Bifurcation Point Research Articles

Hopf bifurcation control for the traffic flow model considering the tail light effect

The research on traffic congestion control has seen rapid development in recent years. Investigating the bifurcation characteristics of traffic flow and designing control schemes for unstable bifurcation points can offer new methods for alleviating traffic congestion. This paper focuses on studying the bifurcation characteristics and nonlinear control of traffic flow based on the continuous model and the taillight effect. Firstly, the traffic flow model is transformed into a stability model suitable for branching analysis through the use of the traveling wave transform. This transformation facilitates the analysis of stability that reflects unstable traffic characteristics such as congestion. Based on this stability model, the existence condition of Hopf bifurcation is proved and some bifurcation points of the traffic system are identified. Secondly, the congestion and stability mutation behaviors near equilibrium and branching points are studied to understand the formation mechanism of traffic congestion. Finally, control schemes are designed using Chebyshev polynomial approximation and stochastic feedback control to delay or eliminate unstable bifurcation points and relieve traffic congestion. This improved traffic flow model helps explain changes in system stability through bifurcation analysis and identify unstable bifurcation points. It can also effectively manage these points by designing a feedback controller. It is beneficial for controlling sudden changes in traffic system stability behavior and mitigating traffic congestion, with important theoretical significance and practical application value.

Investigating the invariant solutions of (1+1)-dimensional Sawada–Kotera model using Lie symmetries analysis

Here attention is focused on the (1+1)-dimensional Sawada–Kotera (SK) model that is prominent in mathematical physics and engineering to analyze plasmas and coherent systems for communication. Our techniques offer fresh perspectives on the model’s attributes and structure, deepening our comprehension of the underlying dynamics. First, the SK model is reduced to ordinary differential equations by constructing the Lie symmetries and using the associated transformation. Graphs are used to build and display invariant solutions. This strategy has caused the revelation of novel constant solutions that have not been found in the previous works. We offer new understandings of nature and changes observed during the SK derivation by taking advantage of the Lie symmetries powerful tools. Next, the fluctuating layout of proposed framework is examined from several perspectives such as sensitivity and bifurcation analysis. We examined the bifurcation analysis of planar dynamical system by using bifurcation theory. We also include an external periodic perturbation term that breaks regular patterns in the perturbed dynamical system. Graphical structures are provided to display the invariant solutions. The sensitivity of the SK model is determined to be strong after sensitivity analysis under different initial conditions. These results are fascinating, fresh, and conceptually useful for understanding the suggested framework. In mathematics and the applied sciences, forecasting and learning about new technologies are greatly aided by the dynamic aspect of system processing.

Effect of Hopf-Hopf bifurcation on the post-flutter behavior of a three-degree-of-freedom airfoil

The post-flutter dynamics of a three-degree-of-freedom nonlinear airfoil with unsteady aerodynamics are investigated based on the Hopf-Hopf bifurcation theory. Many prior works have relied on Hopf bifurcation theory to predict flutter behavior in airfoil systems. Although this approach facilitates flutter prediction and characterization of post-flutter behavior in numerous scenarios, it may be invalid in specific instances, such as when the Hopf bifurcation of the system degenerates. Therefore, this study focuses on a classical degenerate case of Hopf bifurcation in airfoil systems, specifically the Hopf-Hopf bifurcation. We show that the system undergoes various Hopf-Hopf bifurcations under specific parameter conditions as the center of gravity shifts. The local dynamics near the Hopf-Hopf bifurcation points are represented, including quasiperiodic oscillations on a three-dimensional torus. The airfoil begins to oscillate quasiperiodically after the airflow speed crosses specific Hopf-Hopf bifurcation points. The study also uncovers complex quasiperiodic crises and quasiperiodic hysteresis loops, which have not been reported in previous studies of aeroelastic systems. Then, many singularities and bifurcation curves are obtained near the Hopf-Hopf bifurcation point by semiglobal unfolding. Furthermore, the influence of stall effects on the bifurcation structure of the system is represented. It is shown that the types of Hopf-Hopf bifurcations may vary with the changes of stall effects, influencing the system's semiglobal bifurcation structures consequently. For all Hopf-Hopf bifurcation scenarios, stall effects affect one of the Neimark-Sacker bifurcation curve structures unfolded from the Hopf-Hopf bifurcation point significantly, while the other Neimark-Sacker bifurcation curve experiences minimal impact from stall effects. Moreover, a large nonlinear stall coefficient will postpone the onset of quasiperiodic/chaotic oscillations.

