Bifurcation Phenomena Research Articles

Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance

This paper deals with the analysis of nonlinear vibration phenomena in the arm system of an industrial robot. The presented mathematical model of the robot takes into account the flexibility of the joints, which results in low-frequency vibrations in the arm. The study of vibration phenomena has been carried out using analytical, numerical and experimental methods. Special attention has been paid to bifurcation phenomena. The influence of two bifurcation parameters on the behaviour of the robot arm was studied. It was shown that the amplitude and frequency of the harmonic excitation create a nonlinear effect on the vibration amplitudes of the links. In bifurcation regions, the vibration amplitudes can even differ several times depending on the direction of changes in the bifurcation parameter. In general, the results indicate that, contrary to common practice, it is worth paying attention to nonlinear phenomena when analysing robot vibrations.

Dynamic analysis of a new 4D fractional-order financial system and its finite-time fractional integral sliding mode control based on RBF neural network

In this paper, considering the combined effect of interest rate and investment demand on the price index, a new 4D fractional-order financial system is established. Based on the local stability theory, the stability of equilibrium points and static bifurcation phenomena are discussed. Subsequently, to further analyze the dynamic behaviors of the new financial system, numerical tools such as bifurcation diagrams, Lyapunov exponents, phase diagrams, 0–1 test diagrams, and complexity diagrams are employed. It is found that, for a suitable selection of the differential order or system parameter, the proposed system can generate periodic oscillations with different period and chaotic attractors with different shapes. Furthermore, a new controller combining radial basis function (RBF) neural network with fractional-order integral sliding mode control is proposed to realize the finite-time synchronization for a class of fractional-order systems. Finally, the effectiveness of the presented control scheme is verified by numerical simulations and the influence of RBF neural network parameters on the control performance is analyzed.

Dynamical behaviors and optimal control of delayed S2IS rumor propagation model with saturated conversion function over complex networks

Nowadays, online information is disseminated in multiple languages on different platforms, and research on the dissemination trend and control of online rumors is a topic worth discussing. In the bilingual environment, we study the S2IS rumor propagation model with time delay and saturated conversion function which implies the limitation of resources in homogeneous and heterogeneous networks. First and foremost, the analysis of the boundedness of the solutions and the existence of the rumor-spreading equilibrium points is given in our work, and we figure out the basic reproduction number. Next, we conduct the local stability analysis of the rumor-free equilibrium point and the rumor-spreading equilibrium points according to the characteristic equation. The global stability of the rumor-free equilibrium point is shown based on the Lyapunov stability theory. Besides, we study the forward and backward bifurcation phenomenon and optimal control problem in theory, which is seldom discussed in the bilingual environment. Finally, numerical simulations are performed to verify the correctness of our main results.

Assessing the Impact of Time-Varying Optimal Vaccination and Non-Pharmaceutical Interventions on the Dynamics and Control of COVID-19: A Computational Epidemic Modeling Approach

Vaccination strategies remain one of the most effective and feasible preventive measures in combating infectious diseases, particularly during the COVID-19 pandemic. With the passage of time, continuous long-term lockdowns became impractical, and the effectiveness of contact-tracing procedures significantly declined as the number of cases increased. This paper presents a mathematical assessment of the dynamics and prevention of COVID-19, taking into account the constant and time-varying optimal COVID-19 vaccine with multiple doses. We attempt to develop a mathematical model by incorporating compartments with individuals receiving primary, secondary, and booster shots of the COVID-19 vaccine in a basic epidemic model. Initially, the model is rigorously studied in terms of qualitative analysis. The stability analysis and mathematical results are presented to demonstrate that the model is asymptotically stable both locally and globally at the COVID-19-free equilibrium state. We also investigate the impact of multiple vaccinations on the COVID-19 model’s results, revealing that the infection risk can be reduced by administrating the booster vaccine dose to those individuals who already received their first vaccine doses. The existence of backward bifurcation phenomena is studied. A sensitivity analysis is carried out to determine the most sensitive parameter on the disease incidence. Furthermore, we developed a control model by introducing time-varying controls to suggest the optimal strategy for disease minimization. These controls are isolation, multiple vaccine efficacy, and reduction in the probability that different vaccine doses do not develop antibodies against the original virus. The existence and numerical solution to the COVID-19 control problem are presented. A detailed simulation is illustrated demonstrating the population-level impact of the constant and time-varying optimal controls on disease eradication. Using the novel concept of human awareness and several vaccination doses, the elimination of COVID-19 infections could be significantly enhanced.

