Bifurcations Of Equilibrium Points Research Articles

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Turing instability in a network-organized epidemic model with delay

In this paper, we show the impact of both network and time delays on Turing instability and demarcate the role of diffusion in the epidemic. The stability and bifurcation of equilibrium points are analyzed to reveal the epidemic state, which is the precondition of pattern formation. The network could lead to the transition from the endemic to the periodic outbreak via negative wavenumber, which provides a way to prevent significant harm or decrease the damage of the epidemic to humans by the delay, the connection rate, and the infection rate. Also, the threshold value of time delay is proportional to the minimum eigenvalue of the network matrix, which provides a way to control the periodic behavior. Finally, numerical simulations validate these analytical results and the mechanisms of frequent outbreaks.

The bifurcation of constrained optimization optimal solutions and its applications

<abstract> <p>The appearance and disappearance of the optimal solution for the change of system parameters in optimization theory is a fundamental problem. This paper aims to address this issue by transforming the solutions of a constrained optimization problem into equilibrium points (EPs) of a dynamical system. The bifurcation of EPs is then used to describe the appearance and disappearance of the optimal solution and saddle point through two classes of bifurcation, namely the pseudo bifurcation and saddle-node bifurcation. Moreover, a new class of pseudo-bifurcation phenomena is introduced to describe the transformation of regular and degenerate EPs, which sheds light on the relationship between the optimal solution and a class of infeasible points. This development also promotes the proposal of a tool for predicting optimal solutions based on this phenomenon. The study finds that the bifurcation of the optimal solution is closely related to the bifurcation of the feasible region, as demonstrated by the 5-bus and 9-bus optimal power flow problems.</p> </abstract>

Dynamic behavior of self-acting gas-lubricated long journal bearing

Planar dynamics of a rotor supported by self-acting gas-lubricated journal bearing is investigated theoretically. Whirling of the shaft is one of the common problems in a gas-lubricated journal bearing system. As whirling increases, it becomes large enough to cause a destructive contact between the bearing surfaces. It is essential to predict the parameter range for the safe operation of the gas-lubricated journal bearing systems. An analytical model of the long, self-acting gas-lubricated journal bearing system is numerically integrated for analysis of steady-state solutions and limit cycle oscillations. Dynamic behavior of the system is analyzed with the variation of dimensionless bearing compressibility parameter Λ and dimensionless bearing stability parameter Λ2Ω. For a set of stability parameters Λ2Ω ranging from 0.01 to 1000, the critical value of Λ is determined which corresponds to angular speed ω of the rotor. With the decrease of parameter Λ2Ω, critical value Λ decreases. Limit cycles bifurcating from the equilibrium point at critical values are unstable for a low value of Λ2Ω=0.01 and stable for other higher values. In both the cases of stable and unstable limit cycles, limit cycles disappear beyond a compressibility parameter Λ and no further fold bifurcation of the limit cycle is detected. Results are presented as equilibrium point bifurcation and bifurcation of the limit cycle. The dynamics of the system and safe operating range are presented with diagrams. The results obtained are validated through trajectories of the journal center by numerical integration of the system of equations for different initial conditions presented in the figures.

Critical slowing down close to a global bifurcation of a curve of quasi-neutral equilibria

Critical slowing down arises close to bifurcations and involves long transients. Despite slowing down phenomena have been widely studied in local bifurcations i.e., bifurcations of equilibrium points, less is known about transient delay phenomena close to global bifurcations. In this paper, we identify a novel mechanism of slowing down arising in the vicinity of a global bifurcation i.e., zip bifurcation, identified in a mathematical model of the dynamics of an autocatalytic replicator with an obligate parasite. Three different dynamical scenarios are first described, depending on the replication rate of cooperators, (L), and of parasites, (K). If K<L the system is bistable and the dynamics can be either the out-competition of the parasite or the two-species extinction. When K>L the system is monostable and both species become extinct. In the case K=L coexistence of both species takes place in a Curve of Quasi-Neutral Equilibria (CQNE). The novel slowing down mechanism identified is due to an underlying ghost CQNE for the cases K≲L and K≳L. We show, both analytically and numerically, that the delays caused by the ghost CQNE follow scaling laws of the form τ∼|K−L|−1 for both K≲L and K≳L. We propose the ghost CQNE as a novel transient-generator mechanism in ecological systems.

Stability analysis of a fractional-order novel hepatitis B virus model with immune delay based on Caputo-Fabrizio derivative

Abstract In mathematical epidemiology, mathematical models play a vital role in understanding the dynamics of infectious diseases. Therefore, in this paper, a novel mathematical model for the hepatitis B virus (HBV) based on the Caputo-Fabrizio fractional derivative with immune delay is introduced, while taking care of the dimensional consistency of the proposed model. Initially, the existence and uniqueness of the model solutions are proved by Laplace transform and the fixed point theorem. The positivity and boundedness of the solutions are also discussed. Sumudu transform and Picard iteration were used to analyze the stability and iterative solution of the fractional order model of HBV. Further, using the stability theory of fractional order system, the stability and bifurcation of equilibrium point are discussed. Finally, results are presented for different values of the fractional parameter.