Enhancing capture efficiency of drug-carrier particles in the carotid sinus by utilizing a combination of magnetic fields: A numerical approach

Magnetic drug targeting is a relatively new method to treat vascular occlusion in different body parts. However, the effectiveness of this method can be affected due to the severity and location of the occlusion. This can lead to the injection of high dosages of drugs, which can cause serious side effects due to the deposition of drugs in unwanted parts. To mitigate these effects, this study investigates the potential of a guiding magnetic field in enhancing drug absorption for vascular occlusion treatment. The method relies on guiding magnetic nanoparticles (NPs) loaded with drugs toward the occlusion site using two external magnetic fields. Blood flow was modeled as non-Newtonian, considering shear-rate-dependent viscosity and unsteady at the inlet. To test this idea, a computational fluid dynamic (CFD) coupled with a discrete phase model (DPM) approach has been employed to simulate drug delivery in three-vessel structures with varying degrees of occlusion (45 %, 60 %, and 90 %). To avoid the escape of drug carriers, a secondary magnetic field was applied at the bifurcation point to direct the NPs to the site of blockage where the primary magnetic field acts. Then, the states with or without a guiding source at the bifurcation site are compared based on the capture efficiency of each structure. The simulation demonstrated a significant increase in NP capture at the target site, ranging from 2 % to 15 %, depending on the NP size. However, the severity of occlusion substantially impacted the secondary magnetic field's effectiveness. In the 90 % occlusion scenario, the method's efficiency decreased significantly from 26 % to 16 % for NP sizes exceeding 1.5μm. This study highlights the potential of guiding magnetic fields in improving drug delivery to target sites in vascular occlusion.

Long-living periodic solutions of complex cubic-quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering: A bifurcation and numerical study.

We have found long-living periodic solutions of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) perturbed with intrapulse Raman scattering. To achieve this we have applied a model system of ordinary differential equations (SODE). A set of the fixed points of the system has been described. A complete phase portrait as well as phase portraits near the fixed points have been built for a proper choice of parameters. The behavior of the model system near the fixed points has been determined. We have presented a detailed description of the subcritical Poincaré-Andronov-Hopf bifurcation due to the intrapulse Raman scattering that appears at one of the fixed points. We have established that there appears an unstable limit cycle in the SODE. To check the validity of the obtained results from the model system we have compared them with the results of the numerical solution of the CCQGLE perturbed with intrapulse Raman scattering. There has been found a remarkable correspondence between the obtained numerical results for the amplitude and frequency of the soliton pulses and the results for these parameters of the bifurcation theory. We have observed that the numerical characteristics of the propagating solitonlike pulses-amplitude, frequency, width, and position-periodically change if we change the distance with a period determined by the bifurcation analysis.

Local bifurcation structure and stability of the mean curvature equation in the static spacetime

We consider the curvature equation in the static spacetime, $$ \text{div} \Big(\frac{f(x)\nabla u}{\sqrt{1-f^2(x)| \nabla u|^2}}\Big) +\frac{\nabla u \nabla f(x)}{\sqrt{1-f^2(x)| \nabla u|^2}}=\lambda NH \quad\text{in }\Omega, $$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq 1\); the function \(H\) gives the mean curvature. We investigate the local bifurcation structure and stability of the solutions to this equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/48/abstr.html

BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS OF FRACTIONAL DATE–JIMBO–KASHIWARA–MIWA EQUATION

Based on the bifurcation theory, the exact solutions of fractional Date–Jimbo–Kashiwara–Miwa equation are constructed. In terms of [Formula: see text]-derivative, a nonlinear integer-order ODE is obtained, the equation is transformed into a plane dynamical system. The phase portraits are obtained under different bifurcation conditions. According to the orbits of the phase portraits, new bright and dark solitary solutions, periodic solutions, periodic-singular solutions, and singular solutions are established, enriching the diversity of solutions. In addition, the dynamical behaviors of various types of solutions are shown by selecting appropriate parameters, which is useful for understanding the fluctuation behavior in physical models. Compared with previous literature, this paper focuses on the combination of qualitative analyses and qualitative calculations, which makes the paper more systematic and comprehensive, and fills the gap of using bifurcation theory to solve the FDJKM equation. The dynamical behavior of new solutions obtained will provide more and more attractive and complex physical phenomena to explore more mysteries.

