Equilibrium Bifurcation Research Articles

Analysis of global stability and bifurcation for an HIV infection model with cell to cell transmission

Cell-to-cell infection cannot be ignored in the development of HIV in the host. The mathematical difficulty in (Wang et al. in J. Biol. Dyn. 11:455–483, 2016) is mainly due to the assumption of the equality of two parameters, in which they are the proportions of infection that lead to latency caused by virus-to-cell infection and cell-to-cell transmission, respectively. To overcome the restricted condition, we propose a more general HIV development model with virus-to-cell and cell-to-cell infection patterns with logistic growth and saturation incidence. By constructing a proper Lyapunov function we obtain the global stability of the disease-free equilibrium without this restricted condition, thereby the main result in (Wang et al. in J. Biol. Dyn. 11:455–483, 2016) removing the restricted condition is proved by using our method even if two parameters are not equal. We also investigate the existence of Hopf bifurcation of diseased equilibrium in four cases.

A mathematical model to study the role of dystrophin protein in tumor micro-environment

In this research work, the authors have developed a mathematical model to examine the interaction between dystrophin protein and tumor. The authors formulated a system of ordinary differential equations to describe the dynamics of the dystrophin-tumor interaction system. Jacobian matrix and Routh–Hurwitz stability techniques were used to determine equilibrium points, perform stability and bifurcation analysis, and establish the conditions required for the stability of the proposed model. Numerical simulations are performed using Euler’s method to investigate the temporal evolution of the proposed model under different parameter values, such as tumor growth rate and feedback strength of dystrophin protein. The numerical results are presented in tables, and corresponding to each table, a graphical analysis is done. The graphical analysis includes creating phase portraits to visually represent stability regions around the equilibrium points, bifurcation diagrams to identify critical points, and time series analysis to highlight the behavior of the proposed model. The authors explore how variations in dystrophin expression impact tumor progression, identifying potential therapeutic implications of maintaining higher dystrophin levels. This comprehensive analysis enhances our understanding of the dystrophin-tumor interaction, providing a basis for further experimental validation and potential therapeutic strategies.

Synchronization of Chaotic Satellite Systems with Fractional Derivatives Analysis Using Feedback Active Control Techniques

This research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system’s dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems.

Consensus Control of Leader–Follower Multi-Agent Systems with Unknown Parameters and Its Circuit Implementation

With the development and progress of Internet and data technology, the consensus control of multi-agent systems has been an important topic in nonlinear science. How to effectively achieve the consensus of leader–follower multi-agent systems at a low cost is a difficult problem. This paper analyzes the consensus control of complex financial systems. Firstly, the dynamic characteristics of the financial system are analyzed by the equilibrium points, bifurcation diagrams, and Lyapunov exponent spectra. The behavior of the financial system is discussed by different parameter values. Secondly, according to the Lyapunov stability theorem, the consensus of master–slave systems is proposed by linear feedback control, wherein the controllers are simple and low cost. And an adaptive control method for the consensus of master–slave systems is investigated based on financial systems with unknown parameters. In theory, the consensus of the leader–follower multi-agent systems is proved by the parameter identification laws and linear feedback control method. Finally, the effectiveness and reliability of the consensus of leader–follower multi-agent systems are verified through the experimental simulation results and circuit implementation.

Stability and bifurcations for a 3D Filippov SEIS model with limited medical resources

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models.

Oscillations in a tumor-immune system interaction model with immune response delay.

In this paper we consider a tumor-immune system interaction model with immune response delay, in which a nonmonotonic function is used to describe immune response to the tumor burden and a time delay is used to represent the time for the immune system to respond and take effect. It is shown that the model may have one, two or three tumor equilibria, respectively, under different conditions. Time delay can only affect the stability of the low tumor equilibrium and local Hopf bifurcation occurs when the time delay passes through a critical value. The direction and stability of the bifurcating periodic solutions are also determined. Moreover, the global existence of periodic solutions is established by using a global Hopf bifurcation theorem. We also observe the existence of relaxation oscillations and complex oscillating patterns driven by the time delay. Numerical simulations are presented to illustrate the theoretical results.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov–Hopf, Bogdanov–Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-2$$\\end{document}, where m is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov–Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks.

