable dynamics are also triggered by Generalized Hopf bifurcation when the susceptible-Coronavirus interaction rate and one of the following parameters, i.e. increasing rate of Coronavirus originating from infected subpopulation or infected subpopulation recovery rate vary.

Hysteresis behavior and generalized Hopf bifurcation in a three-degrees-of-freedom aeroelastic system with concentrated nonlinearities

A three-degrees-of-freedom aeroelastic system with concentrated structural nonlinearities is considered. The aerodynamic loads in the model are simulated by using unsteady aerodynamic forces. First, we evaluate the linear stability region of the aerodynamic system employing the Lienard–Chipart criterion so that the stability boundaries of the center of gravity position and the uncoupled pitch natural frequency are obtained. Next, we present the distribution of the first Lyapunov coefficient l1 in the parameter space by using the Poincaré projection method to determine the type of Hopf bifurcation (subcritical or supercritical). We identify three types of hysteresis loops associated with Hopf bifurcations: the first type contains one stable equilibrium point and one stable limit cycle, the second type consists of one stable equilibrium point and two stable limit cycles, and the third type comprises two stable limit cycles. Nonlinear analysis shows that the variations in hysteresis loop types are mainly caused by the nonlinear flutter critical points, namely the saddle–node bifurcation points of limit cycles. Furthermore, we determine a degenerate Hopf bifurcation type known as generalized Hopf bifurcation (Bautin bifurcation) and its corresponding second Lyapunov coefficient, l2. Subsequently, we conduct nonlinear flutter analyses for positive and negative values of l2 by using bifurcation diagrams and phase portraits. The results show that when the uncoupled pitch natural frequency is used as the control parameter, the occurring generalized Hopf bifurcations are all supercritical (l2<0). However, when the center of gravity position is used as the control parameter, the generalized Hopf bifurcations that occur have both supercritical and subcritical (l2>0) modes.

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 autonomous and nonautonomous modified Leslie-Gower models.

In ecological systems, the predator-induced fear dampens the prey's birth rate; yet, it fails to extinguish their population, as they endure and survive even under significant fear-induced costs. In this study, we unveil a modified Leslie-Gower predator-prey model by incorporating the fear of predators, cooperative hunting, and predator-taxis sensitivity. We embark upon an exploration of the positivity and boundedness of solutions, unearthing ecologically viable equilibrium points and their stability conditions governed by the model parameters. Delving deeper, we unravel the scenario of transcritical, saddle-node, Hopf, Bogdanov-Takens, and generalized-Hopf bifurcations within the system's intricate dynamics. Additionally, we observe the bistable nature of the system under some parametric conditions. Further, the nonautonomous extension of our model introduces the intriguing interplay of seasonality in some crucial parameters. We establish a set of sufficient conditions that guarantee the permanence of the seasonally driven system. By conducting a numerical study on the seasonally forced model, we observe a myriad of phenomena manifesting the predator-prey dynamics. Notably, periodic solutions, higher periodic solutions, and bursting patterns emerge, alongside intriguing chaotic dynamics. Specifically, seasonal variations of the predator-taxis sensitivity and hunting cooperation can lead to the extinction of prey species and even the control of chaotic (higher periodic) solutions through the generation of a simple periodic solution. Remarkably, the seasonal forcing has the capacity to govern the chaotic behavior, leading to an exceptionally quasi-periodic arrangement in both prey and predator populations.

Effect of motor suspension parameters on bifurcations for a nonlinear bogie system

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie.

Generalized Hopf Bifurcation in a Delay Model of Neutrophil Cells Model

Generalized Hopf Bifurcation in a Delay Model of Neutrophil Cells Model

Modulation of SK Channels via Calcium Buffering Tunes Intrinsic Excitability of Parvalbumin Interneurons in Neuropathic Pain: A Computational and Experimental Investigation.