Bifurcation analysis of combined double-diffusive natural convection and thermal radiation under a non-uniform magnetic field in a wavy enclosure

In the current work, the bifurcation phenomenon of a double-diffusive natural convection flow is analyzed for a non-uniform magnetic field applied to a wavy enclosure in the presence of radiation, for the first time. The finite volume method has been utilized to simulate volumetric radiation. To solve the continuity, momentum, energy, and species equations, the multiple-time relaxation lattice Boltzmann technique is adopted. In addition, the immersed boundary approach has been applied to identify the complex boundaries inside the Cartesian coordinates system. The effects of specific pertinent parameters, including the optical thickness (0.1≤τ≤100), Buoyancy ratio (-5≤N≤5), Rayleigh number (104≤Ra≤105), wave amplitude (0≤λ≤0.2), and Hartmann number (0≤Ha≤50) on thermohydrodynamic characteristics of the system have been investigated. The results indicated that an increase in Hartmann number disappears the bifurcation phenomenon. In addition, it was observed that the initial condition of hot temperature and high concentration leads to instability in the flow. In the radiation case for λ>0.1, 5<τ<30, and 15≤Ha≤30, the flow appears unsteady. Bifurcation occurs in all values of wavy amplitude, and optical thickness studied. Heat and mass transfers decrease when the initial condition is cold temperature and low concentration. Also, Radiation does not affect the critical Rayleigh number.

Assessing the role of active case detection on visceral leishmaniasis control: A case study

In this paper, we formulate and analyze a compartmental model of visceral leishmaniasis (VL). We validate our model by calibrating it to the yearly VL incidence data for India and Bangladesh. The proposed model’s basic reproduction number ([Formula: see text]) has been derived and estimated. We have proved the existence of backward bifurcation in our system. The phenomenon of backward bifurcation has public health implications because the classical requirement of [Formula: see text], while necessary, is no longer sufficient for effective disease control (or elimination). In such a (backward bifurcation) situation, the initial sizes of the model’s subpopulations (state variables) would determine the effectiveness of disease control or elimination. As a result, it is the first attempt to represent and study VL disease dynamics using active case detection (ACD) as a control strategy, although a few experimental studies have been conducted to evaluate ACD in disease transmission. We use sensitivity analysis to investigate the effects of the model’s controllable parameters for the basic reproduction number. We found that [Formula: see text] (monitoring rate to infected individuals) have negative impacts on [Formula: see text]. The numerical result suggests that the ACD strategy can be useful for the VL elimination program. Due to a small increment in the monitoring rate, the dynamic behavior of infected individuals dramatically decreased. Successful employment of ACD strategy may reduce more than 60–80% symptomatic and post kala-azar dermal leishmaniasis (PKDL) individuals. We found that healthcare organizations should prioritize ACD in symptomatic and PKDL individuals over asymptomatic individuals. We also observed that the use of only culling effect to the reservoirs is not beneficial to society in the control of VL, but spraying of insecticides and the use of treated bednets can be effective control strategies to curtail the outbreak of the disease.

