Types Of Bifurcations Research Articles

Basic reinfection number and backward bifurcation.

Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number $ {{{\cal R}_{0}}} $ is below $ 1 $, through a phenomenon known as a backward bifurcation. Causes for this phenomenon include exogenous reinfection, super-infection, relapse, vaccination exercises, heterogeneity among subpopulations, etc. To measure the reinfection forces, this paper defines a second threshold: the basic reinfection number. This number characterizes the type of bifurcation when the basic reproduction number is equal to one. If the basic reinfection number is greater than one, the bifurcation is backward. Otherwise it is forward. The basic reinfection number with the basic reproduction number together gives a complete measure for disease control whenever reinfections (or relapses) matter. We formulate the basic reinfection number for a variety of epidemiological models.

A discrete-time fractional-order Rosenzweig-Macarthur predator-prey model involving prey refuge

In this article, the dynamical behaviors of a discrete-time fractional-order Rosenzweig-MacArthur model with prey refuge are studied. The piecewise constant arguments scheme is applied to obtain the discrete-time model. All possible fixed points and their existence conditions are investigated as well as the local behavior of nearby solutions in various contingencies. Numerical simulations such as the time series, phase portraits, and bifurcation diagrams are portrayed. Three types of bifurcations are shown numerically namely the forward, the period-doubling, and Neimark-Sacker bifurcations. Some phase portraits are depicted to justify the occurrence of those bifurcations.

Stability, convergence and bifurcation of a reduced Boissonade-De Kepper model with delayed feedback

We analyze the local stability, convergence and bifurcation properties of a reduced Boissonade-De Kepper (BD) model, in the presence of time-delayed feedback. We derive conditions to ensure stability, and also investigate the impact of model parameters on the convergence characteristics of the system. The model undergoes a Hopf bifurcation when the conditions for local stability get violated. Using Poincaré normal forms and the center manifold theory, we are able to derive explicit analytic expressions for determining the type of the Hopf bifurcation and the stability of the emerging limit cycles. We also considered the impact of either having a quadratic or a cubic term in the model. Our findings show that the quadratic term always yields a sub-critical Hopf bifurcation, whereas with a cubic term, the model can switch between a sub-critical and a super-critical bifurcation. The insights exhibit the delicate relationship between the form of the non-linear terms and the resulting system dynamics. Some of the analytical insights are corroborated with numerical computations.

Investigation of the effect of moisture on the critical parameters of stability of frame-rod structural systems made of wood

The work of end moments and shear forces is taken as a criterion influencing the value of the limiting compressive force. It was found that the form of bifurcation of an individual element (active or passive) has a significant effect on the value of the critical parameters of stability of the entire system as a whole. A frame-rod structural system made of wood is considered. The influence of humidity on the type of bifurcation of its compressed elements (constrained or forced) and the critical parameters of stability of the system as a whole under various modes of force loading are determined. It is shown that with a change in humidity, individual elements can go from passive loss of stability to active, which will change the value of the critical force by an average of 20 %. To check the obtained analytical dependencies, experimental tests of frame-bar structures made of wood were carried out under their power static loading and variable humidity. At the same time, it was noted that the results of numerical and laboratory studies showed satisfactory agreement, on average by 95 %, which indicates the reliability of the developed methodology.

Results of surgical treatment of arterial aneurysms of the basilar artery bifurcation