Bistable click mechanism for dipteran flight robot

The flight mechanism of the dipteran insects is commonly used to design the flapping robot. However, the nonlinear dynamics of the buckling click mechanism of the flapping robot with bistability still unclear. In this paper, a novel flapping robot model with the click mechanism proposed based on the bistability of snap-through buckling. The nonlinear governing equations of the flight robot are obtained by using the Euler–Lagrange equation. The nonlinear potential energy, moment of the elastic force, equilibrium bifurcation, as well as equilibrium stability are investigated to show the bistable characteristics. The transitional and persistent sets of bifurcation regions in the parameter plane and the corresponding phase portraits are obtained with single and double well behaviors. Then, the periodic response of the free flight are defined by the analytical solution of three kinds of elliptical functions, as well as the amplitude frequency responses are investigated by numerical integration. Based on the topological equivalent, the chaotic thresholds of the homo-clinic and heteroclinic orbits for the chaotic vibration of the perturbed robot system are derived. Finally, the prototype of nonlinear flapping robot is manufactured and the experimental system is setup. The nonlinear static moment of force curves, periodic response and dynamic flight vibration of dipteran robot system are carried out. It is shown that the test results are agree well with the theoretical analysis and numerical simulation. Those results have the potential application for the structure design of the efficient flight robot.

Frequency switching leads to distinctive fast–slow behaviors in Duffing system

The slowly forced Duffing system has been found to exhibit unique fast–slow behaviors in descending frequency switching scheme, the typical of which is the sliding bursting, which is instructive for understanding the dynamics of ubiquitous frequency switching systems in scientific research and practical engineering. This paper is devoted to further refining the fast–slow dynamics of the slowly forced Duffing system with another commonly encountered frequency switching scheme, i.e., the ascending frequency switching, characterized by switching the frequency synchronously according to the increase or decrease of state variable values. Taking the forcing amplitude as an example, this paper provides a theoretical way to fully summarize the fast–slow behaviors in frequency switching systems with the variation of parameter values based on the proposed superposition analysis of one- and two-parameter bifurcations of subsystems. As a result, two typical bifurcation structures and eight threshold windows with distinct switching vector fields therein are identified, inducing up to ten different bursting patterns. Among them, several novel fast–slow dynamics, such as multiple jumps hysteresis loop formed by boundary equilibrium bifurcations and the switching failure phenomena, are presented and investigated. In particular, the underlying mechanism of threshold modulation, i.e., the evolution of unconventional bifurcations, is also found. These findings contribute to complementing and contrasting the existing studies on the fast–slow dynamics of frequency switching systems.

Mosquito suppression via Filippov incompatible insect technique

Mosquito-borne diseases persist as a global health challenge despite ongoing control efforts, necessitating the exploration of alternative control approaches. This research proposes a Filippov incompatible insect technique (IIT) model with a threshold policy control for suppressing mosquito population. The model implements biological and chemical control strategies only when wild mosquito density surpasses an economic threshold. The biological strategy involves releasing Wolbachia-infected male mosquitoes, offering advantages such as a shortened lifespan and disease blocking. Simultaneously, moderate insecticide and larvicide spraying serve as a chemical approach to expedite the control process. The model provides economic advantages by minimizing control costs and environmental benefits through the integration of biological strategies, thus reducing reliance on chemicals. By employing Filippov's convex method, we identify necessary conditions for the existence of sliding segments and determine the dynamics of the switching line. Theoretical analysis reveals the model's long-term dynamics, including sliding bifurcations, pseudo-equilibria, and their stability. The model exhibits boundary equilibrium bifurcations and transcritical bifurcations in both free and controlled systems. Varying threshold values significantly impact the control outcomes effectiveness. The main findings demonstrate that integrated strategies efficiently decrease the population density of wild mosquitoes.

Jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator for low-frequency vibration energy harvesting

In pursuit of enhancing the output power of energy harvesters, this paper introduces a novel jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator (JIB-PTHG) with low potential energy barrier characteristics and low-frequency, broadband energy harvesting capabilities. The JIB-PTHG comprises two flexible beams, and two rigid links, connected via a spring. Utilizing the harmonic balance method and Runge-Kutta fourth-order algorithm, a comprehensive analysis of the mechanical properties of the low-barrier bistable structure is conducted. Static equilibrium bifurcation and dynamic response analysis are performed. Subsequently, an electromechanical coupling model incorporating piezoelectricity and triboelectricity is developed to theoretically predict the output voltage of the two modules, revealing a positive correlation between displacement and output voltage. Experiment validation demonstrated that adjusting the length of the rigid links can effectively increase the deformation of the piezoelectric beam and the relative displacement of the triboelectric layers, thus improving the generator's output performance. Specifically, the TENG module exhibited exceptional energy harvesting performance within the frequency range of 3.5–6.5 Hz, achieving a voltage of 1050 V, a power of 1.84 mW, and the ability to illuminate 70 LED lights under an excitation amplitude of 17.5 mm and a frequency of 5.5 Hz. The PEG module, displayed high output performance within the frequency range of 4-7 Hz, reaching a voltage of 6.2 Vs and a power of 12.8 μW at an excitation frequency of 7 Hz and an amplitude of 17.5 mm. Both theoretical and experimental findings indicate that the JIB-PTHG has achieved improved output power within the frequency range of 3.5–6.5 Hz. Furthermore, the JIB-PTHG has the potential to harness bridge vibration energy and convert it into useful electrical energy for powering wireless sensor nodes in bridge health monitoring systems.

