Bifurcation Branches Research Articles

Interior Crisis Route to Extreme Events in a Memristor-Based 3D Jerk System

In dynamical systems, events that deviate significantly from usual or expected behavior are referred to as extreme events. This paper investigates the mechanism of extreme event generation in a 3D jerk system based on a generalized memristive device. In addition, regions of coexisting parallel bifurcation branches are explored as a way of investigating the multistability of the memristive system. The system is examined using bifurcation diagrams, Lyapunov exponents, time series, probability density functions of events, and inter-event intervals. It is found that extreme events occur via a period-doubling route and are due to an interior crisis that manifests itself as a sudden shift from low-amplitude to high-amplitude oscillations. Multistability is also identified when both control parameters and initial values are modified. Finally, an analog circuit based on the memristive jerk system is designed and experimentally realized. To our knowledge, this is the first time that extreme events have been reported in a memristive jerk system in particular and in jerk systems in general.

Bifurcation structure of steady states for a cooperative model with population flux by attractive transition

AbstractThis paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation.

Unilateral global structure for discrete Dirichlet problem with Minkowski-curvature operator

In this paper, we are concerned with the unilateral global bifurcation results for the Minkowski-curvature discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = λa ( t ) u ( t ) + f ( t , u ( t ) , λ ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where λ ∈ R is a parameter, a : [ 1 , T ] Z → ( 0 , ∞ ) , f ∈ C ( [ 1 , T ] Z × R 2 , R ) , Δ is the forward difference operator with Δ u ( t ) = u ( t + 1 ) − u ( t ) and ∇ is the backward difference operator with ∇ u ( t ) = u ( t ) − u ( t − 1 ) , ϕ ( s ) = s 1 − s 2 is a ϕ-Laplacian operator. We shall show that there are two distinct unbounded continua C + and C − , consisting of the bifurcation branch C if f is not necessarily differentiable at the origin with respect to u . As the applications of the above result, we shall investigate the existence and asymptotic behaviours of one-sign solutions for the half-quasilinear discrete problem { − ∇ [ ϕ ( Δu ( t ) ) ] = α ( t ) u + ( t ) + β ( t ) u − ( t ) + μa ( t ) F ( u ( t ) ) , t ∈ [ 1 , T ] Z , u ( 0 ) = u ( T + 1 ) = 0 , where μ ≠ 0 is a parameter, a : [ 1 , T ] Z → ( 0 , + ∞ ) , α , β : [ 1 , T ] Z → R , u + = max { u , 0 } , u − = − min { u , 0 } , F ∈ C ( R , R ) satisfies sF ( s ) > 0 for s ≠ 0 . We will use the obtained unilateral global bifurcation theory and the approximation of connected components to prove main results.

Backward bifurcation arising from decline of immunity against emerging infectious diseases

Decline of immunity is a phenomenon characterized by immunocompromised host and plays a crucial role in the epidemiology of emerging infectious diseases (EIDs) such as COVID-19. In this paper, we propose an age-structured model with vaccination and reinfection of immune individuals. We prove that the disease-free equilibrium of the model undergoes backward and forward transcritical bifurcations at the critical value of the basic reproduction number for different values of parameters. We illustrate the results by numerical computations, and also find that the endemic equilibrium exhibits a saddle–node bifurcation on the extended branch of the forward transcritical bifurcation. These results allow us to understand the interplay between the decline of immunity and EIDs, and are able to provide strategies for mitigating the impact of EIDs on global health.