Cusp bifurcation in a metastatic regulatory network

Understanding the potential for cancers to metastasize is still relatively unknown. While many predictive methods may use deep learning or stochastic processes, we highlight a long standing mathematical concept that may be useful for modeling metastatic breast cancer systems. Ordinary differential equations (ODEs) can model cell state transitions by considering the pertinent environmental variables as well as the paths systems take over time. Bifurcation theory is a branch of dynamical systems which studies changes in the behavior of an ODE system while one or more parameters are varied. Many studies have applied concepts in one-parameter bifurcation theory to model biological network dynamics, and cell division. However, studies of two-parameter bifurcations are much more rare. Two-parameter bifurcations have not been studied in metastatic systems. Here we show how a specific two-parameter bifurcation phenomenon called a cusp bifurcation separates two qualitatively different metastatic cell state transitions modalities and propose a new perspective on defining such transitions based on mathematical theory. We hope the observations and verification methods detailed here may help in the understanding of metastatic potential from a basic biological perspective and in clinical settings.

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

Online Deterministic Annealing for Classification and Clustering.

Inherent in virtually every iterative machine learning algorithm is the problem of hyperparameter tuning, which includes three major design parameters: 1) the complexity of the model, e.g., the number of neurons in a neural network; 2) the initial conditions, which heavily affect the behavior of the algorithm; and 3) the dissimilarity measure used to quantify its performance. We introduce an online prototype-based learning algorithm that can be viewed as a progressively growing competitive-learning neural network architecture for classification and clustering. The learning rule of the proposed approach is formulated as an online gradient-free stochastic approximation algorithm that solves a sequence of appropriately defined optimization problems, simulating an annealing process. The annealing nature of the algorithm contributes to avoiding poor local minima, offers robustness with respect to the initial conditions, and provides a means to progressively increase the complexity of the learning model, through an intuitive bifurcation phenomenon. The proposed approach is interpretable, requires minimal hyperparameter tuning, and allows online control over the performance-complexity tradeoff. Finally, we show that Bregman divergences appear naturally as a family of dissimilarity measures that play a central role in both the performance and the computational complexity of the learning algorithm.

Steady-State Bifurcation and Spatial Patterns of a Chemical Reaction System

This paper studies the Lengyel–Epstein chemical reaction system with nonlinear functional response and no-flux boundary conditions. We first investigate the existence of steady-state bifurcation solutions of the system. Then, the stability of bifurcation solutions is analyzed. Meanwhile, some spatial patterns induced by steady-state bifurcation are simulated numerically and depicted graphically. It is well known that the classical bifurcation theory in nonlinear dynamical systems is based on simple eigenvalues. However, this is not always the case. Sometimes, the kernel of an objective operator is of two or more dimensions. For such cases, there is no existing theory to deal with them. In this paper, by using the space decomposition technique and implicit function theorem, we analyze the bifurcation phenomenon of the system in the case of a two-dimensional kernel of some certain objective operator. The results show that the chemical reaction between activator iodide and inhibitor chlorite can proceed stably under certain conditions.

Modelling the impacts of media campaign and double dose vaccination in controlling COVID-19 in Nigeria

Corona virus disease (COVID-19) is a lethal disease that poses public health challenge in both developed and developing countries of the world. Owing to the recent ongoing clinical use of COVID-19 vaccines and non-compliance to COVID-19 health protocols, this study presents a deterministic model with an optimal control problem for assessing the community-level impact of media campaign and double-dose vaccination on the transmission and control of COVID-19. Detailed analysis of the model shows that, using the Lyapunov function theory and the theory of centre manifold, the dynamics of the model is determined essentially by the control reproduction number (Rmv). Consequently, the model undergoes the phenomenon of forward bifurcation in the absence of the double dose vaccination effects, where the global disease-free equilibrium is obtained whenever Rmv≤1. Numerical simulations of the model using data relevant to the transmission dynamics of the disease in Nigeria, show that, certain values of the basic reproduction number ((R0≥7)) may not prevent the spread of the pandemic even if 100% media compliance is achieved. Nevertheless, with assumed 75% (at R0=4)) media efficacy of double dose vaccination, the community herd immunity to the disease can be attained. Furthermore, Pontryagin's maximum principle was used for the analysis of the optimized model by which necessary conditions for optimal controls were obtained. In addition, the optimal simulation results reveal that, for situations where the cost of implementing the controls (media campaign and double dose vaccination) considered in this study is low, allocating resources to media campaign-only strategy is more effective than allocating them to a first-dose vaccination strategy. More so, as expected, the combined media campaign-double dose vaccination strategy yields a higher population-level impact than the media campaign-only strategy, double-dose vaccination strategy or media campaign-first dose vaccination strategy.

