Bifurcation Phenomena Research Articles (Page 1)

This study investigates stochastic bifurcation in a generalized tristable Rayleigh–Duffing oscillator with fractional inertial force under both additive and multiplicative recycling noises. The system’s dynamic behavior is influenced by its inherent spatial symmetry, represented by the potential function, as well as by temporal symmetry breaking caused by fractional memory effects and recycling noise. First, an approximate integer-order equivalent system is derived from the original fractional-order model using a harmonic balance method, with minimal mean square error (MSE). The steady-state probability density function (sPDF) of the amplitude is then obtained via stochastic averaging. Using singularity theory, the conditions for stochastic P bifurcation (SPB) are identified. For different fractional derivative’s orders, transition set curves are constructed, and the sPDF is qualitatively analyzed within the regions bounded by these curves—especially under tristable conditions. Theoretical results are validated through Monte Carlo simulations and the Radial Basis Function Neural Network (RBFNN) approach. The findings offer insights for designing fractional-order controllers to improve system response control.

This study investigates the dynamics and control of an infectious disease using an age-structured Susceptible-Vaccinated-Infected (SVI) model, incorporating imperfect vaccination and therapeutic treatment. We identify a backward bifurcation phenomenon, driven by treatment rates, which allows disease persistence even when the basic reproduction number R0 < 1, complicating eradication e orts. To address this, we formulate an optimal control problem with time-dependent vaccination and treatment rates to minimize infected individuals and control costs. Using bifurcation theory and integrated semigroup methods, we establish the model's well-posedness, equilibria stability, and bistability conditions. First-order optimality conditions are derived to characterize optimal controls. Numerical simulations,solved via the Forward-Backward Sweep method, demonstrate that combined vaccination and treatment signi cantly reduces disease prevalence, with early intervention being critical. These ndings underscore the importance of strategic resourceallocation in managing complex epidemic dynamics.

This paper aims to investigate the influence of amplitude modulation and coexisting attractors on the formation mechanism of complex bursting oscillations in slow–fast coupled nonlinear systems. Taking a modified three-dimensional van der Pol–Duffing system as the research object, a parametric excitation with frequency far less than the natural frequency is introduced into the system, establishing a slow–fast system model with two-scale coupling characteristics in the frequency domain. Regarding the excitation term as a slow-varying parameter, the equilibrium points, limit cycles, and their bifurcation structures in the fast subsystem are systematically analyzed, revealing the coexistence conditions of different attractors and their impact on the dynamics of the full system. When the modulation amplitude is relatively small, the slow evolution process of the system becomes notably extended, showing typical quasi-static relaxation–oscillation characteristics. As the modulation amplitude increases, these oscillations vanish rapidly, and the system transitions to a steady equilibrium or an aperiodic oscillatory state. Further investigation reveals that the coexistence of multiple attractors in the fast subsystem can result in diverse bursting behaviors of the full system: trajectories may remain within a single attraction basin or sequentially traverse several basins, thereby generating merged bursting oscillations. Moreover, due to the inertia effect, trajectories may directly cross parameter regions associated with specific attractors, leading to the disappearance of corresponding bifurcation phenomena. When the excitation frequency decreases to a sufficiently low value, previously vanished attractors may re-emerge. If the interval between two types of codimension-1 bifurcation points is extremely narrow, the system may also display compound bursting patterns analogous to those caused by codimension-2 bifurcations. The research in this paper reveals the synergistic influence mechanism of coexisting attractors in slow-fast systems on relaxation oscillations and bursting dynamics under amplitude modulation conditions, enriching the multiscale dynamical theory of nonlinear oscillators and providing new theoretical references for the study of modulation-induced dynamical evolution in other complex nonlinear systems.