Parvalbumin-expressing interneurons (PVINs) play a crucial role within the dorsal horn of the spinal cord by preventing touch inputs from activating pain circuits. In both male and female mice, nerve injury decreases PVINs' output via mechanisms that are not fully understood. In this study, we show that PVINs from nerve-injured male mice change their firing pattern from tonic to adaptive. To examine the ionic mechanisms responsible for this decreased output, we used a reparametrized Hodgkin-Huxley type model of PVINs, which predicted (1) the firing pattern transition is because of an increased contribution of small conductance calcium-activated potassium (SK) channels, enabled by (2) impairment in intracellular calcium buffering systems. Analyzing the dynamics of the Hodgkin-Huxley type model further demonstrated that a generalized Hopf bifurcation differentiates the two types of state transitions observed in the transient firing of PVINs. Importantly, this predicted mechanism holds true when we embed the PVIN model within the neuronal circuit model of the spinal dorsal horn. To experimentally validate this hypothesized mechanism, we used pharmacological modulators of SK channels and demonstrated that (1) tonic firing PVINs from naive male mice become adaptive when exposed to an SK channel activator, and (2) adapting PVINs from nerve-injured male mice return to tonic firing on SK channel blockade. Our work provides important insights into the cellular mechanism underlying the decreased output of PVINs in the spinal dorsal horn after nerve injury and highlights potential pharmacological targets for new and effective treatment approaches to neuropathic pain.SIGNIFICANCE STATEMENT Parvalbumin-expressing interneurons (PVINs) exert crucial inhibitory control over Aβ fiber-mediated nociceptive pathways at the spinal dorsal horn. The loss of their inhibitory tone leads to neuropathic symptoms, such as mechanical allodynia, via mechanisms that are not fully understood. This study identifies the reduced intrinsic excitability of PVINs as a potential cause for their decreased inhibitory output in nerve-injured condition. Combining computational and experimental approaches, we predict a calcium-dependent mechanism that modulates PVINs' electrical activity following nerve injury: a depletion of cytosolic calcium buffer allows for the rapid accumulation of intracellular calcium through the active membranes, which in turn potentiates SK channels and impedes spike generation. Our results therefore pinpoint SK channels as potential therapeutic targets for treating neuropathic symptoms.

Hopf Bifurcation Analysis of Mixed Traffic and Its Implications for Connected and Autonomous Vehicles

Capturing the evolution mechanism of traffic dynamics is of great significance when alleviating traffic congestion and improving traffic efficiency. While several studies have presented stability analyses of various traffic systems, most have relied on linear analysis methods, which cannot capture the complex nonlinear dynamic behavior of heterogeneous traffic flow. To address this shortcoming, this paper derives a generic Hopf bifurcation structure that can be applied to multiclass traffic models. The proposed bifurcation structure is investigated to understand the characteristics of heterogeneous traffic flow in a connected and autonomous environment as case studies. The results, based on selected on-field calibrated traffic models, show that (1) the linear analysis results deviate significantly from the actual instability of the mixed traffic system, which illustrates the necessity of a bifurcation analysis; and (2) connected human-driven vehicles and connected autonomous vehicles have the ability to alleviate the formation and propagation of traffic oscillations. Finally, the theoretical analysis results are verified through simulation experiments.

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.

Co-dimension two bifurcations of limit cycles starting from generalized Hopf bifurcation for the dynamical system of nuclear reactors

Nonlinear stability becomes critical when periodic and aperiodic oscillations arise and are important for the safe operation of nuclear reactors. The linear stability analysis and Hopf bifurcation are well studied in the context of nuclear reactors and fail to detect higher-order nonlinear entities. A reduced-order model which couples neutron dynamics with thermal-hydraulics is used in this work. Hopf bifurcation and limit cycles were reported earlier; however, the bifurcation analysis of limit cycles with two free parameters was not provided in the past and is investigated here. These co-dimension two bifurcations of limit cycles, which define the origin of several bifurcations such as (limit point (LPC), period doubling (PD), and Neimark-Sacker (NS) bifurcation of limit cycles), are our focus. We analyze the bifurcation starting from the Generalized Hopf (GH) and observe R1 resonance bifurcations along with the cusp bifurcation of limit cycles (CPC). We reveal dynamics near the R1 and quasiperiodic behavior are present in the vicinity. We observe cascading of R1 bifurcation due to multiple LPCs occurring in the supercritical Hopf region. We also detect CPC, which changes the direction of the LPC curve, which we call the global stability boundary. We present aperiodic and uncertain oscillations near R1, and for a safer operation, we should understand the existence of higher-order bifurcations.

