Types Of Bifurcations Research Articles

Generalized Hopf bifurcation analysis of a towed caster wheel system

This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system, in which the non-smoothness factor is considered. The analysis can be divided into two cases by the coefficient of quadratic term. If the coefficient of quadratic term is equal to zero, the type of generalized Hopf bifurcation can be determined by the stability of first-order fine focus. Otherwise, we have to construct a Poincaré map for illustrating the type of generalized Hopf bifurcation, which is the heart of our discussion. Besides, we put forward another convenient way to judge the bifurcation types due to the symmetrical characteristic of this wheel system. Numerical simulations show the feasibility of theoretical analysis.

Stability and Bifurcation Analysis of a Discrete Singular Bioeconomic System

The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model.

Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

A quantitative mathematical model of neurons should not only include enough details to consider the dynamics of single neurons but also minimize the complexity of the model so that the model calculation is convenient. The two-dimensional Prescott model provides a good compromise between the authenticity and computational efficiency of a neuron. The dynamic characteristics of the Prescott model under external electrical stimulation are studied by combining analytical and numerical methods in this paper. Through the analysis of the equilibrium point distribution, the influence of model parameters and external stimulus on the dynamic characteristics is described. The occurrence conditions and the type of Hopf bifurcation in the Prescott model are analyzed, and the analytical determination formula of the Hopf bifurcation type in the neuron model is obtained. Washout filter control is used to change the Hopf bifurcation type, so that the subcritical Hopf bifurcation transforms to supercritical Hopf bifurcation, so as to realize the change of the dynamic characteristics of the model.

METHODS OF COMPARATIVE STATICS AND DYNAMICS IN THE THEORY OF ECONOMIC CYCLES

The subject of this work is the problem of describing the dynamic behavior of the price in the market of one product. The typical balance of interaction of supply and demand functions depending on the price is considered. The dynamic model of price evolution is based on the assumption that the demand function at a given time depends on the supply function at all previous points in time, i.e. there is a process with an aftereffect. The core of the integral transformation is a characteristic of the second order, which can initiate periodic regimes in price variables. The aim of the work is to synthesize a mathematical model of price changes in the market of one product and study the stability of its equilibrium states with the manifestation of the structure of marginal cycles. The task of the study is to demonstrate the degree of connection between the problem of stability of equilibrium and the problem of obtaining fruitful results in comparative statics. This duality is the principle of conformity of P. Samuelson. The basic mathematical model of the studied process of price dynamics is a system of two nonlinear differential equations of the first order. The research methods are the nonlinear theory of analysis of dynamical systems, the mathematical theory of stability of systems of differential equations, the conceptual apparatus of analysis of typical bifurcations of birth (death) of the boundary cycle, known as the Andronov-Hopf bifurcation. As a result of a detailed analysis of the properties and parameters of self-oscillating modes, a double cycle is revealed, i.e. there is a fact of coexistence around the equilibrium state of stable and unstable limit cycles. Subsequent mathematical transformations prove that the line of demarcation of these two cycles is completely determined by the static parameters of the studied system, which illustrates the mechanism of action of the principle of conformity of P. Samuelson. Conclusions: on the example of the functioning of the labor market, a comprehensive analysis of the stability of two positive equilibrium states, which characterize the effects of substitution and income. Computer simulations are used to perform computational experiments that demonstrate self-oscillating modes of price changes. As a result of the analysis of the obtained numerical results, it is possible to draw a conclusion about the stability of the limit cycle in the vicinity of the equilibrium state, which corresponds to the substitution effect.

Characterization of bifurcations in the Zeeman catastrophe machine using singular value decomposition

Stable limit cycles of the dynamic Zeeman Catastrophe Machine are calculated and classified. As the geometry of the device is changed, they evolve into different types at discrete bifurcation points. The process is normally presented as a bifurcation diagram using a stroboscopic map, but this does not provide enough information to uniquely identify the bifurcation event. Three types of bifurcation are common: period-doubling, symmetry-breaking, and turning points. A method is developed to distinguish between these types by calculating the response to a change in device geometry. Assuming that the limit cycle has already been calculated, the response to a change in geometry is defined as the solution of a linear second-order differential equation. The equation is solved in finite-difference form by defining a matrix which contains the second-order response to a change in the limit cycle, as well as an augmented matrix which contains the response to a change in device geometry. At a bifurcation, the solution can be analyzed using singular value decomposition of these two matrices. The resulting singular values contain enough information to uniquely identify, and characterize, each type of event. The corresponding singular vectors can also be used to uniquely characterize the symmetry changes that occur in the response to change at each event.

Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response

Abstract In this work, the dynamics of a predator-prey system considering square root type functional response for prey herd behaviour and nonlinear prey harvesting has been analyzed. The conditions under which all equilibria exist as well as the stability of every equilibrium point of the system have been investigated. The proposed model conditionally posses two types of bifurcations, Hopf bifurcation, and saddle-node bifurcation. The saddle-node bifurcation has been analyzed, where the bifurcation parameter is harvesting rate. The existence of a maximum sustainable yield to ensure both populations coexist has been discussed. The results give a clear idea that, if the harvesting rate is chosen at a proper value lesser than the maximum sustainable yield then both populations will coexist and the ecological balance will be maintained. The calculation of the first Lyapunov number provides the Hopf bifurcation direction. To verify our analytical results, several numerical simulations have been carried out.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

Global dynamics and control strategies of an epidemic model having logistic growth, non-monotone incidence with the impact of limited hospital beds.

In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin’s maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases.

Analysis of Geometric Invariants for Three Types of Bifurcations in 2D Differential Systems

Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

Bifurcation Analysis of a Bounded Rational Duopoly Game with Consumer Surplus

The nonlinear dynamical behaviors of economic models have been extensively examined and still represented a great challenge for economists in recent and future years. A proposed boundedly rational game incorporating consumer surplus is introduced. This paper aims at studying stability and bifurcation types of the presented model. The flip and Neimark–Sacker bifurcations are analyzed via applying the normal form theory and the center manifold theorem. This study helps determine an appropriate choice of decision parameters which have significant influences on the behavior of the game. The duopoly game that is formed by considering bounded rationality and consumer surplus is more realistic than the ordinary duopoly game which only has profit maximization. And then, some numerical simulations are provided to verify the theoretical analysis. Finally, we compare the dynamical behaviors of the built model with that of Bischi–Naimzada model so as to better understand the performance of the duopoly game with consumer surplus.

A new type of nonlinear hysteretic model for magnetorheological elastomer and its application

In this paper, a new type of nonlinear hysteretic model for MRE is proposed. Natural frequency is introduced to the hysteretic model to offset the influence of the loading rate on the hysteresis loop. A kind of developed multi-scale method is proposed to solve the natural frequency under strong nonlinear conditions. Based the new hysteretic model, the dynamic characteristics of a MRE sandwich plate with four sides simply supported and by the in-plane stochastic load are discussed. The simulation results show that the new hysteretic model is hardly affected by the loading rate and should be an ideal model for viscoelastic materials; there are abundant types of stochastic bifurcation in the system.

Coronary Artery Bifurcation Angle Degree; Bifurcation Lesion Development and the Prognostic Impact

Abstract Background: Coronary artery bifurcation lesion is one of the challenges in coronary arteries revascularization. Hemo-dynamic factors, as bifurcation angle is widely recognized for its involvement in atherosclerotic plaque formation. Aim of Work: Our aim of this study was to detect the effect of coronary artery bifurcation angle degree on the development of bifurcation atherosclerosis using Computed Tomography Coronary Angiography [CTCA] and its prognos-tic impact. Patients and Methods: The study was included 221 patients having low and intermediate pretest probability for Coronary Artery Disease [CAD]. Coronary arteries were investigated with CTCA for detection of bifurcation lesions, assessment of bifurcation angle and bifurcation lesion type, then 1 year follow-up period for Acute Coronary Syndrome (ACS). Results: 138 patients were males (62%) and 83 were females (38%). The meanage group was 55.7±9.8 years old. Bifurcation lesions were higher at LAD-LCx (125 patients, 56.5%) than at LAD-major diagonal (104 patients, 47%), than at LCx-OM (56 patients, 25%) (p-value <0.0001). At LAD-LCx sites there were higher bifurcation angles in patients with atherosclerotic bifurcation lesions than patients without bifurcation lesions (77.32o±18.1o vs. 62.24o±18.2o, p-value 0.001), while there was no statistically significant difference as regard bifurcation angels at LAD-diagonal and LCx-OM bifurcation sites. There was no relation between bifurcation angle and type of atherosclerotic plaque at the three studied bifurcation sites. Patients with wider bifurcation angels regardless lesions severity or management protocol had higher rates of ACS. Conclusion: A strong relation between LM bifurcation angel and the development of atherosclerotic lesions was noted, where wider bifurcation angles are more prone for atherosclerotic lesions, and more prone to develop ACS.