Abstract To investigate the bifurcation phenomena induced by asymmetric Hopf bifurcation points on the pitchfork equilibrium 
branch in a nonlinear system, this paper designs a novel jerk-like autonomous system based on the classical differential 
equation form of the pitchfork bifurcation. As the bifurcation parameter increases, the pitchfork bifurcation point and two 
Hopf bifurcation points act on the system sequentially, driving transitions between different system states. Upon 
introducing a slowly varying periodic excitation, a non-autonomous system with coupled fast-slow dynamics is obtained. 
As the excitation amplitude increases, the system exhibits different types of bursting oscillations. The bifurcations of the 
fast subsystem reveal the mechanisms underlying these distinct bursting phenomena. The study finds that the sequential 
action of the asymmetric Hopf bifurcation points on the system leads to the formation of homoclinic orbits. The existence 
of these homoclinic orbits is rigorously demonstrated through numerical simulations based on the Smale-Birkhoff 
homoclinic theorem and Poincaré mapping methodology, providing valuable insights for controlling multi-path bifurcation 
behaviors in systems.

This study employs the sum of squares programming (SOSP) method to explore the maximum potential of rear-wheel steering (RWS) in enhancing vehicle stability by expanding the stability region. Firstly, an SOSP-based controller synthesis framework is developed, in which a shape function updating strategy is introduced to reduce the conservatism of stability region estimation. Secondly, to address the input-nonaffine nature of the RWS system, a state feedback control approach based on rear-wheel lateral force is proposed. This ensures an input-affine structure while minimising modelling error. In addition, a constraint reformulation technique is adopted to enhance the feasibility of the SOSP problem under rational system representations. Then, phase portraits and eigenvalue analyses based on SOSP solution reveal that the RWS input induces a dynamical bifurcation phenomenon, in which original saddle points are transformed into locally stable equilibria, thereby enhancing stability near the vehicle’s handling limits. The effectiveness of the synthesised control law is further evaluated under actuator constraints, demonstrating its feasibility within practical rear-wheel steering angle limits. Finally, the differential impact of RWS and active front steering (AFS) on stability region expansion is investigated, revealing fundamental differences in their stabilisation capabilities..

In this work, we construct a multiple solutions theory based on a membrane shape equation. The membrane shape of a vesicle or a red blood cell is determined using the Zhongcan–Helfrich shape equation. These spherical solutions, which have an identical radius but different center positions, can be described by the same equation: . A degeneracy for the spherical solutions exists, leading to multisphere solutions with the same radius. Therefore, there can be multiple solutions for the sphere equilibrium shape equation, and these need to satisfy a quadratic equation. The quadratic equation has a maximum of two roots. We also find that the multiple solutions should be in a line to undergo rotational symmetry. We use the quadratic equation to compute the sphere radius, together with a membrane surface constraint condition, to obtain the number of small spheres. We ensure matching with the energy constraint condition to determine the stability of the full solutions. The method is then extended into the myelin formation of red blood cells. Our numerical calculations show excellent agreement with the experimental results and enable the comprehensive investigation of cell fission and fusion phenomena. Additionally, we have predicted the existence of the bifurcation phenomenon in membrane growth and proposed a control strategy.

Mathematical modeling is essential for understanding infectious disease dynamics and guiding public health strategies. We study the global dynamics of a susceptible-infectious-recovered-susceptible (SIRS) model with a generalized nonlinear incidence function, revealing a rich array of bifurcation phenomena, including saddle-node, cusp, forward and backward bifurcations, Bogdanov-Takens bifurcations, saddle-node bifurcation of limit cycles, subcritical and supercritical Hopf bifurcations, generalized Hopf bifurcations, homoclinic and degenerate homoclinic bifurcations, as well as isola bifurcation. Using normal form theory, we show that the Hopf bifurcation reaches codimension three, resulting in up to three small-amplitude limit cycles. The involvement of the recovered population enables coexistence of these limit cycles, leading to bistable and tristable dynamics. We employ a one-step transformation method to analyze codimension two and three Bogdanov-Takens bifurcations, confirming a maximum codimension of three. In particular, we identify isolas of limit cycles in an SIRS model involving double exposure, introducing a mechanism for generating limit cycles centered on the isola. The findings may have important public health implications, highlighting how nonlinearities in transmission and immunity can produce recurrent outbreaks or persistent infection despite interventions. The existence of multiple limit cycles suggests that small changes in transmission rates or immune response could cause abrupt shifts in outbreak patterns, emphasizing the need for adaptive and flexible intervention strategies.