Objective ‒ to analyze the results of surgical treatment of basilar artery (BA) bifurcation arterial aneurysms (AA), taking into account the method of BA obliteration, the type of disease, to summarize the results of surgical treatment of BA.Materials and methods. A retrospective analysis of the results of a comprehensive examination and surgical treatment of 132 patients with AA of BA bifurcation in a group of 387 operated patients with posterior cerebral circulation aneurysms, who were hospitalized and operated in the vascular departments of the State Institution Romodanov Neurosurgery Institute National Academy of Medical Sciences of Ukraine in the period from 1998 to 2019 years is made. Patients were divided into groups according to the type of disease and the method of surgical treatment of AA of BA bifurcation. The analysis of the results of surgical treatment AA of BA bifurcation was performed taking into account the method of occlusion and the clinical type.Results. The balloon occlusion technique was used in 16 (12.1 %) patients with AA of BA bifurcation, including hemorrhagic and other types of disease. Radical exclusion of AA of BA bifurcation was achieved only in 1 (6.25 %) patients of this group. Exclusion of AA of BA bifurcation at level IIR was also recorded in only 1 (6.25 %) patient. In 4 (25.0 %) patients, the level of AA occlusion corresponded to IIIR. In 10 (62.5 %) patients, after surgery using the balloon occlusion technique, AA of BA bifurcation remained not excluded. Satisfactory functional results of treatment using the balloon occlusion technique, according to mRs, were obtained in 7 (43.7 %) patients of this group, without taking into account the radicality of AA of BA bifurcation exclusion. Transcranial technique, taking into account hemorrhagic and other types of flow, was used in 5 (3.8 %) patients with AA of BA bifurcation. In 4 (80.0 %) patients of this group, radical exclusion from the bloodstream was achieved by clipping (occlusion level ‒ IR). In 1 (20.0 %) wrapping of aneurysm walls were strengthened (occlusion level ‒ AA of BA bifurcation is strengthened). Satisfactory functional treatment results in this group were obtained in 3 (60.0 %) observations. Using modern methods of aneurysm coil embolization, 111 (84.0 %) patients with AA of BA bifurcation were operated. Radical exclusion of AA from the bloodstream at the level of IR was achieved in 40 (36.0 %) patients. The level of occlusion IIR was observed in 48 (43.2 %) patients. In 26 (23.4 %) cases, the level of occlusion corresponded to IIIR. In 1 (0.9 %) case, the aneurysm was excluded at level IIID. Satisfactory functional results, according to mRs, in this group were obtained in 89 (80.2 %) observations.Conclusions. The main method of treatment of AA of BA bifurcation, regardless of the type of disease, should be considered endovascular with the use of modern methods of endovascular obliteration of aneurysm. Methods of transcranial clipping of aneurysms are possible in the surgical treatment of AA of BA bifurcation, but it is advisable to use them when it is impossible to perform endovascular intervention with modern methods of obliteration. The balloon occlusion technique is extremely ineffective in terms of radical shutdown and quality of life of operated patients with AA of BA bifurcation and can only be considered as a subject of the historical aspect of endovascular treatment of AA of BA bifurcation.

Analysis of Zero-Hopf Bifurcation in Two Rössler Systems Using Normal Form Theory

In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation.

Bifurcation Stenting Using Bioresorbable Vascular Scaffold in an Adult Male With Radiation-Induced Coronary Artery Disease and Hodgkin Lymphoma: A Case Report

INTRODUCTION: Radiation-induced coronary artery disease is associated with absolute risk between 0.1% and 0.5% per year after mediastinal radiotherapy of Hodgkin disease. Risk increases with time and becomes statistically significant after 10 years. This often involves the left anterior descending (LAD) and right coronary arteries with proximal arterial narrowing involving the coronary ostia. This is the first documented case of radiation-induced coronary artery disease in a 25-year-old Filipino man diagnosed with nodular sclerosing-type Hodgkin lymphoma stage IIA treated with percutaneous revascularization using bioresorbable vascular scaffolds in a bifurcation type of lesion admitted at St Luke’s Medical Center for chest pain. Patient reported progressive worsening of exertional angina 16 months after chemotherapy and external beam radiation therapy in the neck and mediastinum. Treadmill exercise test revealed inducible ischemia in the inferior leads at stage 3, 7 metabolic equivalents. Coronary angiogram showed severe stenosis involving the midsegment of the proximal LAD artery and the ostial-to-proximal segment of the first diagonal branch, a bifurcation lesion, Medina 1,1,1. A percutaneous coronary intervention was successfully done using two bioresorbable vascular scaffolds via T-and-protrude bifurcation stenting technique and achieves a brisk TIMI (Thrombolysis In Myocardial Infarction) 3 flow. Follow-up after 6 months showed a normal treadmill exercise test. However, patient still complains of occasional chest pain and epigastric discomfort. Coronary angiogram was repeated demonstrating an 80% stenosis in the proximal LAD without evidence of in-stent stenosis. This was confirmed by optical coherence tomography with adequate stent apposition to the intima with a note of a large chronic red thrombus proximal to the previously deployed stent; thus, percutaneous coronary intervention using drug-eluting stent was performed. KEYWORDS: radiation-induced coronary artery disease, Hodgkin lymphoma, bifurcation stenting, bioresorbable vascular scaffold, optical coherence tomography

Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution.

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

Local and global analysis of a nonlinear duopoly game with heterogeneous firms

In this paper, we introduce a nonlinear duopoly game whose players are heterogeneous and their inverse demand functions are derived from a more general isoelastic demand. The game is modeled by a discrete time dynamic system whose Nash equilibrium point is unique. The conditions of local stability of Nash point are calculated. It becomes unstable via two types of bifurcations: flip and Neimark–Sacker. Some local and global numerical investigations are performed to show the dynamic behavior of game’s system. We show that the system is noninvertible and belongs to Z_{2}-Z_{0} type. We also show some multistability aspects of the system including basins of attraction and regions known as lobes.

Input-driven bifurcations and information processing capacity in spintronics reservoirs

The authors find two types of input-driven bifurcation in the spin-torque oscillator, and they reveal the information processing capacity as a reservoir computing changes according to these bifurcations.

Order-chaos-order and invariant manifolds in the bounded planar Earth–Moon system

In this work, we investigate the Earth-Moon system, as modeled by the planar circular restricted three-body problem, and relate its dynamical properties to the underlying structure associated with specific invariant manifolds. We consider a range of Jacobi constant values for which the neck around the Lagrangian point $L_1$ is always open but the orbits are bounded due to Hill stability. First, we show that the system displays three different dynamical scenarios in the neighborhood of the Moon: two mixed ones, with regular and chaotic orbits, and an almost entirely chaotic one in between. We then analyze the transitions between these scenarios using the Monodromy matrix theory and determine that they are given by two specific types of bifurcations. After that, we illustrate how the phase space configurations, particularly the shapes of stability regions and stickiness, are intrinsically related to the hyperbolic invariant manifolds of the Lyapunov orbits around $L_1$ and also to the ones of some particular unstable periodic orbits. Lastly, we define transit time in a manner that is useful to depict dynamical trapping and show that the traced geometrical structures are also connected to the transport properties of the system.

Analysis of right‐sided ligamentum teres: The novel anatomical findings and classification

The true anatomy of right-sided ligamentum teres (RSLT) has not been fully explained for a century. This study aimed to clarify the exact anatomy of RSLT. The computed tomography data of 17651 surgical patients were observed and 76 patients with RSLT, were classified into the bilateral ligamentum teres (LT) group (type A) and three RSLT groups, (B) bifurcation type, (C) trifurcation type, and (D) independent posterior branch type. Type A had double LT that connected to both the right and left sides of the umbilical portion (UP). Types B-D had a P3+4 rather than a left UP. Type D was anatomically different from types A-C. Upon comparing types A-C and type D, type D had a significantly smaller volume of segments 3+4 (P<.001), and the UP was more often on the left side. The position of the gallbladder fundus in type D was more commonly observed on the right side of the LT compared with that observed in the other types (P=.007). The change in the volume of segments 3+4 and the extent of the RSLT shift create a false perception that the gallbladder changes the position.

Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient

Abstract

On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation

The slow-fast effect in a vector field with a codimension-two double Hopf bifurcation at the origin is investigated in the paper. To explore the typical evolution of the dynamics, the universal unfolding of the normal form of the vector field is taken into consideration. When the parametric excitation is introduced, the frequency of which is far less than the two natural frequencies, slow-fast behaviors may appear. Regarding the whole parametric excitation term as a slow-varying parameter, the equilibrium branches and the bifurcations of the generalized autonomous fast subsystem can be derived. With the variation of the exciting amplitude, different types of bursting oscillations caused by the coupling of two scales in frequency domain may appear, the mechanism of which are obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the fast subsystem. It is found that, for relatively small exciting amplitude, no bifurcation of the fast subsystem occurs, and the system behaves in quasi-periodic oscillations. With the increase of the exciting amplitude, different types of bifurcations may take place, leading to the single-mode bursting oscillations, which may evolve to mixed-mode bursting oscillations. Symmetric breaking bifurcations result in the transitions between a symmetric attractor and two co-existed asymmetric attractors in the system. Furthermore, for the mixed-mode bursting oscillations, the trajectory may alternate between the single-mode oscillations and the mixed-mode oscillations via bifurcations of the equilibrium branches, causing the synchronization and non-synchronization between different state variables.

Towards a classification of bifurcations in Vlasov equations.

We propose a classification of bifurcations of Vlasov equations, based on the strength of the resonance between the unstable mode and the continuous spectrum on the imaginary axis. We then identify and characterize a new type of generic bifurcation where this resonance is weak, but the unstable mode couples with a stable mode and a Casimir invariant of the system to form a size-3 Jordan block. We derive a three-dimensional reduced noncanonical Hamiltonian system describing this bifurcation. Comparison of the reduced dynamics with direct numerical simulations on a test case gives excellent agreement. We finally discuss the relevance of this bifurcation to specific physical situations.

Hepatic volume profiles in potential living liver donors with anomalous right-sided ligamentum teres.

In living liver donors with rare anatomical anomaly of right-sided ligamentum teres (RSLT), right or left hemiliver procurement is commonly contraindicated. The purpose of this study was to evaluate the hepatic volume profiles in potential donors with RSLT using semi-automated CT volumetry (CTV). Among 5535 potential donor candidates in our institution between April 2003 and May 2019, 23 cases of RSLT (0.4%) were included. Proportional liver volumes were measured using semi-automated CTV and compared with those of manual volumetry and intraoperative graft weights (seven surgical cases). The mean percentage volume of the right posterior section was significantly larger than that of the left hemiliver (38.5 ± 8.4% vs. 23.3 ± 5.7%, P < 0.001). Particularly in independent right lateral type, the mean percentage volume of the right posterior section was about two times larger to that of the left hemiliver (41.5% ± 6.5% vs. 21.9% ± 4.4%, P < 0.001), whereas the volume proportions of these two parts were similar between the two parts in bifurcation and trifurcation types (P = 0.810 and 0.979, respectively). Semi-automated CTV of corresponding whole liver, right posterior section, right anterior section, and left hemiliver showed strong correlations with manual CTV (r = 0.989-0.998; P < 0.001). For the seven surgical cases, the graft weights estimated by semi-automated CTV showed a significant correlation with intraoperative graft weights (r = 0.972; P < 0.001). In independent right lateral type of RSLT, the right posterior section tends to be significantly larger than left hemiliver, and may be an alternative option for graft in potential living liver donors with this rare anatomical anomaly.

Re-evaluation of the Couinaud classification for segmental anatomy of the right liver, with particular attention to the relevance of cranio-caudal boundaries

