Bifurcation Point Research Articles

General relationship of local topologies, global dynamics, and bifurcation in cellular networks

General relationship of local topologies, global dynamics, and bifurcation in cellular networks

Numerical Simulations and Bifurcation of Ca2+ Oscillatory Behaviour in the Connection of Neurons and Astrocytes.

Extensive research has demonstrated that astrocytes actively participate in the regulation of synaptic communication. To examine the dynamic behavior of the model, a neuron-astrocyte model has been solved, and a bifurcation analysis has been performed. This paper uses the equilibrium point, stability theory, and the center manifold theorem to theoretically investigate the dynamical analysis of Ca2+ oscillations in the cytosol. The connections at tripartite synapses between the cells have been modeled using IP3 and 2-AG. A mathematical model is used to depict the overall framework of bifurcation and induced Ca2+ dynamics. The differences in the presence and disappearance of Ca2+ oscillations are partially explained by two subcritical Hopf bifurcation points, according to the results. Communication between the cells occurs through the oscillations of Ca2+ concentration. Furthermore, numerical simulations are conducted to confirm the efficacy of the suggested approach. Thus, our findings imply that neuron-astrocyte crosstalk plays a fundamental role in generating a variety of neuronal activities, thereby improving the brain's capacity for information processing.

An ODEs multiscale model with cell proliferation for hepatitis C virus infection treated with direct acting antiviral agents

In a recent study, a mathematically identical ODE model is derived from a multiscale PDE model of hepatitis C virus infection, which helps to overcome the limitations of the PDE model in the analysis. Here, an extended proposed model is formulated for this transformed ODE model by including the hepatocyte proliferation of both uninfected and infected cells. Unlike the transformed model, the proposed model can predict the triphasic viral decline and the virus level after therapy cessation without oscillations. Numerical simulations are performed to investigate the effect of hepatocyte proliferation and therapy with direct-acting antivirals agents (DAAs). The basic reproduction number is obtained, the equilibrium points are specified, and their stability is analysed. A bifurcation analysis is performed to specify the bifurcation points and to study the effect of varying system parameters. Various viral load profiles generated by the model are confirmed to fit with reported data in the literature.

Impact of Honest Signal, Cues, Prey’s Experience Rate and Environmental Toxicity in a Predator–Prey Interaction Model

Biological signaling plays a crucial role in predator–prey interaction systems, serving both intraspecific and interspecific communication purposes. Prey populations utilize signaling internally, employing honest signals among themselves to enhance their defense capabilities. Conversely, they also use signaling for interspecific communication, emitting cues to predators to increase their own chances of survival. Recently, Al-Salman et al. [Appl. Math. Modell. 89, 1405 (2021)] proposed a two-species interaction model incorporating Holling Type II predation along with honest signals and cues. In this paper, we extend and modify their study by incorporating factors such as prey’s experience rate and environmental toxicity in prey populations. We conduct a comprehensive mathematical and numerical analysis of the model. We investigate the positive invariance and dissipativeness of the model, discuss the existence of equilibrium points and analyze the behavior of the system around all equilibrium points. Particularly, we determine conditions for local stability and Hopf bifurcation of the system. Furthermore, we perform a detailed numerical analysis to study periodic solutions through Hopf bifurcations in the proposed system. We plot two-parameter bifurcation diagrams to examine changes in the system with combined effects, identifying critical values for stability switching. There is no chaotic dynamics seen in our investigation. Numerical results indicate that the system tends to stabilize when the experience rate is high, and the rate of error-prone honest signals regulates system stability. Additionally, we analyze how environmental toxicity affects the stability of the system.

Bifurcation Enhances Temporal Information Encoding in the Olfactory Periphery

Living systems continually respond to signals from the surrounding environment. Survival requires that their responses adapt quickly and robustly to the changes in the environment. One particularly challenging example is olfactory navigation in turbulent plumes, where animals experience highly intermittent odor signals while odor concentration varies over many length- and timescales. Here, we show theoretically that olfactory receptor neurons (ORNs) can exploit proximity to a bifurcation point of their firing dynamics to reliably extract information about the timing and intensity of fluctuations in the odor signal, which have been shown to be critical for odor-guided navigation. Close to the bifurcation, the system is intrinsically invariant to signal variance, and information about the timing, duration, and intensity of odor fluctuations is transferred efficiently. Importantly, we find that proximity to the bifurcation is maintained by mean adaptation alone and therefore does not require any additional feedback mechanism or fine-tuning. Using a biophysical model with calcium-based feedback, we demonstrate that this mechanism can explain the measured adaptation characteristics of ORNs. Published by the American Physical Society 2024