This paper presents the analytical and experimental analysis of a spring-coupled triple-pendulum system in which a free spring is attached to the end of the second pendulum. This configuration, which has not been addressed in prior literature, introduces a hybrid mechanical system that combines rigid multi-arm dynamics with nonlinear coupling. We derive the full equations of motion and establish well-posedness and boundedness of the solutions. We prove the existence of nontrivial limit cycles and establish their orbital stability via Floquet theory. We then derive the Poincaré return map and identify a critical parameter value at which the system undergoes a period-doubling (flip) bifurcation. To validate the theoretical results, we construct a physical prototype of the proposed system using three-pendulum set up and a freely suspended elastic spring. Time series data, phase portraits, and Poincaré sections confirm the presence of stable periodic motion, bifurcation phenomena, and chaotic behavior.

Abstract The co-infection of COVID-19 and Monkeypox presents a public health challenge due to their distinct transmission dynamics and potential complications. Monkeypox was mainly limited to Central and West Africa, but COVID-19’s global spread raises concerns about their co-infection. This study presents a comprehensive mathematical model that includes therapy as a control measure and describes the transmission dynamics of COVID-19 and Monkeypox co-infections. We developed and analyzed a compartmental mathematical model incorporating treatment interventions. The model includes sub-models for individual diseases and a comprehensive co-infection framework. Qualitative analysis was performed to determine equilibrium stability, and numerical simulations were conducted to assess parameter sensitivity and intervention effectiveness. The study qualitatively evaluates sub-models for COVID-19 and Monkeypox, demonstrating locally asymptotically stable disease-free equilibrium states when their basic reproduction numbers are less than unity. For the COVID-19 sub-model, $$R_{0}^C < 1$$ ensures disease elimination, while for Monkeypox, $$R_{0}^M < 1$$ guarantees stability. The co-infection model shows local stability at its disease-free equilibrium point under specific conditions, depending on the initial population size, indicating that the introduction of a small number of infected individuals will not result in a significant disease outbreak. The global asymptotic instability of the COVID-19 and Monkeypox co-infection model suggests the possibility of a backward bifurcation phenomenon, which means that the traditional criterion requiring the basic reproduction number to be less than unity is no longer sufficient to control the co-infection of these diseases, although it remains necessary. Numerical simulations investigating the effects of parameter changes reveal that higher contact rates promote disease transmission. Notably, treatment interventions with rates $$\rho _C = 0.8$$ for COVID-19 and $$\rho _M = 0.4$$ for Monkeypox resulted in significant reductions in infected populations, with up to 60% decrease in co-infection cases. The analysis showed that the introduction of a therapy class reduces the incidence of both diseases, while the presence of COVID-19 increases the risk of contracting Monkeypox by approximately 25%. The mathematical model demonstrates that treatment interventions are effective control strategies for managing COVID-19 and Monkeypox co-infections. The backward bifurcation phenomenon indicates that achieving $$R_{0}^M < 1$$ alone is insufficient for disease control, necessitating sustained intervention efforts. These findings provide quantitative insights for public health policy development and emphasize the importance of comprehensive treatment strategies in managing co-infection dynamics.

The mode-locking characteristic and microwave pulses regulation based on a self-mode-locking optoelectronic oscillator (SML-OEO) are analyzed and experimentally demonstrated. The oscillating modes in the SML-OEO cavity can be phase-coherent by means of the inherent oscillatory characterization via the opto-electronic loop. A self-excited microwave frequency comb (MFC) with the characteristics of a periodic rectangular pulse is generated. The phenomenon of nonlinear dynamical bifurcation in the SML-OEO, which leads to an amplification of the time scale in the system, where a single-cycle oscillatory pulse with a period of τ evolves into an oscillatory pulse with a period of 2τ. On this basis, by injecting a microwave signal with a period of 2τ/N (N is an integer) into the OEO loop, the gain in the OEO cavity is regulated so that the SML-OEO forms a new mode-locked state. The temporal domain characteristic forms a periodic rectangular pulse with a period of 2τ/N. The experiment results show that the microwave pulse of SML-OEO can be effectively regulated from 30 ns to 3.75 ns as N is set from 3 to 8.