Dynamic mode decomposition of thermocapillary convection in GaAs melt liquid bridge between unequal ends

The effects of the radius ratio on the stability of thermocapillary convection in the GaAs melt (Pr=0.068) liquid bridge, situated between unequal ends, are studied by three-dimensional direct numerical simulation. The simulated results reveal that, for the first time, the GaAs melt flow transforms from a stable axisymmetric flow into a three-dimensional steady flow through the first bifurcation at 0.5≤Γr≤0.9. The bifurcation phenomenon was previously common in the range of small Prandtl numbers(0.001≤Pr≤0.057). Additionally, we observe that the stability of the thermocapillary liquid bridge is initially weakened and then slowly improved as the radius ratio Γr decreases. The second bifurcation occurs with the further increase of Ma, and the flow becomes oscillatory. Remarkably, two different oscillation forms are identified, depending on the difference in the degree of symmetry of the flow field structure during its oscillation process. Once the flow reaches a stable periodic oscillation, the Dynamic Mode Decomposition (DMD) technique is used to study the inherent spatio-temporal evolution of the temperature field. By combining the spectrum obtained from the Fast Fourier Transform (FFT), we provide evidence for the existence of mixed forms with different azimuthal wavenumbers from a dynamic perspective, intuitively displaying the different components of the oscillation and examining the contribution of different modes to the oscillation. The dynamic model analysis demonstrates that oscillation form I is a mixed form with m = 1 and m = 2, while oscillation form II belongs to the form where m = 2 is absolutely dominant.

Mathematical analysis of the dynamics of a fractional‐order tuberculosis epidemic in a patchy environment under the influence of re‐infection

In human societies, memory has a significant impact on the procedure of progression and control of epidemics. Therefore, fractional differential equations govern an epidemic model, including the memory effect. Since infectious diseases can spread from one place to another, human movement impacts the prevalence of such diseases. In this study, we present a two‐patch model for tuberculosis with fractional order and exogenous re‐infection, in which susceptible individuals can move without any obstacle between the patches. The present work establishes the effect of population movement and the backward bifurcation phenomenon on the prevalence of the tuberculosis epidemic with fractional‐order derivatives. Furthermore, to show that the proposed model has backward bifurcation, we provide some conditions between the model's parameters, so in this case, the model of tuberculosis has numerous boundary points. We show the tuberculosis model can exhibit backward bifurcation neither in case of exogenous re‐infection, with some conditions, nor in the situation that the model is vacant from any re‐infection. In contrast, the disease‐free equilibrium of the model is globally asymptotically stable when . Furthermore, we investigate the fractional‐order derivative on the spread of infection. The results show that can perform the function of adventure or expertise of people about the disease background.

Novel extended mixed controller design for bifurcation control of fractional‐order Myc/E2F/miR‐17‐92 network model concerning delay

MiR‐17‐92 has a vital effect on the adjustment of the Myc/E2F protein in chemistry. In this work, we propose a novel fractional‐order delayed Myc/E2F/miR‐17‐92 network model that revels the relation between miR‐17‐92, E2F, and Myc. Taking advantage of Laplace transform, we obtain the characteristic equation of the established fractional‐order delayed Myc/E2F/miR‐17‐92 network model. By virtue of stability theorem and bifurcation criterion of fractional‐order dynamical equation, a novel delay‐independent condition guaranteeing the stability and the generation of bifurcation phenomenon of the involved model is acquired. By exploiting a proper extended mixed controller (including controller and delayed feedback controller), we efficiently control the stability domain and the emergence of bifurcation for the the involved model. By utilizing another suitable extended mixed controller (including controller and hybrid controller), we can successfully adjust the stability domain and the emergence of bifurcation for the the involved model. The research indicates that delay plays a vital role in stabilizing system and controlling bifurcation of the fractional‐order delayed Myc/E2F/miR‐17‐92 network model. Computer experiment results illustrate the scientificity of the gained key analytical outcomes. The acquired results in this research are totally innovative and own immense theoretical meaning in regulating the concentrations of different proteins.