How Cerebral Vessel Tortuosity Affects Development and Recurrence of Aneurysm: Outer Curvature versus Bifurcation Type.

Background and PurposePrevious studies have assessed the relationship between cerebral vessel tortuosity and intracranial aneurysm (IA) based on two-dimensional brain image analysis. We evaluated the relationship between cerebral vessel tortuosity and IA according to the hemodynamic location using three-dimensional (3D) analysis and studied the effect of tortuosity on the recurrence of treated IA. MethodsWe collected clinical and imaging data from patients with IA and disease-free controls. IAs were categorized into outer curvature and bifurcation types. Computerized analysis of the images provided information on the length of the arterial segment and tortuosity of the cerebral arteries in 3D space. ResultsData from 95 patients with IA and 95 controls were analyzed. Regarding parent vessel tortuosity index (TI; P<0.01), average TI (P<0.01), basilar artery (BA; P=0.02), left posterior cerebral artery (P=0.03), both vertebral arteries (VAs; P<0.01), and right internal carotid artery (P<0.01), there was a significant difference only in the outer curvature type compared with the control group. The outer curvature type was analyzed, and the occurrence of an IA was associated with increased TI of the parent vessel, average, BA, right middle cerebral artery, and both VAs in the logistic regression analysis. However, in all aneurysm cases, recanalization of the treated aneurysm was inversely associated with increased TI of the parent vessels. ConclusionsTIs of intracranial arteries are associated with the occurrence of IA, especially in the outer curvature type. IAs with a high TI in the parent vessel showed good outcomes with endovascular treatment.

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

Optimal harvesting and stability of a predator-prey model for fish populations with schooling behavior.

In this paper, the schooling behavior of prey fish population in a predator-prey interaction is investigated. By taking an economical interest which can be elaborated by the presence of nonselective harvesting into consideration, we studied the dynamical behavior. The existence, positivity and boundedness of solution have been established. The analysis of the equilibrium states is presented by studying the local and the global stability. The possible types of local bifurcation that the system can undergoes are discussed. The effect of fishing effort on the evolution of the species is examined. Further, by using Pontryagin's maximum principle a proper management strategy has been used for avoiding the extinction of the considered species and maximizing the benefits. For the validation of the theoretical result, several of graphical representations have been used.

Two heads are not always better than one: Craniofacial and axial bifurcation in cheloniid embryos and hatchlings (Chelonia mydas and Caretta caretta).

Morphological data on craniofacial and axial bifurcation in sea turtles is not well documented in the literature. Here, we use micro-computed tomography (μ-CT) imaging to describe the body, skull, and vertebral morphology in axially-bifurcated cheloniid sea turtle embryos and hatchlings (Chelonia mydas and Caretta caretta) from south Florida beaches. We describe three types of craniofacial and axial bifurcations: bifacial, bicephalic, and bicephalic with biaxial duplication ranging from facial bones to the sacrum. We predicted smaller body dimensions in bifacial and bicephalic embryos and hatchlings compared with their normal counterparts. In addition, we hypothesized that bicephalic individuals would have greater rostral deviation angles than bifacial animals, and that vertebral dimensions would vary between the control and anomalous embryos and hatchlings. Among hatchlings (developmental Stage 31), we found that maximum curved carapace length and curved carapace width were greatest in the control specimens when compared with the anomalous specimens. Overall, we found that rostral deviations were smaller in bifacial animals compared with their bicephalic counterparts. Right and left rostral deviations were symmetrical or nearly symmetrical in all bifacial and bicephalic specimens. Among C. caretta, we found that bicephalic animals had greater standardized vertebral measurements than their bifacial conspecifics. In bifacial animals, bifurcation extended to either the frontal or parietal skull bones, while duplication extended to C5 vertebrae and T8 vertebrae in bicephalic animals. This study provides an in-depth description of anatomical alterations associated with these abnormalities. Prognosis of these organisms is poor; however, understanding the prevalence of these malformations can allow for better assessments of population health, as numerous environmental factors are known to lead to these changes.