Complex dynamic behaviors in a small network of three ring coupled Rayleigh-Duffing oscillators: Theoretical study and circuit simulation

This work focuses on the dynamics of a small network of three ring-coupled unidirectional Rayleigh-Duffing oscillators. The equations governing the Rayleigh-Duffing oscillator, containing a cubic term, make this study a more interesting and complex case to analyze. Coupling is achieved by perturbing the amplitude of each oscillator with a signal proportional to the amplitude of the other. The sixth-order self-driven nonlinear system obtained after coupling is analyzed, and presents up to twenty seven equilibrium points. Amongst these equilibrium points, we determined which can present the Hopf bifurcation. Also, the effects of the coupling coefficients and damping coefficients are analyzed. It is shown that varying these different coefficients leads to the appearance of extremely complex dynamic phenomena such as: instability and bifurcations (i.e coexistence of bifurcation branches), coexistence of up to fifteen attractors (heterogeneous multistability) and eight spiral chaotic attractor. The investigation of the coupled system is carried out by using to both analytical and numerical tools such as Hopf bifurcation theorem, the phase portraits, bifurcation diagrams, Lyapunov exponent diagram, frequency spectrum, to name but a few. The Routh-Hurwitz criterion is also used to analyze the stability of equilibrium points. We compute basins of attraction to highlight different zones corresponding to coexisting attractors. The implementation of an analog circuit of coupled Rayleigh-Duffing oscillators has enabled us to confirm the analytical and numerical results.

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

Tactics of Surgical Repair of Single Ventricle with Excessive Pulmonary Blood Flow and Obstruction of the Distal Arch of the Aorta in Newborns

Pulmonary artery stenosis with distal aortic arch reconstruction and coarctation of the aorta in newborns is an effective palliative procedure for single ventricle and high pulmonary hypertension on the way to total cavo-pulmonary anastomosis. The aim. To present a case of complex correction of a single ventricle of the heart with tricuspid valve atresia and high pulmonary hypertension, obstruction of the distal aortic arch and coarctation of the aorta, and a final effective Fontan procedure with a good long-term outcome. Case report. On December 15, 2011, a 9-day-old newborn patient M. was admitted for treatment at the Department of Cardiovascular Surgery of Odesa Regional Children’s Clinical Hospital with a diagnosis of: a single ventricle of the heart, transposition of the great arteries, tricuspid atresia, distal arch hypoplasia, coarctation of the aorta, patent ductus arteriosus, high pulmonary hypertension. The first stage of surgical treatment was carried out on December 19, 2011: main pulmonary artery banding, reconstruction of the distal arch of the aorta through modified Amato technique, closure of the patent ductus arteriosus, resection of the coarctation of the aorta and extended end-to-end aortoplasty. At the age of 1 year (December 27, 2012), the second stage was performed: bidirectional Glenn procedure with plastic surgery of bifurcation and right pulmonary artery branch. The third stage was performed at the age of 3 years 11 months (November 19, 2015): Fontan procedure with extracardiac conduit. Conclusions. Early elimination of pulmonary hypertension by pulmonary artery banding ensures the preservation of the pulmonary vascular bed with low resistance, which is a crucial condition for the effective final hemodynamic correction of the single ventricle of the heart – the Fontan procedure. In case of hypoplasia of the distal aortic arch with coarctation in newborns, one of the alternative approaches is the use of modified Amato technique avoiding artificial circulation.

Geometrically parametrised reduced order models for studying the hysteresis of the Coanda effect in finite element-based incompressible fluid dynamics

This article presents a general reduced order model (ROM) framework for addressing fluid dynamics problems involving time-dependent geometric parametrisations. The framework integrates Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) hyper-reduction techniques to effectively approximate incompressible computational fluid dynamics simulations. To demonstrate the applicability of this framework, we investigate the behaviour of a planar contraction-expansion channel geometry exhibiting bifurcating solutions known as the Coanda effect. By introducing time-dependent deformations to the channel geometry, we observe hysteresis phenomena in the solution.The paper provides a detailed formulation of the framework, including the stabilised finite elements full order model (FOM) and ROM, with a particular focus on the considerations related to geometric parametrisation. Subsequently, we present the results obtained from the simulations, analysing the solution behaviour in a phase space for the fluid velocity at a probe point, considered as the Quantity of Interest (QoI). Through qualitative and quantitative evaluations of the ROMs and hyper-reduced order models (HROMs), we demonstrate their ability to accurately reproduce the complete solution field and the QoI.While HROMs offer significant computational speedup, enabling efficient simulations, they do exhibit some errors, particularly for testing trajectories. However, their value lies in applications where the detection of the Coanda effect holds paramount importance, even if the selected bifurcation branch is incorrect. Alternatively, for more precise results, HROMs with lower speedups can be employed.