This paper presents a mathematical model that examines the effect of nonlinear incidence on disease transmission dynamics. Furthermore, we also accommodate newborn and adult vaccination strategy as the prevention strategy to prevent rapid spread of the disease due to nonlinear incidence rate. Assuming a constant population size, the system is reduced to a two-dimensions and nondimensionalized using the average infectious period as the time scale. Analytical results reveal the existence of both disease-free and endemic equilibria, with the possibility of backward bifurcation when the nonlinear incidence parameter exceeds a critical threshold. This implies that disease persistence may still occur even when the basic reproduction number is less than one. Numerical simulations using MATCONT conducted to visualize the occurrence of both forward and backward bifurcations phenomena. Using COVID-19 parameter values, a global sensitivity analysis via Partial Rank Correlation Coefficient - Latin Hypercube Sampling method indicates that newborn vaccination has a stronger impact on reducing the basic reproduction number. These findings provide important insights for designing effective vaccination strategies and understanding the complex dynamics arising from nonlinear transmission and imperfect immunization.

Flow-driven dynamics in a mussel-algae system with nonlinear boundary interactions.

This study investigates the regulation of tissue growth through mathematical modeling of systemic and local feedback mechanisms. Employing reaction-diffusion equations, the models explore the dynamics of tissue growth, emphasizing endocrine signaling and inter-tissue communication. The analysis identifies critical factors influencing the emergence of spatial structures, bifurcation phenomena, the existence and stability of stationary pulse and wave solutions. It also elucidates mechanisms for achieving coordinated tissue growth. In particular, if negative feedback is sufficiently strong, their final finite size is provided by a stable pulse, otherwise they manifest unlimited growth in the form of a wave. These findings contribute to the theoretical insights into biological processes such as embryogenesis, regeneration, and tumor development, while highlighting the role of feedback systems in maintaining physiological homeostasis.

This study aims to precisely analyze period-tripling bifurcations reported in the literature, specifically addressing the challenge of distinguishing between genuine and pseudo-period-tripling bifurcations caused by numerical errors or minor parameter variations. Current research on period-tripling bifurcations is limited, with most studies relying on numerical methods for solutions and identifying period-tripling bifurcations by observing numerical bifurcation diagrams. However, these methods may lead to misjudgments due to numerical and visual inaccuracies. In order to address this issue, this study employs the incremental harmonic balance method for accurate solutions and quantitative analysis of typical nonlinear systems. The Floquet theory, phase diagrams, frequency spectra, bifurcation diagrams, and Poincaré sections are used to analyze two types of pseudo-period-tripling bifurcations meticulously. The results indicate that certain observed period-tripling bifurcation phenomena (e.g. bifurcations from stable period-1 to unstable period-3 and from stable period-2 to stable period-6) are actually pseudo-bifurcations caused by numerical errors or visual misinterpretations rather than real dynamic behavior changes. This study provides a more accurate and reliable method to investigate period-tripling bifurcations, correct misconceptions in existing studies, and promote a deeper understanding of the complex behaviors in nonlinear dynamical systems.

Nonlinear wind-induced vibrations and coupled static–dynamic instabilities pose significant challenges for long-span suspension bridges, especially under large-amplitude and high-angle-of-attack conditions. However, existing studies have yet to fully capture the mechanisms behind large-amplitude torsional flutter. To address this, wind tunnel experiments were performed on H-shaped bluff sections and closed box girders using a high-precision five-component piezoelectric balance combined with a custom support system. Complementing these experiments, a finite element time-domain simulation framework was developed, incorporating experimentally derived nonlinear flutter derivatives. Validation was achieved through aeroelastic testing of a 1:110-scale model of the original Tacoma Narrows Bridge and corresponding numerical simulations. The results revealed Hopf bifurcation phenomena in H-shaped bluff sections, indicated by amplitude-dependent flutter derivatives and equivalent damping coefficients. The simulation results showed less than a 10% deviation from experimental and historical wind speed–amplitude data, confirming the model’s accuracy. Failure analysis identified suspenders as the critical failure components in the Tacoma collapse. This work develops a comprehensive performance-based design framework that improves the safety, robustness, and resilience of long-span suspension bridges against complex nonlinear aerodynamic effects while enabling cost-effective, targeted reinforcement strategies to advance modern bridge engineering.

