Bifurcation Value Research Articles

Hopf bifurcations for a delayed discrete single population patch model in advective environments

Abstract In this paper, we consider a delayed discrete single population patch model in advective environments. The individuals are subject to both random and directed movements, and there is a net loss of individuals at the downstream end due to the flow into a lake. Choosing time delay as a bifurcation parameter, we show the existence of Hopf bifurcations for the model. In homogeneous non-advective environments, it is well known that the first Hopf bifurcation value is independent of the dispersal rate. In contrast, for homogeneous advective environments, the first Hopf bifurcation value depends on the dispersal rate. Moreover, we show that the first Hopf bifurcation value in advective environments is larger than that in non-advective environments if the dispersal rate is large or small, which suggests that directed movements of the individuals inhibit the occurrence of Hopf bifurcations.

Solvability of functional third-order problems of Ambrosetti–Prodi-type

This work presents an Ambrosetti–Prodi alternative for functional problems composed of a fully third-order differential equation with two types of functional boundary conditions. The discussion of existence and non-existence of solution is obtained in a more general case, and the multiplicity of solution is done with restrictive boundary conditions-The main arguments are based on the lower and upper solutions method, together with the Leray–Schauder topological degree theory. We stress that the multiplicity situation requires different speed growths on the variables.An example illustrates the results’ applicability and shows a technique to estimate the bifurcation values of the parameter.

Stability and Hopf Bifurcation in a Modified Sprott C System

Abstract In this article, a Modified Sprott C system is considered. The stability of equilibrium points and the occurrence of Hopf bifurcation in the system are investigated. It has been proved that the system displays a Hopf bifurcation at α = 0. Additionally, by applying normal form theory, the stability, direction and increase (or decrease) of the period of bifurcating periodic solutions of the system are illustrated. It has been shown that the solutions of bifurcating periodic solutions at the bifurcation value α = 0 are unstable. The type of Hopf bifurcation is subcritical and the periods of the bifurcating periodic solutions increase.

Complex Dynamics and Bifurcations Analysis of Discrete-Time Modified Leslie–Gower System

This work introduces a discrete modified Leslie–Gower prey–predator system with Holling type-II functional response. The persistence of the discrete model under certain conditions is discussed. The conditions assuring the existence of fixed points are derived and nonlinear dynamics of system are explored at these fixed points. It has been shown that the system exhibits transcritical bifurcation and flip bifurcation at semi-trivial fixed point under certain bifurcation values. In addition, the center manifold and bifurcation theories are employed to attain the conditions for existence of flip and Neimark–Sacker bifurcations at coexistence fixed point. The system is found to exhibit periodic solutions along with bifurcations leading to wide range of chaotic dynamics. The numerical simulations are performed to confirm the analytical analysis.

Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions.

We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.

A Novel Method for the Measurement of Retinal Arteriolar Bifurcation.

The purpose of this research was to develop protocols for evaluating the bifurcation parameters of retinal arteriole and establish a reference range of normal values. In this retrospective study, we measured a total of 1314 retinal arteriolar bifurcations from 100 fundus photographs. We selected 200 from these bifurcations for testing inter-measurerand inter-method agreement. Additionally, we calculated the normal reference range for retinal arteriolar bifurcation parameters and analyzed the effects of gender, age, and anatomical features on retinal arteriolar bifurcation. The measurement method proposed in this study has demonstrated nearly perfect consistency among different measurers, with interclass correlation coefficient (ICC) for all bifurcation parameters of retinal arteriole exceeding 0.95. Among healthy individuals, the retinal arteriolar caliber was narrowest in young adults and increased in children, teenagers, and the elderly; retinal arteriolar caliber was greater in females than in males; and the diameter of the inferior temporal branch exceeded that of the superior temporal branch. The angle between the two branches of retinal arteriolar bifurcation was also greater in females than in males. When using the center of the optic disc as a reference point, the angle between the two branches of the retinal arteriole at the proximal or distal ends increased. In contrast, the estimated optimum theoretical values of retinal arteriolar bifurcation were not affected by these factors. The method for the measurement of retinal arteriolar bifurcation in this study was highly accurate and reproducible. The diameter and branching angle of the retinal arteriolar bifurcation were more susceptible to the influence of gender, age, and anatomical features. In comparison, the estimated optimum theoretical values of retinal arteriolar bifurcation were relatively stable. Video available for this article.

Analysis of the geometry of the zero-velocity curves in the N-body ring problem depending on the mass ratio parameter

The purpose of the present study is the investigation of the effect of the mass ratio parameter, β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, on the geometry of the zero-velocity curves, when there are N=3,4,…,100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=3,4,\\ldots ,100$$\\end{document} peripheral bodies. It is well known that there is a bifurcation value of the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} parameter in the N-body ring problem that produces a change in the number of stationary solutions in the system from 5N to 3N. By examining the behavior of the critical values of the Jacobi constant that define each of the zones of stationary solutions, we have unveiled the existence of other bifurcations or critical values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} in the scenario with 5N stationary solutions, which cause different changes in the geometry of the zero-velocity curves, which in turn affect the threshold for the total opening of the curves of zero velocity.