Equilibrium bifurcation and extreme risk in the EU carbon futures market

Considering the long-term memory and volatility clustering of the European Union (EU) Carbon Emission Allowances (EUA) futures returns, based on the economy–energy–environment system perspective and the assumption of investors' heterogeneity, this study proposes a joint modeling approach combining the fractionally integrated generalized autoregressive conditional heteroscedasticity model (FIGARCH) and the stochastic cusp catastrophe model (SCC) to examine the equilibrium bifurcations and extreme risks in the EU carbon futures market. The relevant results are threefold. (1) The SCC model has good fitting effect and interpretability, and is an effective method for investigating catastrophe reactions under time-varying volatility conditions. (2) In the EUA futures market, chartists are mainly affected by short-term price and trading volume changes, which leads to the emergence of equilibrium bifurcations, while fundamentalists make investment decisions based on the economy, the energy market, and market supply–demand, which affects the asymmetry of equilibrium bifurcations. (3) Using the catastrophe criterion (i.e., Cardan's discriminant of the equilibrium surface equation), we identify148 equilibrium bifurcation time points in the EUA futures market from December 3, 2009 to September 16, 2020, most of which are concentrated in two upward periods with an average scale of extreme risks is about 32.51 %. Our analysis provides theoretical support for regulatory authorities to stabilize the carbon futures market and build a collaborative extreme risk management framework covering energy and macroeconomics, also proposing suggestions for traders to effectively prevent extreme risks.

SIHQR model with time delay for worm spread analysis in IIoT-enabled PLC network

Industrial Internet of Things (IIoT) is a product of deep integration between Internet of Things (IoT) and industrial control networks. As an important component of IIoT, Programmable Logic Controller (PLC) has become a springboard for many hackers to disrupt industrial networks by spreading viruses. It is known that worm can cause enormous loss. Therefore, revealing the rules of worm spread in PLC networks and exploring methods to inhibit worm spread are increasingly important. To this end, we propose a new worm spread model named SIHQR with time delay. Additionally, we establish a game model between susceptible and infected nodes in order to express the infection rate by several game parameters. Through analysis, we discuss the conditions for the emergence of three scenarios: disease-free equilibrium, endemic equilibrium and Hopf bifurcation. We study the stability at the equilibrium points and obtain an expression for a threshold value which determines whether the system exhibits endemic equilibrium or Hopf bifurcation. In the experimental section, we analyze the impact of initial values of node states and the magnitude of game parameters on worm spread in the three aforementioned scenarios. Finally, we propose a method to mitigate or even eliminate the Hopf bifurcation. The experimental results show that the proposed model has welldone performance.

Two-Parameter Bifurcations and Hidden Attractors in a Class of 3D Linear Filippov Systems

We take into consideration two different kinds of two-parameter bifurcations in a class of 3D linear Filippov systems, namely pseudo-Bautin bifurcation and boundary equilibrium bifurcations for two scenarios. The bifurcation conditions for generating rich dynamic behaviors are established. The main objective is to investigate the effects of two parameters interacting simultaneously on a variety of dynamic phenomena. In order to analyze the pseudo-Bautin bifurcation, we build the Poincaré map and analyze the number of fixed points whose types are related to the crossing limit cycles. In order to analyze boundary equilibrium bifurcations for two scenarios, we perform an analysis on the existence and admissibility of equilibria. Besides, a comprehensive investigation on hidden attractors induced by boundary equilibrium bifurcations is conducted. The novelty resides in overcoming the constraints of previous studies that solely take into account the dynamics of individual parameter variations. We innovatively characterize the two-parameter bifurcation mechanism of a new class of Filippov systems, and qualitatively demonstrate the coexistence of hidden attractor and stable pseudo-equilibrium.