Influence of additional mass and connection of nonlinear energy sinks on vibration reduction performance

The broadband vibration reduction performance of nonlinear energy sink (NES) has attracted wide attention. However, the impact of the NES’s additional mass other than the oscillator and how it is connected to the primary structure has been ignored. More recently, it has been discovered that vibration attenuation through the cellular application of NES can achieve greater efficiency. However, the connection between NES cells and the primary structure, as well as between cells, has not been studied. In this study, by considering the additional mass of the NES cells, the influence of the connection modes of NES cells on the vibration reduction efficiency is investigated theoretically, optimally and experimentally for the first time. The forced vibration models of linear oscillator coupled with NES cells are established by viscoelastic connection and rigid connection respectively. The approximate analysis and numerical analysis show that the vibration reduction efficiency of NES cells is affected by the resonance frequency of the primary structure and the external excitation intensity and shows a nonlinear trend. With the change of the resonant frequency of the primary structure, the viscoelastic connection NES cells can almost always obtain higher vibration reduction efficiency than the rigid connection NES cells. The global bifurcation results show that the strongly modulated responses of the structure can be triggered by the viscoelastic connection. Moreover, the connection modes between NES cells also affect the vibration reduction efficiency. The optimal parameters of the connection damping and connection stiffness are obtained by the particle swarm optimization algorithm. Finally, the viscoelastic connection and rigid connection, and the effect of the connection mode between NES cells on the vibration reduction efficiency are compared by experiments. The conclusions of theoretical research are verified. This work can provide theoretical guidance for the engineering application of NES cells.

Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

<i>Objective: </i>All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. <i>Methods: </i>Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. <i>Results: </i>Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. <i>Conclusions: </i>The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

Dynamics of a discrete Rosenzweig–MacArthur predator–prey model with piecewise-constant arguments

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

Bifurcation Near a Transcritical Singularity in Planar Singularly Perturbed Systems

ABSTRACTWe classify all bifurcation phenomena of the flow near a transcritical singularity in planar singularly perturbed differential systems that do not have a breaking parameter via qualitative analysis and blow‐up technique. Here, the directional blown up vector fields can have several singularities and no first integral that are different from those in the literatures. The obtained local bifurcations are also illustrated by numerical simulations through a modified Leslie–Gower model, whose global dynamics is thereby obtained.

Global bifurcation for monotone fronts of elliptic equations

We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer (1973) and Buffoni–Toland (2003), but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs.As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.

The Diffusive Lotka–Volterra Competitive Model with Advection Term: Exclusion, Coexistence and Multiplicity

Earlier, the research was mainly focused on the classic diffusive two-species Lotka–Volterra competitive system (without advection). Averill et al. [2017] regarded a diffusive two-species Lotka–Volterra competitive system with advection term (with one species having a combination of random dispersal and biased movement upward along the resource gradient, while the other species adopts purely random dispersal). In this paper, we consider a more general diffusive two-species Lotka–Volterra competitive system with advection term (involving both species with a combination of random dispersal and biased movement upward along their respective resource gradients). By utilizing the theory of monotone dynamical systems, spectral analysis, methods of super- and sub-solutions and some nontrivial analytic skills, we finally obtain a rather deep understanding on the global dynamics. Moreover, our results reveal that the dynamical behavior of the system is very complex: exclusion, coexistence and bistability all may happen depending closely on the interspecific competition intensities. More interestingly, by bifurcation theory, we find that the system always admits multiple coexistence steady states when the interspecific competition intensities are near the critical point.

Competing Bifurcations Determine Symmetry Breaking During Droplet Snaps on Smooth Patterned Surfaces.