Islands in the stream: Wood‐induced deposition and erosion in the river corridor

AbstractLarge wood causes and responds to deposition and erosion within a river corridor. We focus on the anastomosing, gravel‐bed Swan River and two meandering, gravel‐bed tributaries in northwestern Montana, USA to explore the temporal dimensions of deposition and erosion associated with channel avulsions and island formation and to introduce the concept of wood levees. Channel avulsion represents isolation of part of the existing floodplain and formation of an anastomosing channel planform, with wood‐induced deposition at the point of channel bifurcation. Islands form at a wood jam that migrates upstream with time as sediment accumulates in the lee of the jam. The island creates only a local interruption of the single‐channel planform. We use tree‐ring and 14C dating to constrain wood‐induced island ages. We interpret the three wood‐induced forms of deposition and erosion that we describe here as reflecting a temporal continuum. Wood levees have primarily non‐woody vegetation and may be transient relative to the other features. Tributary islands appear to persist from a decade to over a century. Tree ages of 100–200 years at the floodplain avulsion site and the characteristics of the secondary channels suggest that these wood‐induced avulsion features can persist for more than a century. Understanding the temporal dynamics of wood‐induced features and spatial variation in erosion and deposition provides insight into the dynamics and spatial heterogeneity of natural river corridors, with implications for river restoration.

Positive steady states in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1)−d1Δu=λu−u2−buv,x∈Ω,−div[d(w)∇v+χ(w)v∇w]=μv−v2−cuv,x∈Ω,−d3Δw=−κw+τu,x∈Ω,u=v=w=0,x∈∂Ωin a bounded smooth domain Ω⊂RN(N≥1), where λ,μ,b,c,d1,d3,κ,τ are positive constants, and (d(w),χ(w))∈[C1[0,∞)]2 with d(w),χ(w)>0 for all w≥0. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with d(w),α(w) being arbitrary functions in w, this makes the analysis of system (0.1) much more difficult. By the W2,p-estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form (u,0,w) with u,w>0 and the semitrivial solution of the form (0,v,0) with v>0, which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states.

Extracting and analyzing the governing model for plastic deformation of metallic glasses

This paper first detects hidden system from plastic deformation of metallic glasses by sparse identification. The extracted model simulates four types of stress-time curves and displays the prediction of serrated events. This interpretation effectively explains various experimental phenomena of repeated yielding. Further, in terms of parametric sensitivity analysis to the model, two parameters are taken as bifurcation parameter, and the analysis of codimension-one and codimension-two bifurcation are carried out to excavate the causes of dynamic transformation, including saddle–node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and cusp bifurcation. Different bifurcation points correspond different types of stress-time curves. The homologous phase diagrams including periodic orbit, unstable orbit and chaotic behavior are presented to show the dynamics diversity of the model. In addition to dynamic analysis, statistical analysis for plasticity values is also applied to excavate the crossover between periodic and chaotic plastic dynamic transitions. Our results provide a novel perspective on the deformation of metallic glasses from the viewpoint of dynamic model and are also important for evaluating the plastic deformation properties of metallic glasses in practical applications.

Effective detection of early warning signal with power spectrum in climate change system

The development of early warning signals (EWS) before abrupt changes can help prevent system collapse. Current EWS, such as increasing autocorrelation (AC) and variance, provide general indicators of impending tipping points by detecting the slowing down of dynamics near transitions. However, these conventional EWS often fail to distinguish between oscillatory behavior (e.g., Hopf bifurcation) and shifts to a distant attractor (e.g., Fold bifurcation). Additionally, traditional EWS are less reliable in systems affected by density-dependent noise. To address these limitations, alternative EWS based on power spectrum analysis, known as spectral EWS, have been proposed. In this study, we apply analytical approximations for EWS as systems approach different types of local bifurcations. This novel method allows us to use spectral EWS that offer enhanced sensitivity to approaching transitions and increased robustness to density-dependent noise. We demonstrate the application of spectral EWS as robust indicators across three general models with different bifurcations. Our analysis also reveals distinct signals preceding transitions in data from sea ice loss. This combined approach underscores the advantages of incorporating spectral EWS into existing methodologies, providing a more comprehensive toolkit for anticipating critical transitions in real-world systems.