Switched Epidemic Models: A Contact Rate Dependent Analysis

This paper introduces a switched SQWVEAIR (Susceptible, Quarantined, Weak, Vaccinated, Exposed, Asymptomatic, Infected, and Recovered) epidemic model considering changing contact rates in an infected society. It also takes into account the presence of vulnerable individuals as frontiers in vaccination, recognizing their societal importance. The qualitative analysis of the model involves two theorems on the positive invariance set and the positivity of the solutions. Additionally, it calculates the equilibria of the proposed model and analyzes their global stabilities using the Lyapunov functional technique, focusing on the basic reproduction number (R0) as a bifurcation value. To achieve globally uniformly exponential stability, it calculates the contact-rate-dependent average dwell time (CRDADT) for each stable and unstable subsystem, developing a Lyapunov function. It also derives sufficient conditions, expressed as linear matrix inequalities (LMIs), to ensure stability. A numerical example is provided to demonstrate the effectiveness of the theoretical analysis and validate our research findings.

Stochastic stability of an autoresonance model with a center–saddle bifurcation

The purpose of this work is to investigate the effect of stochastic perturbations of the white noise type on the stability of capture into autoresonance in oscillating systems with a variable pumping amplitude and frequency such that a center–saddle bifurcation occurs in the corresponding limiting autonomous system. The another purpose is determine the dependence of the intervals of stochastic stability of the autoresonance on the noise intensity. Methods. The existence of autoresonant regimes with increasing amplitude is proved by constructing and justificating asymptotic solutions in the form of power series with constant coefficients. The stability of solutions in terms of probability with respect to noise is substantiated using stochastic Lyapunov functions. Results. The conditions are described under which the autoresonant regime is preserved and disappears when the parameters pass through bifurcation values. The dependence of the intervals of stochastic stability of autoresonance on the degree of damping of the noise intensity is found. It is shown that more stringent restrictions are required to preserve the stability of solutions for the bifurcation values of the parameters. Conclusion. At the level of differential equations describing capture into autoresonance, the effect of damped stochastic perturbations on the center–saddle bifurcation is studied. The results obtained indicate the possibility of using damped oscillating perturbations for stable control of nonlinear systems.

Nonlocal delay gives rise to vegetation patterns in a vegetation-sand model.

The vegetation pattern generated by aeolian sand movements is a typical type of vegetation patterns in arid and semi-arid areas. This paper presents a vegetation-sand model with nonlocal interaction characterized by an integral term with a kernel function. The instability of the Turing pattern was analyzed and the conditions of stable pattern occurrence were obtained. At the same time, the multiple scales method was applied to obtain the amplitude equations at the critical value of Turing bifurcation. The spatial distributions of vegetation under different delays were obtained by numerical simulation. The results revealed that the vegetation biomass increased as the interaction intensity decreased or as the nonlocal interaction distance increased. We demonstrated that the nonlocal interaction between vegetation and sand is a crucial mechanism for forming vegetation patterns, which provides a theoretical basis for preserving and restoring vegetation.

Time-delayed feedback control for chaotic systems with coexisting attractors

<abstract><p>This study investigated the Hopf bifurcation of the equilibrium point of chaotic systems with coexisting attractors under the time-delayed feedback control. First, the equilibrium point and Hopf bifurcation of chaotic systems with coexisting attractors were analyzed. Second, the chaotic systems were controlled by time-delayed feedback, the transversality condition of Hopf bifurcation at the equilibrium point was discussed, and the time-delayed value of Hopf bifurcation at the equilibrium point was obtained. Lastly, the correctness of the theoretical analysis was verified by using the numerical results.</p></abstract>

Bifurcations of a Leslie-Gower predator-prey model with fear, strong Allee effect and hunting cooperation

<p>Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multi-stability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed.</p>

Is azulene's local aromaticity and relative stability driven by the Glidewell-Lloyd rule?

The local aromaticity of azulene has been studied to understand their electronic properties. For this purpose, we have used the magnetic criterion through magnetically induced current density maps, ring current strengths, NICSzz(1), and the bifurcation value of three-dimensional surfaces of NICSzz. On the other hand, the delocalization criterion was used by calculating the MCI and ELFπ. The results show that the five-membered ring (5-MR) is more aromatic than the seven-membered ring (7-MR) and more aromatic than the free C5H5- ring. The opposite case is seen for the seven-membered ring, which is less aromatic than the free C7H7+. The local aromatic rings in azulene are formed due to an intramolecular electron transfer from the 7-MR to the 5-MR. In addition, the proposed resonance structures that allow explaining the properties of azulene, such as the dipole moment or the relative stability (in comparison to other isomers), show a preference for the formation of 5-MRs; for this reason, it is possible to conclude that the aromaticity and relative stability of azulene is driven by the Glidewell-Lloyd rule.