A stage structured predator-prey model with periodic orbits

&lt;p style='text-indent:20px;'&gt;This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference.&lt;/p&gt;

BISTABILITY, GENERALIZED AND ZERO-HOPF BIFURCATIONS IN A PEST CONTROL MODEL WITH FARMING AWARENESS

Introduction of biopesticides and predators in the crop fields has a crucial impact on the pest management as the biopesticides induce infection among the pest population and the infected pests are easy to be captured by the predators. In this paper, we investigate a mathematical model for a pest management system with farming awareness-based inclusion of predators and biopesticides. The level of awareness among the farmers is assumed to grow at a constant rate besides its growth due to an increase in the pest population in the crop field. The proposed model is analyzed mathematically for the feasibility and stability of the equilibrium points; existence of Hopf bifurcation together with direction and stability of the bifurcating periodic solutions are discussed; sufficient conditions are derived for which the system does not exhibit any periodic solution. Our simulation results show that the activeness of farmers for employing biopesticides in the crop field potentially suppresses the oscillations and brings forth stability in the system. Further, we observe that the system showcases the appearances of Generalized Hopf bifurcation, Zero-Hopf bifurcation, transcritical bifurcation and bistability. Overall, our findings manifest that by imparting knowledge among the farmers regarding the use of biopesticides and predators in the crop fields, the pest population can be effectively controlled.

A business-cycle model connecting heterogeneous micro investment behaviors with macro dynamics

To analyze the effect of animal spirits on economic dynamics, we construct a simple Keynesian business-cycle model with animal spirits incorporated into it. Similar to the well-known Kaldor model, we assume that each firm possesses the investment function depending on demand, in which the animal spirits are connected with the degree of the propensity to invest. However, we would like to emphasize that, unlike the Kaldor model, even if the level of demand for each firm is the same, the degree of propensity to invest (i.e., the strength of animal spirits) depends on the change in demand. Thus, we assume that animal spirits in demand-upswing phases are stronger than those in demand-downswing phases. In each phase, the degree of dependence on demand is firm specific. Therefore, we adopt a statistical approach to connect such heterogeneous micro-investment behaviors with a macro-investment function. Through this approach, some kind of nonlinearity is naturally introduced into the macro-investment function. We demonstrate that the nonlinear macro-investment function yields persistent business cycles. We also demonstrate the occurrence of a generalized Hopf bifurcation.

An Investigation of the Bifurcation Behavior of an F-18 Aircraft Model

Bifurcations of equilibria and limit cycles of an F-18 aircraft model have been investigated in this paper based on one or two continuation parameters. First, it is shown that the aircraft can be in a stable straight-and-level flight state by coordinating the elevator deflection and the engine thrust and fixing the other parameters. Second, the aircraft may be disturbed by perturbations out of the longitudinal plane and get into the lateral-directional motion mode by a branch point bifurcation. Third, when the straight flight state loses its stability, body axis angular rates will present periodic or quasi-periodic motion pattern by the Hopf or Neimark–Sacker bifurcation. Finally, bifurcation structure of limit cycles has been exhibited, including the generalized Hopf bifurcation, the Neimark–Sacker bifurcation, the Hopf-Hopf bifurcation and the fold-Neimark–Sacker bifurcation. Meanwhile corresponding bifurcation curves in two-parameter plane have been depicted with the help of numerical continuation techniques.

Generalized Hopf Bifurcation of Semilinear FDEs in General Banach Spaces

In this paper, we investigate the generalized Hopf bifurcation for semilinear functional differential equations in general Banach spaces by applying the Lyapunov–Schmidt reduction process. We not only obtain explicit expressions about the parameters of the original system, but also use these expressions to determine whether there are several periodic solutions when the parameters are changed. Moreover, an example is given to illustrate the theoretical results.