Instability, bifurcation and nonlinear dynamics of Poiseuille flow in fluid overlying an anisotropic and inhomogeneous porous domain

The present study focuses on the finite amplitude analysis of Poiseuille flow in an anisotropic and inhomogeneous porous domain that underlies a fluid domain. The nonlinear interactions are studied by imposing finite amplitude disturbances to the Poiseuille flow. The former interactions in terms of modal amplitudes dictate the fundamental mode, the distorted mean flow, the second harmonic and the distorted fundamental mode. The harmonics are solved progressively in increasing order of the least stable mode obtained from the linear theory to ascertain the cubic Landau equation, which in turn helps to determine the bifurcation phenomena. The presented weakly nonlinear theory predicts the existence of subcritical transition to turbulence of Poiseuille flow in such superposed systems. In general, on moving away from the bifurcation point, it is found that a decrease in the value of inhomogeneity (in terms of Ai), Darcy number (δ) and an increase in the value of depth ratio (dˆ; the ratio of fluid domain thickness to that of porous domain) favours subcritical bifurcation. For the considered variation of parameters, the bifurcation, either subcritical or supercritical, remains the same irrespective of the value of media anisotropy (ξ) in the vicinity of the bifurcation point except for dˆ=0.2,Ai=1. In such a situation, subcritical (supercritical) bifurcation is witnessed for ξ=0.001,0.01,0.1 (1,3). Furthermore, in contrast to isotropic and homogeneous porous media, both subcritical and supercritical bifurcations are observed when moving away from the bifurcation point. A correspondence between the type of mode via linear theory and the type of bifurcation via nonlinear theory is witnessed, which is further affirmed by the secondary flow patterns. Finally, the presented theoretical results reveal an early onset of subcritical transition to turbulence in comparison with isotropic and homogeneous porous media.

Complex dynamical properties and chaos control for a discrete modified Leslie-Gower prey-predator system with Holling II functional response

In this study, the semi-discretization technique is employed to establish a discrete representation of a modified Leslie-Gower prey-predator system that includes a Holling II type functional response. The dynamics of this model are then analyzed through the application of center manifold theory and bifurcation theory. We present comprehensive results for the local stability of the fixed points across the entire parameter space. Additionally, we provide sufficient conditions for the occurrence of flip bifurcation and Neimark-Sacker bifurcation. Besides, the system has experienced a flip bifurcation to chaos controlled using the method of chaos control, viz., state feedback method, pole placement technique, and hybrid control strategy. Furthermore, we provide specific conditions to ensure that bifurcation and chaos can be stabilized. Finally, numerical simulations are conducted to validate theoretical analysis and illustrate several new complex dynamical behaviors between two species.

Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model

Ocean circulation plays an important role in the formation and occurrence of extreme climate events. The study shows that the periodic variation of ocean circulation under strong wind stress is closely related to climate oscillation. Ocean circulation is a nonlinear dynamic system, which shows complex nonlinear characteristics, so the essence behind ocean circulation has not been clearly explained. Therefore, the response and evolution of the circulation system to wind stress are studied based on the bifurcation and catastrophe theories in nonlinear dynamics. First, the quasi-geostrophic gyre equation and the normalized gravity model are introduced and developed to study ocean circulation driven by wind stress, and solved using the Galerkin method. Then, the bifurcation and catastrophe behaviors of the system governed by the low-order ocean circulation model during the change in wind stress intensity and the coexistence of multiple equilibria in the circulation system are studied in detail. The results show that saddle and unstable nodes appear in the system after a cusp catastrophe. With the change in model parameters, the unstable node becomes the unstable focus, and then the subcritical Hopf bifurcation occurs. The system forms a bistable interval when the system undergoes a catastrophe twice, and the system shows hysteresis. In addition, multiple equilibrium states are coexisting in the circulating system, and the unstable equilibrium state always changes into a stable equilibrium state through vortex movement. Therefore, there is a route for the system to induce short-term climate oscillation, that is, in the multi-stable equilibrium state of the system, the vortex oscillates after being subject to small disturbances, and then triggers climate oscillation.

Topographic analysis of retinal and choroidal vascular displacements after macular hole surgery

It has been reported that the retinal vessel and macular region of the retina are displaced after macular hole (MH) surgery. However, there is no detailed information for correlations between retinal and choroidal displacements. We obtained optical coherence tomography angiography (OCTA) and en-face optical coherence tomography (OCT) images from 24 eyes to measure the retinal and choroidal vascular displacement before and after surgery. These images were merged into infrared images using blood vessel patterns. The same vascular bifurcation points were automatically selected for each follow-up image, and the displacements of the bifurcation points were analyzed as a vector unit for prespecified grid regions in a semi-automated fashion. The results showed displacements of the choroidal intermediate vessels and retinal vessels following MH surgery (p = 0.002, p < 0.001). The topographic changes showed inferior, nasal, and centripetal displacement of the retina and inferiorly displaced choroid. The ILM peeling size and basal MH size were significantly associated with the retinal displacement (p < 0.001 and p = 0.010). Additionally, changes in the amount of the choroidal displacement were significantly correlated with that of the retinal displacements (p = 0.002). Clinicians should keep in mind that there might be topographic discrepancies of the displacement between retina and choroid when analyzing them following surgery.