Abstract This study presents a novel approximation approach for the Fractional Rössler System using the Variational Iteration Method (VIM). The Fractional Rössler System, an extension of the classical Rössler System, incorporates fractional-order derivatives to capture more intricate dynamical behaviors. VIM is employed due to its efficiency in handling nonlinear fractional differential equations (FDEs) and its novel application to this system. A comparative analysis with the Adams-Bashforth-Moulton method has been conducted, and numerical values are presented to validate the effectiveness of VIM. The graphical results reveal significant chaotic dynamics, including the presence of strange attractors and sensitivity to initial conditions and bifurcation phenomena, indicating transitions between periodic and chaotic regimes. This work provides a new perspective on approximating fractional chaotic systems, demonstrating the potential of VIM in advancing the study of complex dynamical systems.

Synthetic jets have emerged as a versatile tool for flow control of boundary layers, propulsion, etc. Non-circular synthetic jets offer more complex flow phenomena than circular synthetic jets, thus making them more suitable for flow control applications. The experiments to characterize the flow in a low aspect ratio (AR = 2) elliptical synthetic jet (ESJ) issued in a quiescent environment are conducted for two formation lengths (Lo = 1.4 and 2.6) with respective Reynolds numbers of 280 and 520, utilizing planar time-resolved particle image velocimetry (TRPIV) and laser-induced fluorescence (LIF) flow visualization. The evolution of the synthetic jet is studied using phase-averaged flow structures and vortex core locations, which reveal that at Lo = 1.4, the phenomenon of vortex ring bifurcation occurs, which is generally assumed to be a high aspect ratio phenomenon in the literature. A complete axis-switching is observed for Lo = 2.6. The statistical flow characteristics reveal that the flow fluctuations in the elliptical synthetic jet (ESJ, AR = 2) are more complex than the circular synthetic jet (CSJ, AR = 1). The vortices are more stretched in the case of ESJ than CSJ due to enhanced vortex interaction in the former case, which enables higher momentum exchange and increased flow fluctuations. The vortex strength of ESJ during axis-switching (Lo = 2.6) is higher than the vortex bifurcation case (Lo = 1.4). The fluctuating kinetic energy budget of the elliptical synthetic jet (ESJ) obtained in both minor axis plane (X-Y plane) and major axis plane (X-Z plane) is compared with that of the circular synthetic jet (CSJ). The production term is found to be the most significant one in both minor and major planes. The transport of the energy due to flow fluctuations acts as a source term in the shear layer region of the vortex rings, while it behaves as a sink in the vortex core regions. The two surrogates of dissipation rate obtained based on resolved velocity gradients are compared, and it is found that the axisymmetric dissipation rate resembles the flow better. The present study shows that the elliptical synthetic jet (ESJ) has more complex flow physics than the circular synthetic jet (CSJ) and thus can be useful for various heat transfer and flow control applications.

A co-infection model between HIV and COVID-19 that takes into account COVID-19 vaccination and public awareness is discussed in this article. Rigorous analysis of the model is conducted to establish the existence and local stability conditions of the single-infection models. We discover that when the corresponding reproduction number for COVID-19 and HIV exceeds one, the disease continues to exist in both single-infection models. Furthermore, HIV will always be eradicated if its reproduction number is less than one. Nevertheless, this does not apply to the single-infection COVID-19 model. Even when the fundamental reproduction number is less than one, an endemic equilibrium point may exist due to the potential for a backward bifurcation phenomenon. Consequently, in the single-infection COVID-19 model, bistability between the endemic and disease-free equilibrium may arise when the basic reproduction number is less than one. From the co-infection model, we find that the reproduction number of the co-infection model is the maximum value between the reproduction number of HIV and COVID-19. Our numerical continuation experiments on the co-infection model reveal a threshold indicating that both HIV and COVID-19 may coexist within the population. The disease-free equilibrium for both HIV and COVID-19 is stable only if the reproduction numbers are less than one. Additionally, our two-parameter continuation analysis of the bifurcation diagram shows that the condition where both reproduction numbers equal one serves as an organizing center for the dynamic behavior of the co-infection model. An extended version of our model incorporates four different interventions: face mask usage, vaccination, and public awareness for COVID-19, as well as condom use for HIV, formulated as an optimal control problem. The Pontryagin's Maximum Principle is employed to characterize the optimal control problem, which is solved using a forward-backward iterative method. Numerical investigations of the optimal control model highlight the critical role of a well-designed combination of interventions to achieve optimal reductions in the spread of both HIV and COVID-19.