Non linear stability analysis of supercritical water cooled reactor: Parallel channels configuration

The present work is devoted to the study of non-linear dynamics of the supercritical water reactor (SCWR) with a parallel channel configuration. To fulfil this objective, point reactor kinetics has been coupled with a parallel channel thermal-hydraulic model. The reduced ordered model is developed by coupling point kinetics equations with one group of delayed neutrons, fuel heat transfer and 1-D fluid flow model. The existing studies are commonly limited to linear stability analysis, therefore, present work focused on to demonstrate the nonlinear dynamics of the SCWR with parallel channel reactor core. The stability boundaries are shown in thermal hydraulic (pseudo phase-change and pseudo subcooling number) and intrinsic reactivity feedbacks (Doppler and density reactivity feedback) parametric spaces. Several bifurcation characteristics associated to density wave oscillations namely, subcritical, supercritical and generalized Hopf bifurcations are observed, which is also confirmed by calculating first Lyapunov coefficients. Corresponding to these phenomena, several numerical simulations are carried out at different parametric values, which confirms the existence and nature of the limit cycle behavior associated with these bifurcations. In addition, parametric sensitivity analysis on the stability characteristics are also carried out.

Endemic bubble and multiple cusps generated by saturated treatment of an SIR model through Hopf and Bogdanov–Takens bifurcations

The current study presents complex dynamics of an SIR epidemic model that incorporates a saturated type incidence rate as well as treatment. We provide here rigorous results for asymptotic stability of equilibrium states of the proposed system. Several bifurcations including Hopf, Generalized Hopf, saddle–node, transcritical and Bogdanov–Takens are also discussed. The stability of bifurcated periodic solutions is verified with the help of first Lyapunov number. Extensive numerical simulations are performed to validate these results. In a numerical example it is observed that if the saturation factor increases slowly, then the unique endemic equilibrium state is asymptotically stable for a certain range. The further increase in the value of saturation parameter, the endemic equilibrium state loses its stability and periodic solutions appear through Hopf bifurcation. It is also observed that the increase in saturation parameter beyond Hopf bifurcation threshold, results in regaining the stability of the endemic equilibrium state, which forms an interesting dynamical phenomenon in the bifurcation diagram named as an endemic bubble. It is pointed out that in the case of two endemic equilibrium states, one of these two is always saddle, whereas, the other one becomes unstable through Hopf bifurcation. In this scenario, the periodic solution is initially stable and it becomes unstable through generalized Hopf bifurcation. In numerical example for Bogdanov–Takens bifurcation two pairs of feasible bifurcation thresholds exist for the same set of parameters value. The bifurcation diagrams and equilibrium surfaces are also plotted to observe the combined effects of medication and saturation parameters.

Bifurcation analysis of a sheep brucellosis model with testing and saturated culling rate.

Testing-culling is a very effective measure for the prevention and control of animal diseases. In this paper, based on sheep brucellosis control policies and animal testing characteristics and considering the limitation of culling resources, a dynamic model is established to study the impact of testing-culling measure. Theoretical analysis reveals that the model may have one or three positive equilibria. The equilibrium in the middle is always unstable, and the model shows saddle-node bifurcation, generalized Hopf bifurcation and Bogdanov-Takens bifurcation. Moreover, the theoretical results are verified via numerical analysis. These results reveal that testing and culling strategies can induce complex transmission dynamics that can help us develop appropriate prevention and control measures for animal brucellosis.

Different Environment Feedback in Fast-Slow Eco-Evolutionary Dynamics and Resulting Limit Cycles

The fast-slow dynamics of an eco-evolutionary system are studied, where we consider the feedback actions of environmental resources that are classified into those that are self-renewing and those externally supplied. We show although these two types of resources are drastically different, the resulting closed-loop systems bear close resemblances, which include the same equilibria and their stability conditions on the boundary of the phase space, and the similar appearances of equilibria in the interior. After closer examination of specific choices of parameter values, we disclose that the global dynamical behaviors of the two types of closed-loop systems can be fundamentally different in terms of limit cycles: the system with self-renewing resources undergoes a generalized Hopf bifurcation such that one stable limit cycle and one unstable limit cycle can coexist; the system with externally supplied resources can only have the stable limit cycle induced by a supercritical Hopf bifurcation. Finally, the explorative analysis is carried out to show the discovered dynamic behaviors are robust in even larger parameter space.