The Poynting Vector Field Singularities: Effects of Symmetry and Its Violation

AbstractThe phenomenological theory revealing the generic effects of the problem symmetry, its violation, and energy conservation law on the singularities of the Poynting vector field is presented. The bifurcation scenario of formation (annihilation) of the singularities under variations of the problem parameters is elucidated. The results describe the singularities in scattering a linearly polarized plane electromagnetic wave. However, they are valid for any configuration of the incident beam at its scattering by a subwavelength particle. The author shows that topological changes in the pattern of the Poynting vector field occur through a finite number of pitchfork bifurcations. It means that the patterns are topologically stable under variations of the problem parameter(s) that lie between the bifurcation values. The latter ensures that the discussed topological properties of the problem are robust to weak symmetry violation, which is inevitable in any actual experiment. The general consideration is illustrated by a detailed study of singularities in scattering by an infinite right circular germanium cylinder. The results open the possibility of fitting and controlling radiation patterns on subwavelength scales important for various nanotechnologies.

Effect of fractional order on the stability and the localisation of the critical Hopf bifurcation value in a fractional chaotic system

Effect of fractional order on the stability and the localisation of the critical Hopf bifurcation value in a fractional chaotic system

Bifurcation analysis of a delayed reaction–diffusion–advection Nicholson’s blowflies equation

In this paper, we investigate the dynamics of a reaction-diffusion Nicholson’s blowflies equation with advection. The stability of positive steady state and existence of Hopf bifurcation are obtained by analyzing the distribution of the eigenvalues. Moreover, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction and stability of the Hopf bifurcation is derived. Meanwhile, we find out that the bifurcation value is increasing with respect to the advection rate. Finally, numerical results demonstrate that the advection term causes the population to move from upstream to downstream, which also indicates that advection term plays a key role in the description and interpretation of some common natural phenomena.

Rate and memory effects in bifurcation-induced tipping.

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution, or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call, here, the shifted stability exchange. We perform a systematic study on how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical simulations and partly analytical results for different types of bifurcations and different paradigmatic systems. We show that the nonautonomous dynamics can be split into two regimes. Depending on whether we exceed the numerical or experimental precision or not, the system may enter the nondeterministic or the deterministic regime. This is determined solely by the conditions of the drift. Finally, we deduce the scaling laws governing this phenomenon and we observe very similar behavior for different systems and different bifurcations in both regimes.

HIDDEN BOUNDARY OF GLOBAL STABILITY IN THE KAPRANOV CONJECTURE ON THE PULL-IN RANGE

Within the framework of the development of the theory of hidden oscillations, the problem of determining the boundary of global stability and revealing its hidden parts corresponding to the non-local birth of hidden oscillations is considered. For a phase-locked loop with a proportional-integrating filter and a piecewise-linear phase detector characteristic, effective methods for determination of bifurcations of the global stability loss, for obtaining analytical formulas of the bifurcation values, and for constructing trivial and hidden parts of the global stability boundary are suggested.

Automatic explanation of the classification of Spanish legal judgments in jurisdiction-dependent law categories with tree estimators

Automatic legal text classification systems have been proposed in the literature to address knowledge extraction from judgments and detect their aspects. However, most of these systems are black boxes even when their models are interpretable. This may raise concerns about their trustworthiness. Accordingly, this work contributes with a system combining Natural Language Processing (nlp) with Machine Learning (ml) to classify legal texts in an explainable manner. We analyze the features involved in the decision and the threshold bifurcation values of the decision paths of tree structures and present this information to the users in natural language. This is the first work on automatic analysis of legal texts combining nlp and ml along with Explainable Artificial Intelligence techniques to automatically make the models’ decisions understandable to end users. Furthermore, legal experts have validated our solution, and this knowledge has also been incorporated into the explanation process as “expert-in-the-loop” dictionaries. Experimental results on an annotated data set in law categories by jurisdiction demonstrate that our system yields competitive classification performance, with accuracy values well above 90%, and that its automatic explanations are easily understandable even to non-expert users.

Revisiting hollow all-carbon structures: Aromaticity and kinetic stability

In this contribution, the aromaticity of the so-called hollow carbon structures, proposed as an allotrope of carbon, has been revisited through the bifurcation values of the electron localization function (ELF) as a complement to the previous results obtained through the magnetic criterion for Gaudiene (C72). Additionally, we studied systems with 162, 216, and 648 carbon atoms were studied. Since, in large systems, the use of methodologies such as DFT can often be prohibitive, the GFN2-xTB method has been applied, whose wavefunction has been used to calculate the ELF. To the authors' knowledge, this is the first time an ELF study has been carried out with large systems and this method. On the other hand, the kinetic stability has been evaluated using molecular dynamics simulations at different temperatures. The results show that the hollow carbon structures present aromatic character, leading to high thermodynamic and kinetic stability, making them good candidates for exploring diverse and interesting applications. Finally, it is demonstrated that GFN2-xTB is a method that allows the successful study of large systems where the analysis using DFT or post-Hartree-Fock calculations would have a high computational cost.