To explore the influence of mine roof on the damage and failure of sandstone surrounding rock under different pressure rates, mechanical experiments with different strain rates were carried out on sandstone rock samples. The strength, deformation, failure, energy and damage characteristics of rock samples with different strain rates were also discussed. The research results show that with the increases in the strain rate, peak stress, and elastic modulus show a monotonically increasing trend, while the peak strain decreases in the reverse direction. At a low strain rate, the proportion of the mass fraction of complete rock blocks in the rock sample is relatively high, and the shape integrity is good, while rock samples with a high strain rate retain more small-sized fragmented rock blocks. This indicates that under high-rate loading, the bifurcation phenomenon of secondary cracks is obvious. The rock samples undergo a failure form dominated by small-sized fragments, with severe damage to the rock samples and significant fractal characteristics of the fragments. At the initial stage of loading, the primary fractures close, and the rock samples mainly dissipate energy in the forms of frictional slip and mineral fragmentation. In the middle stage of loading, the residual fractures are compacted, and the dissipative strain energy keeps increasing continuously. In the later stage of loading, secondary cracks accelerate their expansion, and elastic strain energy is released sharply, eventually leading to brittle failure of the rock sample. Under a low strain rate, secondary cracks slowly expand along the clay–quartz interface and cause intergranular failure of the rock sample. However, a high strain rate inhibits the stress relaxation of the clay, forces the energy to transfer to the quartz crystal, promotes the penetration of secondary cracks through the quartz crystal, and triggers transgranular failure. A constitutive model based on energy damage was further constructed, which can accurately characterize the nonlinear hardening characteristics and strength-deformation laws of rock samples with different strain rates. The evolution process of its energy damage can be divided into the unchanged stage, the slow growth stage, and the accelerated growth stage. The characteristics of this stage reveal the sudden change mechanism from the dissipation of elastic strain energy of rock samples to the unstable propagation of secondary cracks, clarify the cumulative influence of strain rate on damage, and provide a theoretical basis for the dynamic assessment of surrounding rock damage and disaster early warning when the mine roof comes under pressure.

ABSTRACTPredator‐prey models, such as the Leslie‐Gower model, are essential for understanding population dynamics and stability within ecosystems. These models help explain the balance between species under natural conditions, but the inclusion of factors like the Allee effect and intraspecific competition adds complexity and realism to these interactions, enhancing our ability to predict system behavior under stress. To detect early indicators of population collapse, this study investigates the intricate dynamics of a modified Leslie‐Gower predator‐prey model with both Allee effect and intraspecific competition. We analyze the existence and stability of equilibria, as well as bifurcation phenomena, including saddle‐node bifurcations of codimension 2, Hopf bifurcations of codimension 2, and Bogdanov‐Takens bifurcations of codimension at least 4. Detailed transitions between bifurcation curves–specifically saddle‐node, Hopf, homoclinic, and limit cycle bifurcations–are also examined. We observe a novel transition phenomenon, where a system jumps from saddle‐node bifurcation to homoclinic and limit cycle bifurcations. This suggests that burst oscillations may serve as an early warning of system collapse rather than simply a tipping point. Our findings indicate that moderate levels of intraspecific competition or Allee effect support coexistence of both populations, while excessive levels may destabilize the entire biological system, leading to collapse. These insights offer valuable implications for ecological management and the early detection of risks in population dynamics.