Bifurcation analysis for a two-dimensional peristaltic driven flow of power-law fluid in asymmetric channel

In the present article, the bifurcations of equilibrium points and their streamlined patterns for the peristaltic transport of shear-thinning and shear-thickening fluids through an asymmetric channel are studied by incorporating a power-law model. An exact solution in the wave frame of reference is obtained under the vanishing Reynolds number and long wavelength approximations. A system of non-linear autonomous differential equations is developed to locate the equilibrium points in the flow. The qualitative nature of equilibrium points and their bifurcations are investigated through the dynamical system method. There exist three distinct flow conditions (backward flow, trapping, and augmented flow). It is observed that the shifting of these flow phenomena corresponds to bifurcations where non-hyperbolic degenerate points are conceived. The impacts of various embedded parameters on flow phenomena and their bifurcations are demonstrated through graphical representations. It is found that the trapping phenomenon manifests at a high flow rate for shear-thinning fluids. That is, the backward flow region shrinks for large values of the power-law index. Trapping in mechanical devices can be diminished by enlarging the phase difference of channel walls, while an opposite trend is observed for amplitude ratios. The obtained results are concluded through global bifurcation diagrams. At the end, findings of this analysis are verified by making a comparison with the existing literature.

Impact of Chloride Channel on Spiking Patterns of Morris-Lecar Model

In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl- fluctuations on firing patterns of this neuron model. After adding Cl- channel in the original Morris-Lecar model, the dynamics of the original model such as its bifurcations of equilibrium points would be changed and they occurred at different values compared to the primary model. We discover these qualitative changes in the point of dynamical systems and neuroscience. We will conduct the co-dimension two bifurcations analysis with respect to different control parameters to explore the complicated behaviors for this new neuron model.

Dynamics of an epidemic model with impact of toxins

We have considered a toxin-dependent dynamical model to study the effects of an environmental toxin on the spread of infectious diseases in the population. The well-posedness of the model is discussed by showing the positivity and boundedness of the state variables. Model analysis is performed as well as the stability and bifurcation of equilibrium points are also established. Numerical simulation is performed to verify the theoretical results and it shows that the system possesses a transcritical bifurcation around the disease-free equilibrium. Our analytical results indicate that there exists a threshold value of the environmental toxin, i.e., if the environmental toxin amount is lower than the threshold, the system has a stable disease-free equilibrium and if the environmental toxin amount is higher than the threshold value, then the system has a unique endemic equilibrium. Numerical simulation also shows that the environmental toxin plays a crucial role in spreading of infectious diseases. An optimal control problem has been formulated in the later part to minimize the cost and disease fatality by choosing the treatment and the depuration of the toxin as control variables. Numerical analysis indicates that if only control via depuration is used, then it will be more economical for the time period whereas treatment works well but with a lesser intensity. Moreover, simultaneous use of both the control interventions is more useful for the system dynamics and it reduces the number of infective individuals and also minimizes the economic cost generated from disease burden. This study gives a clear view of the impact of toxins on the spread of infectious diseases in the population and it helps the disease control agencies and governments to take suitable precautions to control the disease.

Equilibria and orbits in the dynamical environment of asteroid 22 Kalliope

Abstract This paper studies the orbital dynamics of the potential of asteroid 22 Kalliope using observational data of the irregular shape. The zero-velocity surface are calculated and showed with different Jacobian values. All five equilibrium points are found, four of them are outside and unstable, and the other one is inside and linearly stable. The movement and bifurcations of equilibrium points during the variety of rotation speed and density of the body are investigated. The Hopf bifurcations occurs during the variety of rotational speed from ω=1.0ω 0 to 0.5ω 0, and the Saddle-Node bifurcation occurs during the variety of rotational speed from ω=1.0ω 0 to 2.0ω 0. Both unstable and stable resonant periodic orbits around Kalliope are coexisting. The perturbation of an unstable periodic orbit shows that the gravitational field of Kalliope is strongly perturbed.

Parameters and bifurcations of equilibrium points in the gravitational potential of irregular-shaped bodies subjected to a varying external shape

In this work, we investigate variations in the positions, eigenvalues, Jacobi integrals, topological cases, as well as the stability of equilibrium points around the asteroid 243 Ida when varying the external shape of the body. First, we employ a polyhedral shape model to calculate the surface height, gravitational force acceleration on the surface, and effective potential on the surface of 243 Ida. We then adopt the homotopy analysis method to generate variations in the external shape of a generic body within a continuum between the modeled external shape of 243 Ida and that of a sphere while maintaining a constant volume. Then, we calculated the positions, eigenvalues, Jacobi integrals, as well as the Hessian matrices of the equilibrium points in the gravitational potential of the generic body. We analyzed the topological cases and stability of the equilibrium points based on the obtained eigenvalues and Hessian matrices. It is concluded that the positions, eigenvalues, and Jacobi integrals change when varying the external shape of the body. For the four external equilibrium points E1–E4, the norm of the position vectors increases; however, the norm of the position vector for the internal equilibrium point E5 decreases. Equilibrium points E2 and E4 are subject to Hopf bifurcation when varying the external shape of the body, and the topological cases and stability of E2 and E4 are correspondingly changed. In contrast, the topological cases and stability of equilibrium points E1, E3, and E5 remain unchanged when varying the external shape of the body.