Cycle Skipping-Induced Bifurcation Analysis of Single-Inductor Bipolar-Output DC–DC Converters With One-Cycle Control

Due to strong nonlinear coupling effect amongst input/output ports and special cycle skipping effect, single-inductor bipolar-output (SIBO) DC-DC converters with one-cycle control may exhibit various bifurcation phenomena with complex border collision. This brief aims to investigate the cycle skipping effect and explore the occurrence mechanism of these nonlinear behaviors for stability improvement. On account of the cycle skipping effect, a set of the discrete-time mathematical expressions are derived to describe the dynamic characteristics of the one-cycle controlled (OCC) SIBO converter in discontinuous state space. Then, bifurcation analysis is performed by numerical simulation and eigenvalue trajectories. It is shown that cycle skipping effect activates more multi-cycle modes except single-cycle modes. Interestingly, the same slow-scale bifurcations in different operation region present various oscillatory form, which leads to hybrid-scale instability. Especially, the stability boundaries are presented in multi-mode space to guide optimal design. Finally, some experiment results are provided to verify the theoretical analysis.

Mathematical modeling and analysis of COVID-19 and TB co-dynamics

This study proposes a mathematical model for examining the COVID-19 and tuberculosis (TB) co-dynamics thoroughly. First, the single infection dynamics: COVID-19 infection and TB infection models are taken into consideration and examined. Following that, the co-dynamics with TB and COVID-19 is also investigated. In order to comprehend the developed model dynamics, the basic system attributes including the region of definition, theory of nonnegativity and boundedness of solution are investigated. Further, a qualitative analysis of the equilibria of the formulated model equations is performed. The equilibria of both infection models are globally asymptotically stable if their respective basic reproductive number is smaller than one. As the associated reproductive number reaches unity, they experience the forward bifurcation phenomenon. Additionally, it is demonstrated that the formulated co-dynamics model would not experience backward bifurcation by applying the center manifold theory. Moreover, model fitting is done by using daily reported COVID-19 cumulative data in Ethiopia between March 13, 2020, and May 31, 2022. For instance, the non-linear least squares approach of fitting a function to data was performed in the fitting process using scipy.optimize.curve_fit from the Python. Finally, to corroborate the analytical findings of the model equation, numerical simulations were conducted.

Synchronous mass detection based on 1:1:2 internal resonance of piezo-driven resonator

The bifurcation-based mass sensor can take advantage of amplitude jumps between multiple steady states to realize quantitative mass detection and mass warning. Compared with the classical detection based on the frequency shift induced by an added mass, sudden jumps in amplitude make the detection of a very small mass possible. Most research has been focused on the mass detection of a single analyte, but less research has been done on the detection of double analytes. In this study, the concept of synchronous mass detection for double analytes based on bifurcation induced by 1:1:2 internal resonance is first proposed and verified experimentally. Firstly, a piezo-driven three-degree-of-freedom (3-DOF) magnetic coupling resonant structure with the potential of 1:1:2 internal resonance is designed. Through driving the high-frequency resonant beam, the parametric vibration of two low-frequency resonant beams is realized, which causes the modal coupling vibration phenomenon in the three frequency intervals. The physical conditions of multi-mode coupling vibration are also deduced theoretically. Typically, by measuring the shifts of the upper and the lower critical bifurcation frequencies, the quantitative mass detection of the double analytes can be carried out simultaneously. Beside, a conceptual scheme with multiple mass warnings for double analytes is also creatively proposed. The experimental results show that with the increase of the adsorption mass, the resonant amplitude of the system obviously changes near the equilibrium point, accompanied by a bifurcation phenomenon. The research results provide a systematic framework for the development of resonant mass sensing devices based on multi-mode coupled vibration.