BackgroundAlthough the Couinaud classification of liver segments has been challenged by several studies, whether the cranio-caudal boundaries can be delineated in the right liver has not yet been assessed. This study scrutinized the third-order branching pattern of the portal vein in the right liver with attention to the validity of cranio-caudal segmentation. MethodsThree-dimensional reconstruction of the portal vein and hepatic vein, using non-contrast-enhanced magnetic resonance imaging was performed in 50 healthy participants. ResultsIn the right paramedian sector, the portal vein ramified into 2 thick P8s (P8vent and P8dor) in all the participants. Additional thick P8s that ran laterally and/or medially (P8lat and/or P8med) were observed in 18 (32%) participants. In contrast, multiple thin P5s, ranging in number from 2 to 6 (median, 4), branched from the right paramedian trunk, the right portal trunk, and/or even from P8s. In the right lateral sector, an arch-like type in which multiple P6s ramified from a single thick P7 was observed in 26 (52%) participants. A bifurcation type composed of a single P7 and a single P6 was observed in 23 (46%) participants, and a trifurcation type was observed in 1 participant. ConclusionNo clear cranio-caudal intersegmental plane could be delineated in the right liver in most of the participants. The resection of a whole Couinaud segment in the right liver should not be regarded as a systematic, anatomic resection from an oncologic viewpoint. In contrast, detailed information on the third-order portal vein ramification pattern is likely to be helpful when performing smaller anatomic resections.

Collapse Behavior of Tensegrity Barrel-Vault Structures Based on Di-Pyramid (DP) Units

In this study, the collapse behavior of a family of tensegrity structures, i.e. di-pyramid (DP) barrel-vaults that can offer promising solutions for civil engineering applications is analyzed. Depending on whether struts’ snap or cables’ rupture dictate the occurrence of overall collapse, two different designs are considered. The effects of geometric parameters, self-stress properties, loading type, boundary conditions and strengthening schemes on the structural behavior are discussed. It is found that the structures with symmetric and ridge loading types undergo bifurcation type instability instead of limit point which is encountered in the cases with asymmetric loading type. Constraint against lateral thrust is beneficial in improving strength and initial stiffness of the studied cases, by as much as 60% and 90%, respectively. In most cases, the rate of strength variation associated with increasing self-stress levels is quite slow, while the slackness load increases by at least 400% being the primary achievements. By using non-uniform self-stress distribution, the initial stiffness of these structures can be increased up to 240%. Increasing the rise-to-span ratio improves the initial stiffness and collapse strength of the structure significantly at the expense of expedition of cables slackness. Significant gains in collapse resistance of these structures under symmetric loading are obtained with strengthened critical struts or cables, depending on which collapse case dominates, but the initial stiffness is generally not influenced by these schemes.

Thermal postbuckling analysis of FG-CNTRC doubly curved panels with elastically restrained edges using Reddy's higher order shear deformation theory

For the first time, postbuckling behavior of thick doubly curved panels made of carbon nanotube reinforced composite (CNTRC), under preexisting external pressure and subjected to uniform temperature rise is analyzed in this paper. Carbon nanotubes (CNTs) are reinforced into matrix through functionally graded (FG) distribution patterns, and effective properties of CNTRC are determined according to extended rule of mixture. Formulations are based on a higher order shear deformation theory including Von Karman-Donnell nonlinearity, initial geometrical imperfection and elasticity of tangential constraints of boundary edges. Analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is used to obtain nonlinear load-deflection relation. Taking into account temperature dependence of material properties, postbuckling temperature-deflection paths are traced through an iteration process. The effects of preexisting external pressure, CNT volume fraction, tangential edge constraints, initial geometrical imperfection and curvature ratios on thermal postbuckling behavior of CNTRC doubly curved panels are analyzed through numerical examples. The study reveals that thermally loaded panels experiences a quasi-bifurcation response due to the presence of preexisting external pressure. For the most part, perfect panels are deflected toward convex side at the onset of undergoing thermal load. Particularly, imperfect panels may exhibit a bifurcation type buckling response when imperfection size satisfy a special condition.

Bifurcations and chaos in nonlinear Lindblad equations

The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the dynamics of the density matrix is of a quantum system results not only from the interaction with an external environment, but also with other quantum systems of the same type, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. As specific examples that display the various types of dynamical behavior, we suggest explicit models based on systems of interacting spins at finite temperature and exposed to amagnetic field that can change in dependence of the magnetization. Due to the interaction between spins, which is treated at mean-field level, the Hamiltonian as well as the transition rates of the Lindblad equation become dependent on the density matrix.