Pulse vaccination in a SIR model: Global dynamics, bifurcations and seasonality

We analyze a periodically-forced dynamical system inspired by the SIR model with impulsive vaccination. We fully characterize its dynamics according to the proportion p of vaccinated individuals and the time T between doses. If the basic reproduction number is less than 1 (i.eRp<1), then we obtain precise conditions for the existence and global stability of a disease-free T-periodic solution. Otherwise, if Rp>1, then a globally stable T-periodic solution emerges with positive coordinates.We draw a bifurcation diagram (T,p) and we describe the associated bifurcations. We also find analytically and numerically chaotic dynamics by adding seasonality to the disease transmission rate. In a realistic context, low vaccination coverage and intense seasonality may result in unpredictable dynamics. Previous experiments have suggested chaos in periodically-forced biological impulsive models, but no analytic proof has been given.

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.

Maternal serum PlGF associates with 3D power doppler ultrasound markers of utero-placental vascular development in the first trimester: the rotterdam periconception cohort.

Circulating angiogenic factors are used for prediction of placenta-related complications, but their associations with first-trimester placental development is unknown. This study investigates associations between maternal angiogenic factors and utero-placental vascular volume (uPVV) and utero-placental vascular skeleton (uPVS) as novel imaging markers of volumetric and morphologic (branching) development of the first-trimester utero-placental vasculature. In 185 ongoing pregnancies from the VIRTUAL Placenta study, a subcohort of the ongoing prospective Rotterdam Periconception cohort, three-dimensional power Doppler ultrasounds of the placenta were obtained at 7-9-11weeks gestational age (GA). The uPVV was measured as a parameter of volumetric development and reported the vascular quantity in cm3. The uPVS was generated as a parameter of morphologic (branching) development and reported the number of end-, bifurcation- crossing-or vessel points and total vascular length. At 11weeks GA, maternal serum biomarkers suggested to reflect placental (vascular) development were assessed: placental growth factor (PlGF), soluble fms-like tyrosine kinase-1 (sFlt-1) and soluble endoglin (sEng). sFlt-1/PlGF and sEng/PlGF ratios were calculated. Multivariable linear regression with adjustments was used to estimate associations between serum biomarkers and uPVV and uPVS trajectories. Serum PlGF was positively associated with uPVV and uPVS development (uPVV: β = 0.39, 95% CI = 0.15;0.64; bifurcation points: β = 4.64, 95% CI = 0.04;9.25; crossing points: β = 4.01, 95% CI = 0.65;7.37; total vascular length: β = 13.33, 95% CI = 3.09;23.58, all p-values < 0.05). sEng/PlGF ratio was negatively associated with uPVV and uPVS development. We observed no associations between sFlt-1, sEng or sFlt-1/PlGF ratio and uPVV and uPVS development. Higher first-trimester maternal serum PlGF concentration is associated with increased first-trimester utero-placental vascular development as reflected by uPVV and uPVS. Clinical trial registration number Dutch Trial Register NTR6854.

Hopf-Hopf bifurcation, period [formula omitted] solutions, slow-fast phenomena, and chimera of an optoelectronic reservoir computing system with single delayed feedback loop

In this paper, we investigate the co-dimension two bifurcations and complicated dynamics of an optoelectronic reservoir computing (RC) system with single delayed feedback loop. We focuses primarily on its underlying system’s Hopf-Hopf bifurcation. Firstly, we apply DDE-BIFTOOL built in Matlab to sketch the bifurcation diagrams with respect to two bifurcation parameters, namely feedback strength β and time delay τ, and find the existence of the Hopf-Hopf bifurcation points. Then, using the multiple scales method, we obtain their normal forms, and using the normal form method, we unfold and classify their local dynamics. Then numerical simulations are conducted to verify these results. We discover rich dynamical behaviors of the system in specific regions. Besides, other complicated dynamics, such as fast-slow phenomena, Period n solutions, and chimera, are found in the system. All these rich dynamical phenomena can provide excellent performance potentially for this optoelectronic reservoir computing system with single delayed feedback loop.

Quantum theory of polariton weak lasing and polarization bifurcations

Quantum theory of polariton weak lasing and polarization bifurcations