Improving system loadability with the consideration of multiple bifurcations

The introduction of large numbers of power electronic converters along with increasing load demand will create voltage stability issues in future power systems. Most present analysis methodologies neglect the non-linearities within these systems. Bifurcation analysis is an increasingly popular method which not only incorporates system non-linearities, but also efficiently provides information about the system trajectory towards multiple instability mechanisms. This paper develops a new concept of combined system loadability with the consideration of saddle-node bifurcations (SNB), limit-induced bifurcations (LIB), and Hopf bifurcations (HB) simultaneously. These concepts have shown great promise in analyzing complex, multi-origin stability phenomena which are starting to appear in power systems and other areas, and are difficult to study using other methods. An optimization approach is applied to identify the system maximum loadability with these different types of bifurcation points which have not previously been considered at the same time. This creates a new system loadability boundary concept consisting of multiple types of bifurcation which is more representative of the actual power system limit boundary. By maximizing the shortest distance to the composite loadability boundary, the system loading margin is increased. Three illustrative systems are used to verify the proposed methodologies.

Flows with collective dynamics on a sphere

In this article different properties of flow codes are studied and a diagram is constructed as a whole topological invariant of them. In particular, flows with no more than 6 saddles are described. Two types of simple bifurcations: positive and negative – are considered as well. Summarizing the results on compact surfaces with boundary remains an interesting question for future works.

A novel fractional piecewise linear map: regular and chaotic dynamics

In this paper, a new piecewise linear map of the plan and its fractional version deduced from the Lozi map is introduced and analysed. The main attention is paid to the study of fixed points and their stability, cycles of period two and their stability regions and the type of bifurcation that occur in the dynamical behaviours of this map. The routes to chaos and some chaotic attractors that exist in the behaviour of the integer map are discussed. Finally, the chaotic behaviour of the associated proposed fractional map is analysed by means of bifurcations diagrams.

Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Flexure Pivot Tilting Pad Journal Bearing in Comparison to a Fixed Profile Journal Bearing With Experimental Verification

Abstract In this paper, an efficient numerical method consisting of the real mode component mode synthesis (CMS) model reduction, shooting method with parallel computing, and Floquet analysis was developed for nonlinear rotordynamics analysis of a flexible rotor supported by a four-lobe flexure pivot tilting pad journal bearing (FPTPJB) in load-on-pad (LOP) and load-between-pad (LBP) orientations in comparison to a fixed profile journal bearing (JB) of the same pad geometry. The method used the rotor's finite elements and bearing forces obtained from directly solving the Reynolds equation to determine the limit cycles and Hopf bifurcation types. For the investigated rotor and bearing parameters, the numerical results indicated that the onset speed of instability (OSI) of FPTPJB is considerably higher than that of JB of the same orientation. Also, FPTPJB in LOP orientation yielded substantially higher OSI than the LBP one, whereas the OSI of JB in LOBP orientation was higher than the LOP counterpart. Nonlinear calculation results indicated that all bearing types and orientations gave subcritical Hopf bifurcation. The FPTPJB in LOP orientation produced the largest stable operating region, whereas the JB in LBP configuration yielded the smallest one. The experiment showed subcritical Hopf bifurcation occurred at speed close to the calculated OSI in all cases except FPTPJB in LOP orientation that the OSI is higher than the maximum test rig speed. The whirling orbit had the same frequency as the first critical speed and was precessed in the shaft rotation direction.