The shape and stability of a droplet in contact with a solid surface is affected by the chemical composition and topography of the solid, and crucially, by the droplet's size. During a variation in size, most often observed during evaporation, droplets on smooth patterned surfaces can undergo sudden shape and position changes. Such changes, called snaps, are prompted by the surface pattern and arise from fold and pitchfork bifurcations which respectively cause symmetric and asymmetric motions. Yet, which type of snap is likely to be observed is an open fundamental question that has relevance in the rational design of surfaces for managing droplets. Here we show that the likelihood of observing symmetric or asymmetric snaps depends on the distance between fold and pitchfork bifurcation points and on how this distance varies for droplets that grow or shrink in size on surfaces patterned with a smooth topography. Our results can help develop strategies to control droplets by exploiting smooth surface patterns but also have broader relevance in situations where different types of bifurcations compete in determining the stability of a system, for instance in snap-through instabilities observed in elastic media.

Quantifying risk of a noise-induced AMOC collapse from northern and tropical Atlantic Ocean variability

Abstract The Atlantic Meridional Overturning Circulation (AMOC) exerts a major influence on global climate. There is much debate about whether the current strong AMOC may collapse as a result of anthropogenic forcing and/or internal variability. Increasing the noise in simple salt-advection models can change the apparent AMOC tipping threshold. However, it's not clear if `present-day' variability is strong enough to induce a collapse.
Here, we investigate how internal variability affects the likelihood of AMOC collapse. We examine internal variability of basin-scale salinities and temperatures in four CMIP6 pre-industrial simulations. We fit this to an empirical, process-based AMOC box model, and
find that noise-induced AMOC collapse (defined as a decade in which the mean AMOC strength falls below $5 \ \rm{Sv}$) is unlikely, however, if the AMOC is pushed closer to a bifurcation point due to external climate forcing, noise-induced tipping becomes more likely.
Surprisingly, we find a case where forcing temporarily overshoots a stability threshold but noise decreases the probability of collapse. Accurately modelling internal decadal variability is essential for understanding the increased uncertainty in AMOC projections.

Optical solitons, qualitative analysis, and chaotic behaviors to the highly dispersive nonlinear perturbation Schrödinger equation

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

A stochastic mechanism causing long or short transients near a bifurcation point

When a bifurcation parameter in a deterministic model becomes a stochastic process, additional new stochastic processes may arise near a critical parameter value. By examining the sample paths of a population model with a strong Allee effect, in which the bifurcation parameter is defined as an Ornstein–Uhlenbeck (OU) process, we find that transient times before extinction may be extended or shortened according to the excursions of the OU process. We find the distributions of transient times as they depend on process parameters.

Research on nonlinear instability of blended-wing-body aircraft based on experimental bifurcation analysis

Focusing on the nonlinear instability problem of the blended-wing-body aircraft, first, numerical bifurcation analysis is adopted to obtain catastrophe points and branches where the elevator is taken as a continuous parameter. Second, based on virtual flight tests and bifurcation theory, an experimental bifurcation analysis method is developed. Open-loop and closed-loop experiments are carried out, and the corresponding experimental bifurcation diagrams are obtained. The comparative analysis shows that the experimental bifurcation analysis enables the aircraft to make a response that is close to reality under the influence of nonlinear and unsteady aerodynamics and has a more intuitive embodiment of instability characteristics, providing an accurate prediction of departure. The open-loop experiment predicts a sudden longitudinal nose-up at an angle of attack of 7.4°. The closed-loop experiment can transform the open-loop unstable branch into a stable branch. However, roll instability still occurs at a critical angle of attack of 16.0°. These research results provide important dynamic information for the stable flight and the design of the controller.

Bifurcation and Controller Design of 5D BAM Neural Networks With Time Delay

ABSTRACTAll the time delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In the current article, we study a class of 5D delayed bidirectional associative memory (BAM) neural networks that conform to objective reality. First of all, we prove that the solution of the delayed 5D BAM neural networks exists and is unique by virtue of fixed point theorem and some inequality techniques. Secondly, the Hopf bifurcation and stability of the delayed 5D BAM neural networks are investigated by exploiting the stability criterion and bifurcation theory. Once more, Hopf bifurcation control strategy of the delayed 5D BAM neural networks is explored by virtue of two different hybrid controllers. By adjusting the parameters of the controllers, we can control the stability domain and Hopf bifurcation onset. Eventually, the correctness of the theoretical results was verified through numerical simulations. The conclusions obtained in this paper are new and have important theoretical value in neural network area.