Bending–torsion coupled wave in thin-walled mono-symmetric metabeam: A non-dimensional analysis

The influence of the periodic resonators on wave dispersion of a thin-walled mono-symmetric beam is studied in this paper. The primary focus is to develop a non-dimensional framework to investigate the coupled bending–torsion wave propagation analytically. The resonators are placed eccentrically and are capable to form a subwavelength attenuation band in the beam due to its periodicity. The governing equations corresponding to the bending and torsional motion of a metabeam are presented in a dimensionless form, including the warping effect. The transfer matrix approach is adopted to obtain the dispersive phenomena using the Bloch–Floquet principle. The modal characteristics of a mono-symmetric beam without a resonator are computed from the frequency response function and are validated with finite element results. Subsequently, the impact of various non-dimensional parameters on the attenuation band and band merging, conversion, and bifurcation phenomena are investigated through a comprehensive parametric study. It is observed that the occurrence of the band gap primarily depends on the resonator properties and its location. A substantial band gap is achieved in the low-frequency range, which will be useful for the application in the field of vibration control and acoustics.

Tanh-like models for analysis and prediction of time-dependent flow around a circular cylinder at low Reynolds numbers

When employing traditional low-order approximation equations to forecast the Hopf bifurcation phenomenon in the wake of a circular cylinder at low Reynolds numbers, inaccuracies may arise in estimating the phase. This is due to the fact that, in this transition process, the frequency varies with time. In this paper, we propose a method for analyzing and predicting the vortex shedding behind a cylinder at low Reynolds numbers. The proposed method is based on coordinate transformation and description function and is demonstrated using data from computational fluid dynamics simulation of flow around a cylinder at Reynolds number 100. The resulting governing equations explicitly contain the flow amplitude and implicitly contain the flow frequency. The proposed method is found to have higher accuracy compared to other methods for nonlinear identification and order reduction. Finally, the method is extended to predict nonlinear vortex shedding in the Reynolds number range of 80–200.

Modeling the dynamics of COVID-19 in the presence of Delta and Omicron variants with vaccination and non-pharmaceutical interventions

Since its inception in December 2019, many safe and effective vaccines have been invented and approved for use against COVID-19 along with various non-pharmaceutical interventions. But the emergence of numerous SARS-CoV-2 variants has put the effectiveness of these vaccines, and other intervention measures under threat. So it is important to understand the dynamics of COVID-19 in the presence of its variants of concern (VOC) in controlling the spread of the disease. To address these situations and to find a way out of this problem, a new mathematical model consisting of a system of non-linear differential equations considering the original COVID-19 strain with its two variants of concern (Delta and Omicron) has been proposed and formulated in this paper. We then analyzed the proposed model to study the transmission dynamics of this multi-strain model and to investigate the consequences of the emergence of multiple new SARS-CoV-2 variants which are more transmissible than the previous ones. The control reproduction number, an important threshold parameter, is then calculated using the next-generation matrix method. Further, we presented the discussion about the stability of the model equilibrium. It is shown that the disease-free equilibrium (DFE) of the model is locally asymptotic stable when the control reproduction is less than unity. It is also shown that the model has a unique endemic equilibrium (EEP) which is locally asymptotic stable when the control reproduction number is greater than unity. Using the Center Manifold theory it is shown that the model also exhibits the backward bifurcation phenomenon when the control reproduction number is less than unity. Again without considering the re-infection of the recovered individuals, it is proved that the disease-free equilibrium is globally asymptotically stable when the reproduction threshold is less than unity. Finally, numerical simulations are performed to verify the analytic results and to show the impact of multiple new SARS-CoV-2 variants in the population which are more contagious than the previous variants. Global uncertainty and sensitivity analysis has been done to identify which parameters have a greater impact on disease dynamics and control disease transmission. Numerical simulation suggests that the emergence of new variants of concern increases COVID-19 infection and related deaths. It also reveals that a combination of non-pharmaceutical interventions with vaccination programs of new more effective vaccines should be continued to control the disease outbreak. This study also suggests that more doses of vaccine should provide to combat new and deadly variants like Delta and Omicron.