Non-smooth dynamics of a fishery model with a two-threshold harvesting policy

The non-linear dynamical systems theory helps implement regulatory measures to control the growth and evolution of various populations. While invasion by alien fish species is an emerging threat to native fish species in marine ecosystems, a suitable fishery management protocol needs to be incorporated in marine protected areas (MPAs) to mitigate the problem. We propose a policy of selective harvesting whenever the density of a species crosses a predefined threshold. An ecosystem with such a harvesting policy is modelled by a piece-wise smooth prey-predator type fishery model, where a control function defined by two thresholds evokes signals to cease or commence harvesting. This work reports the dynamics and bifurcations of a Filippov system, including the possibility of sliding mode dynamics. We show that the variations in the growth rate and the harvesting rate of the invasive fish species can lead to hysteresis through saddle–node and transcritical bifurcations and the appearance of a stable homoclinic orbit. We obtain the conditions for sliding domains, regular or virtual equilibria, boundary equilibria, pseudo-equilibria, and tangent points. Depending on the parameter choice, the system can stabilize at a regular equilibrium, a pseudo-equilibrium, or a pseudo-attractor, exhibiting multistability. We report the occurrence of regular or virtual equilibrium bifurcation, boundary node bifurcation and pseudo-saddle–node bifurcation. Our findings indicate that proper harvesting strategies can prevent the proliferation of invasive fish species in MPAs.

ANALYSIS OF THE COOPERATIVE LOTKA-VOLTERRA MODEL IN THE CASE OF TWO AUTOMOTIVE INDUSTRIES IN INDONESIA

The automotive industry sector is one of the main sectors contributing to the national economy. Companies in the automotive industry work together to increase productivity and win the market competition. This research aims to construct the cooperative interaction between two companies into the cooperative Lotka-Volterra model. The cooperative Lotka-Volterra model was analyzed for stability at the equilibrium point and bifurcation analysis was performed. Sales data for the Calya 1.2 G product from PT Toyota Motor Manufacturing Indonesia and sales data for the New Sigra 1.2 R MT product from PT Astra Daihatsu Motor are applied to the model. The results of the study show that there is a model that explains the cooperative interaction of the two companies, namely the cooperative Lotka-Volterra model. Four equilibrium points are obtained with three equilibrium points being unstable and the fourth equilibrium point being stable with conditions. The Hopf bifurcation analysis of the model shows that there are no parameters that cause a change in stability from initially stable to unstable. The data simulation shows that the cooperation of the two companies is mutually beneficial because it increases the number of sales and creates balanced market conditions.

Regularization of the Boundary Equilibrium Bifurcation in Filippov System with Rich Discontinuity Boundaries

This paper studies a particular type of planar Filippov system that consists of two discontinuity boundaries separating the phase plane into three disjoint regions with different dynamics. This type of system has wide applications in various subjects. As an illustration, a plant disease model and an avian-only model are presented, and their bifurcation scenarios are investigated. By means of the regularization approach, the blowing up method, and the singular perturbation theory, we provide a different way to analyze the dynamics of this type of Filippov system. In particular, the boundary equilibrium bifurcations of such systems are studied. As a consequence, the nonsmooth fold bifurcation becomes a saddle-node bifurcation, while the persistence bifurcation disappears after regularization.

A novel nonlinear oscillator consisting torsional springs and rigid rods

In this paper, we propose an archetypal double-winged novel nonlinear oscillator composed of torsional springs and rigid rods which behaves smooth and discontinuous dynamics and a kind of special collision pendulum depending on the varying of a geometrical parameter. The system behaves complicated equilibrium bifurcations with different geometrical states, leading to a series of buckling phenomenon and the loss of uniqueness near the limit points. A ‘collision’ parameter is introduced to determine the motions near the limit points which causes a discontinuous jump of velocity and exhibits the standard dynamics as an inverted pendulum, even though the mechanic model is rather different from a pendulum. We also distinguish the coexistence of periodic and chaotic motions and also a smooth period 1 located in a small region under the presence of damping and external force using amplitude–frequency analysis and numerical simulations. This research enriches the complicated dynamical behaviors of geometrical nonlinear oscillator and provide a novel structure which is of great potential in vibration isolation engineering.

DISPERSAL IN PREY–PREDATOR SYSTEMS WITH PREY’S FEAR AND PREDATOR’S ADAPTIVE SEARCH

This paper considers predator–prey systems in which the predator adapts the search speed to prey’s density. The prey admits the cost of fear because of the predator and moves between source–sink patches to escape from predation. Using dynamical systems theory, we demonstrate equilibrium stability, uniform persistence and bifurcations in the system. It is shown that enhancing the search speed of predator and/or decreasing prey’s dispersal from the source to sink could promote persistence of the system, while increasing the fear cost would stabilize the system. Moreover, dispersal could make the prey persist in both source–sink patches, even drive the predator into extinction. The dispersal could also make the prey reach a total population abundance larger than that without dispersal, even larger than the carrying capacity. Asymmetry in dispersal plays a role in the persistence and abundance. A novel finding of this work is that dispersal may lead to results reversing those without dispersal. Numerical simulations illustrate and extend our results. This work is important in understanding complexity in predation systems.