MODELING PROBLEMS OF LOGICAL THINKING IN PRIMARY CLASS MATHEMATICS LESSONS

Models are created for various purposes (analysis of the studied processes, verification or demonstration of a new idea, method, system; to study and change reality, as a means of forecasting) and in a clear and concise form the studied system, process, characteristics of the idea; allows you to get important things about the connections and relations of the object. V.V. According to Davydov, the model will contain an image of the studied object. According to the nature of the description of the modeled system, there are modeling of the structure (structural models) and modeling of the operation of the processes occurring in it (flow or process models). To solve the second problem, mathematical modeling is the most effective. Modeling involves the use of abstraction and idealization. Reflecting the essential features of the original, the model acts as a kind of implementation of abstraction. The effectiveness of modeling as an achievement of consistency between model and reality is complicated by uncertainty, an important feature of models in the humanities. E. N. Gusinsky, studying the characteristics of social and humanitarian systems, emphasizes that they always have conscious and unconscious motives, are determined not only by past experience, but also are influenced by many factors of the current environment and the external environment. A.N. Dakhin emphasizes the complexity of modeling humanitarian systems, because they do not have clear components that make up the system - each of the components can "over time" become a pedagogical bifurcation point and dominate the design of goals and teaching technology. This article talks about the modeling of logical issues, the need for its application, the diversity of educational content, and the increase in requirements for the quality of education.

Geometric flow control in lateral flow assays: Macroscopic two-phase modeling

Lateral flow assays (LFAs) are widely employed in a diverse range of applications, including clinical diagnostics, pharmaceutical research, forensics, biotechnology, agriculture, food safety, and environmental analysis. A pivotal component of LFAs is the porous polymeric membrane, which facilitates the capillary-driven movement of fluids, known as “imbibition,” in which a wetting fluid displaces a non-wetting fluid within the pore space of the membrane. This study presents a multi-scale modeling framework designed to investigate the imbibition process within LFAs. The framework integrates microscopic membrane characteristics into a macroscopic two-phase flow model, allowing the simulation of imbibition in membranes with different micro-scale properties and macro-scale profiles. The validity of the model was established through comparative analysis with documented case studies, a macro-scale single-phase flow model, and experimental observations, demonstrating its accuracy in simulating the imbibition process. The study also examines imbibition in various geometric configurations, including bifurcated (Y-shaped) and multi-branch geometries commonly found in multiplexed LFAs. The influence of geometric features such as length ratio, width ratio, branching angle, bifurcation point location, and asymmetry on fluid transport is investigated. Results indicate that membranes with larger branching angles exhibit slower imbibition. In addition, the influence of membrane type on macroscopic flow patterns is evaluated, showing that membranes with lower permeability require longer imbibition times. The insights gained from this research support a data-driven strategy for manipulating wetting behavior within LFAs. This approach can be leveraged to optimize the performance of LFAs and increase their effectiveness in various applications.

Overcoming stretching and shortening assumptions in Euler–Bernoulli theory using nonlinear Hencky’s beam models: Applicable to partly-shortened and partly-stretched beams

This paper addresses the challenges of the Euler–Bernoulli beam theory regarding shortening and stretching assumptions. Certain boundary conditions, such as a cantilever with a horizontal spring attached to its end, result in beams that partly shorten or stretch, depending on the spring stiffness. The traditional Euler–Bernoulli beam model may not accurately capture the geometrical nonlinearity in these cases. To address this, nonlinear Hencky’s beam models are proposed to describe such conditions. The validity of these models is assessed against the nonlinear Euler–Bernoulli model using the Galerkin method, with examples including cantilever and clamped-clamped configurations representing shortened and stretched beams. An analysis of a cantilever with a horizontal spring, where stiffness varies, using the nonlinear Hencky’s model, indicates that increasing horizontal stiffness stiffens the system. This analysis reveals a transition from softening to linear behavior to hardening near the second resonance frequency, suggesting a bifurcation point. Despite the computational demands of nonlinear Hencky’s models, this study highlights their effectiveness in overcoming the inherent assumptions of stretching and shortening in Euler–Bernoulli beam theory. These models enable a comprehensive nonlinear analysis of partly shortened or stretched beams.