Low-Profile Visualized Intraluminal Support Device for Y-Stent-Assisted Coiling of Wide-Neck Intracranial Aneurysms: A Single-Center Experience

BackgroundThe Low-Profile Visualized Intraluminal Support (LVIS) device has been frequently used as intracranial stent for treating intracranial aneurysms. However, the feasibility and efficacy of LVIS devices in Y-stent-assisted coiling (Y-SAC) have remained contentious. This study aimed to evaluate long-term angiographic and clinical outcomes of Y-SAC using LVIS devices. MethodsWe retrospectively reviewed the clinical presentation and angiography data of patients treated with Y-SAC using LVIS stents. The vascular angle geometry between the parent and the two branch vessels, before and after stent deployment and after coiling were analyzed. Based on Raymond-Roy Occlusion Classification (RROC), aneurysm occlusion status was classified. Clinical outcomes were assessed using the modified Rankin Scale (MRS). ResultsForty patients with 40 aneurysms were included in this study. Immediate postprocedural angiograms revealed complete/near-complete occlusion (RROC 1 and 2) in 31 aneurysms (77.5%). The long-term follow-up angiographic studies were available in 32 patients revealing RROC class 1 and 2 in 93.8% of patients. Y-SAC with LVIS devices significantly decreased the angle between the bifurcation branches from 171.90 ° ± 48.0° (SD) to 130.21° ± 46.3° (SD) (p<0.0001). Periprocedural complications occurred in five patients (12.5%) including four in-stent thrombosis (10.5%). Thirty-six patients (90.0%) had favorable clinical outcomes at the final follow-up. Univariate analysis depicted that WFNS of 3-5, thick of subarachnoid hemorrhage on head CT scan, intraprocedural complications, and in-stent thrombosis were predictors of poor outcome. ConclusionY-SAC using LVIS device for intracranial bifurcation aneurysms is a feasible and relatively safe procedure with favorable long-term angiographic and clinical outcomes.

Multiparameter Bifurcation Analysis of Power Systems Integrating Large-Scale Solar Photovoltaic and Wind Farms Power Plants

In this paper, we propose a novel codimension-three-parameter bifurcation analysis of equilibria and limit cycles when integrating Renewable Energy Sources (RESs) power plants with an exponential static load model. The study investigates the effect of solar photovoltaic generation margin, wind power generation margin, and loading factor on the local bifurcation of the modified IEEE nine-bus system. The proposed technique considers the real case of the West System Coordination Council (WSCC), the western states of the USA, by using specific models of RES power plants and static loads. The proposed technique helps to create a set of linearly varying parameters for critical operating points of nonlinear systems. The study explores detailed voltage stability analysis through the examination of bifurcation diagrams. The Hopf, limit-induced, and saddle-node bifurcation branches are identified, defining the parameter space’s stable and unstable operational regions. Additionally, the stability regions surrounding the equilibrium points are diligently explored, clarifying the consequences that various bifurcations may exert on these regions. The study offered in this proposed work aids in determining the best ways to monitor and improve these margins while considering system variables and load design.

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

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.

Stability of Hydroelastic Waves in Deep Water

Two-dimensional periodic travelling hydroelastic waves on water of infinite depth are investigated. A bifurcation branch is tracked that delineates a family of such solutions connecting small amplitude periodic waves to the large amplitude static state for which the wave is at rest and there is no fluid motion. The stability of these periodic waves is then examined using a surface-variable formulation in which a linearised eigenproblem is stated on the basis of Floquet theory and solved numerically. The eigenspectrum is discussed encompassing both superharmonic and subharmonic perturbations. In the former case, the onset of instability via a Tanaka-type collision of eigenvalues at zero is identified. The structure of the eigenvalue spectrum is elucidated as the travelling-wave branch is followed revealing a highly intricate structure.