Dynamical study of a chaotic predator-prey model with an omnivore

In this paper, the dynamics and bifurcations of a three-species predator-prey model with an omnivore are further investigated. The food web considered in this work comprises prey, predator and a third species, which consumes the carcasses of the predator along with predation of the original prey. The conditions for existence, uniqueness and continuous dependence on initial conditions for the solution of the model are derived. Analytical and numerical bifurcation studies reveal that the system undergoes transcritical and Hopf bifurcations around its equilibrium points. Further, the Hopf bifurcation curves in the parameters’ space along with codimension two bifurcations of equilibrium points and bifurcation of limit cycles that arise in the system are investigated. In particular, the occurrence of generalized Hopf, fold Hopf and Neimarck-Sacker bifurcations is unveiled and illustrates the rich dynamics of the model. Finally, bifurcation diagrams, phase portraits and Lyapunov exponents of the model are presented.

Bifurcation of equilibrium points in the potential of heterogeneous Itokawa

Bifurcation of equilibrium points in the potential of heterogeneous Itokawa

Qualitative effects of introducing nonlinear birth and death rates for the predator in a predator-prey type model

In this paper, we study how introducing nonlinear birth and death rates for the predator might affect the qualitative behavior of a mathematical model, describing predator-prey systems. We base our investigations on a known model, exhibiting anti-predator behavior. We propose a generalization of the latter by introducing generic birth and death rates for the predator and study the dynamics of the resulting system. We establish existence and uniqueness of positive model solutions, their uniform boundedness, existence, local stability and bifurcations of equilibrium points as well as global stability properties of the solutions. Most of the solution properties are demonstrated numerically and graphically by various numerical examples. Based on the obtained results, we show that the model with nonlinear birth and death rates can describe a much more complex behavior of the predator-prey system than the classical model (i.e., with linear rates) does.

Neimark-Sacker Bifurcation in Demand-Inventory Model with Stock-Level-Dependent Demand

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

Bifurcation of equilibrium points in the potential field of asteroid 101955 Bennu

The stability and topological structure of equilibrium points in the potential field of the asteroid 101955 Bennu have been investigated with a variable density and rotation period. A dimensionless quantity is introduced for the nondimensionalization of the equations of motion, and this quantity can indicate the effect of both the rotation period and bulk density of the asteroid. Using the polyhedral model of the asteroid Bennu, the number and position of the equilibrium points are calculated and illustrated by a contour plot of the gravitational effective potential field. The topological structure and the stability of the equilibrium points are also investigated using the linearized method. The results show that there are nine equilibrium points in the potential field of the asteroid Bennu, eight in the exterior of the body and one in the interior of the body. Moreover, bifurcation will occur with a variation of the density and rotation period. Different equilibrium points will encounter each other and mix together. Thus, the number of equilibrium points will change. The stability and topological structure of the equilibrium points will also change because of the variation of the density and rotation period of the asteroid. When considering the error of the density of Bennu, the range of the dimensionless quantity covers the critical values that will lead to bifurcation. This means that the stability of the equilibrium points is uncertain, making the dynamical environment of Bennu much more complicated. These bifurcations can help better understand the dynamic environment of an irregular-shaped asteroid.

Equilibrium Point Bifurcation and Singularity Analysis of HH Model with Constraint

We present the equilibrium point bifurcation and singularity analysis of HH model with constraints. We investigate the effect of constraints and parameters on the type of equilibrium point bifurcation. HH model with constraints has more transition sets. The Matcont toolbox software environment was used for analysis of the bifurcation points in conjunction with Matlab. We also illustrate the stability of the equilibrium points.

Duffing-van der Pol-type oscillator systems

In this paper, under certain parametric conditions we areconcerned with the first integrals of the Duffing-van derPol-type oscillator system, which include the van der Poloscillator and the damped Duffing oscillator etc as particularcases. After applying the method of differentiable dynamics toanalyze the bifurcation set and bifurcations of equilibrium points,we use the Lie symmetry reduction method to find two nontrivialinfinitesimal generators and use them to construct canonicalvariables. Through the inverse transformations we obtain the firstintegrals of the original oscillator system under the givenparametric conditions, and some particular cases such as thedamped Duffing equation and the van der Pol oscillator system areincluded accordingly.

Singular Casimir Elements: Their Mathematical Justification and Physical Implications

Bifurcation of equilibrium points in fluids or plasmas is studied using the notion of Casimir foliation that occurs in the noncanon- ical Hamiltonian formalism of the ideal dynamics. The nonlinearity of the system makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and creates a singularity where the nullity of the Poisson operator changes. The problem is an infinite-dimensional generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.

SYMMETRIC BIFURCATIONS IN A RING OF COUPLED DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS

Symmetry bifurcations of equilibrium points of three coupled differential equation with piecewise continuous arguments (EPCA) oscillators are studied. The system is equivariant under dihedral group D3 with order 6. This causes several types of symmetrical bifurcations.