Heterogeneous multistability and antimonotonicity for a new 3D system with a triple well nonlinearity: theoretical study, control and microcontroller implementation

We propose a new 3D autonomous multistable jerk-like system with a nonlinear term consisting of a six-order triple well function. The presence of six equilibrium points with symmetrical locations along the x-axis represents one of the main distinguishing properties of the new system. Strikingly, the stability analysis of equilibria reveals a cascade of Hopf bifurcations at three specific values of a single control parameter, which results in several forms of complexity. Accordingly, various forms of coexisting attractors such as stable fixed points, limit cycles of diverse periodicities, and chaotic attractors are depicted for some special parameter values. Moreover, It is found that the new jerk-like system with six order triple well polynomial function exhibit extremely complex nonlinear behaviors such as anti-monotone bifurcations, hysteresis and parallel bifurcation branches. These latter aspects explain the presence of multiple (i.e. up to four) coexisting asymmetric attractors for some special rank of parameters. In the presence of multiple competing dynamics, we resort to basins of attraction in order to highlight the how the state space is magnetized. The combination of dynamic features discussed in the new jerk-like system with triple well polynomials nonlinearity introduced in this article is unique and rarely reported. An electronic version of the new system with triple well polynomial nonlinearity is implemented in PSpice. Moreover, a hardware digital implementation of the system is also carried out using an Arduino microcontroller. A very good agreement is captured between PSpice simulation results, the laboratory measurements and the theoretical predictions.

Radial solutions and a local bifurcation result for a singular elliptic problem with Neumann condition

We study the problem − Δ u = λ u − u − 1 with a Neumann boundary condition; the peculiarity being the presence of the singular term − u − 1 . We point out that the minus sign in front of the negative power of u is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution ( u λ , λ ) with u λ approaching the trivial constant solution u = λ − 1 / 2 and λ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions u with λ fixed.

Possibilities of Surgical Interventions in Severe Lower Limb Ischemia in the Stage of Trophic Disorders

Occlusive pathology in the branches of aortic bifurcation or linear remains an urgent task of endovascular surgery in all highly developed countries throughout the world due to atherosclerotic narrowing of the vessels especially in the lower extremities. We have described the possibilities of surgical elimination of critical limb ischemia (CLI) in the stage of trophic disorders through the cohort study, which was conducted in the Grodno university clinic. The hospital and university committee approved the study. Within the last two years 2020-2022 in the hospital a total number of 68 patients were surgically treated for the CLI with trophic disorders. All of them were staged III-IV according to R.Fontine - A.V. Pokrovsky staging and ankle-brachial index (ABI). Aorto-femoral, Infrainguinally femoro-popliteal-tibial revascularization and transplantation of autologous venous allograft for the reconstruction of the lower extremities in the above mentioned patients were carried out. We obtained a very positive results, in 97.1% the patients had a limb with positive dynamics and its functional state was preserved. Only in 2 patients out of 66 which is 2.9%, the limb was amputated due to complications developed and inadequate revascularization. Based on our clinical observation we have proposed that, the use of autogenous venous material, modern original deobliterating reconstructions of arterial vasculature and transplantable allograft structures may be promising in the amelioration of occlusive pathology in severe life threatening lower limb ischemia in the stage of trophic disorders

Short delay limit of the delayed Duffing oscillator.

The delayed Duffing equation, x^{″}+ɛx^{'}+x+x^{3}+cx(t-τ)=0, admits a Hopf bifurcation which becomes singular in the limit ɛ→0 and τ=O(ɛ)→0. To resolve this singularity, we develop an asymptotic theory where x(t-τ) is Taylor expanded in powers of τ. We derive a minimal system of ordinary differential equationsthat captures the Hopf bifurcation branch of the original delay differential equation. An unexpected result of our analysis is the necessity of expanding x(t-τ) up to third order rather than first order. Our work is motivated by laser stability problems exhibiting the same bifurcation problem as the delayed Duffing oscillator [Kovalev etal., Phys. Rev. E 103, 042206 (2021)2470-004510.1103/PhysRevE.103.042206]. Here we substantiate our theory based on the short delay limit by showing the overlap (matching) between our solution and two different asymptotic solutions derived for arbitrary fixed delays.

Theoretical study and circuit implementation of three chain-coupled self-driven Duffing oscillators.

In this paper, we describe the scenario from the birth of oscillations to multi-spiral chaos in a novel system composed of three chain-coupled self-driven Duffing oscillators. Eight of the equilibrium points develop (multiple) Hopf bifurcation when varying a parameter (e.g., coupling coefficient). Considering the computer integration of the state equations, the combined exploitation of Lyapunov exponent plots, bifurcation diagrams, basins of attraction, and phase portraits, unusual and attractive features were highlighted including the coexistence of eight bifurcation branches, Hopf bifurcations, a multitude of coexisting types of oscillations and a six-spiral chaotic attractor, just to cite a few. Using basic electronic components, the electronic circuit of the three chain-coupled Duffing oscillator system is performed. Orcad-PSpice simulated dynamics of the proposed chain-coupled analog circuit confirm the theoretically disclosed features. Moreover, the practical feasibility of the coupled system is demonstrated by considering microcontroller-based hardware realization.

The COBRA II (COmplex Bifurcation lesions: RAndomized comparison of a fully bioresorbable modified-T stenting strategy versus bifurcation reconstruction with a dedicated self-expanding stent in combination with bioresorbable scaffolds) study: Final 5-year follow-up

BackgroundIn order to advocate further research in bioresorbable scaffold (BVS) technology we report the final 5-year outcomes of the COBRA II study, the only randomized controlled trial (RCT) performed to investigate the safety, feasibility, and performance of Absorb BVS (Abbott Vascular) in true coronary bifurcations. MethodsCOBRA II was a prospective single-center RCT. Fifteen patients with true coronary bifurcation lesions were randomized to bifurcation treatment with self-expanding biolimus-eluting Axxess bifurcation device (Biosensors International) combined with additional bioresorbable everolimus-eluting Absorb BVS in the bifurcation branches on (Axxess group) or to 2-stent mod-T stenting technique with Absorb BVS (mod-T group). Optical coherence tomography was performed post-procedure and at 30-months. The primary endpoint was change in minimal luminal area (MLA) on OCT from baseline to 30-months. ResultsFifteen patients with coronary bifurcation lesions were randomized to the Axxess group (n = 8) or Modified-T group (n = 7). At 30 months, MLAs were significantly smaller than post-procedure in the majority of bifurcation segments treated with BVS due to neointima formation, while MLAs in the proximal Axxess segment remained stable (primary endpoint).Five-year clinical follow-up was available for all patients. Only 1 major adverse cardiac event occurred; a patient underwent target lesion revascularization at 30 months in the Axxess group. There were no cases of cardiac death, spontaneous MI, or stent/scaffold thrombosis. ConclusionsIn this small RCT bifurcation study, BVS luminal dimensions were significantly smaller at 30 months, with acute strut discontinuities and late Intraluminal dismantling frequently observed, although acceptable clinical outcomes were noted at 5 years.

Collective dynamics of two coupled Hopfield inertial neurons with different activation functions: theoretical study and microcontroller implementation

This work deals with the regular and chaotic dynamics of a system made up of two Hopfield-type neurons with two different activation functions: the hyperbolic tangent function and the Crespi function. The mathematical model is in the form of an autonomous differential system of order four with odd symmetry. The analysis highlights nine equilibrium points and four of these points experience a Hopf bifurcation at the same critical value of a control parameter which can be either the diss1ipation parameter or one of the coupling coefficients. This makes plausible the presence of four parallel bifurcation branches as well as the coexistence of multiple attractors in the behavior of the system. One of the highlights revealed in this work is the coexistence of three double-scroll type attractors of particular topology as well as the presence of a four-spiral attractor. Furthermore, the coexistence of both self-excited and hidden dynamics is also reported. All this plethora of dynamics is elucidated by making use of the usual tools for analyzing nonlinear systems such as bifurcation diagrams, the maximum of Lyapunov exponent, basins of attractions as well as phase portraits. A physical implementation of the microcontroller-based system is envisaged in order to confirm the plethora of behaviors observed theoretically.