Bifurcation Point Research Articles

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

Synergistic Aspects Of The Transformation Of Energy Security Management Mechanisms In Conditions Of Geopolitical Instability

The article systematizes the basic provisions of the concept of energy security, in which the emphasis is placed on uninterrupted access to energy sources, their diversification, environmental friendliness, and ensuring resistance to external threats. The content of the concept of «energy security», which acts as a system-forming element of national (economic, ecological, technical) security, orienting this system to the goals of sustainable energy development thanks to the effective use of fuel and energy resources in the context of ensuring economic growth, social progress, and environmental well-being, has been studied. It is proved that for the interpretation of the logic of energy development, it is advisable to use the system-synergistic approach, which explains the transformation of a complex system by passing bifurcation points, the formation of order attractors, and the emergence of self-organization mechanisms. Among the factors that determine the level of energy security, the geopolitical aspect associated with a wide variety of fluctuating influences that can bring the system out of equilibrium is highlighted in the first place. Any imbalance in the energy system directly affects the economic and social spheres, the state of the environment, as well as the international competitiveness of the state. Therefore, special attention is focused on the development of effective institutions, their consolidation in the daily activities of business entities that are part of the fuel and energy complex, and timely adaptation to changes in the operating environment. For the formation of attractors of sustainable energy development, it is necessary to apply effective management mechanisms that combine institutional and market components. Energy security is defined as a management object in which certain functional units, regulatory mechanisms, and algorithms for making and implementing management decisions are distinguished. The trends considered in the course of the study confirm the high energy interdependence of states in the conditions of globalization and, accordingly, the need to develop international energy security management mechanisms. As a result of combining the principles of flexibility, adaptability to modern geopolitical trends, requirements for socio-economic efficiency and environmental standards, on the one hand, and the characteristics of resistance, stability (to ensure national stability), on the other hand, a special regime of sustainable energy development at the level of individual states, integration associations and in the global dimension.

Stability Analysis Method of a Hierarchical Control Structure Microgrid Based on a Small-Signal Model and Hopf Bifurcation Theory

To the oscillation and stability problem caused by multi-scale and broadband electromagnetic dynamics among many isomerized power electronic devices in a microgrid, a small-signal model of hierarchical control structure microgrid and stability analysis method based on oscillation trajectories are proposed. Moreover, a hierarchical control structure is used as the research object in microgrid, and the analysis is performed based on the small-signal model and Hopf bifurcation theory. First, the small-signal model of the hierarchical control structure microgrid is established, combining the dominant eigenvalue and participation factor analysis methods, to analyze the influence of the controller and related sensitive parameters on the dynamic performance of the system. Then, based on the small-signal model and Hopf bifurcation theory, a stability analysis method based on oscillation trajectories is proposed. The relationship between different oscillation modes and limit cycles in the microgrid system and the influence of the sag control parameter values on the oscillation trajectory and stability domain of the system are researched on. Combining Hopf bifurcation theory and oscillation trajectory to delineate the stable domain of parameter trajectories, the influence of secondary control on the system stability is analyzed under different oscillation trajectories and load disturbances, revealing the corresponding relationship between the state trajectories of different oscillation modes and system stability of a hierarchical control structure microgrid. Based on the proposed oscillating trajectories, the stability analysis method has sufficient universality for studying the impact of system stability, establishing the parameter selection standards for the design of microgrids. Finally, according to the verification results, the correctness and applicability of above methods are verified.

Early warning signals for bifurcations embedded in high dimensions

Recent work has highlighted the utility of methods for early warning signal detection in dynamic systems approaching critical tipping thresholds. Often these tipping points resemble local bifurcations, whose low dimensional dynamics can play out on a manifold embedded in a much higher dimensional state space. In many cases of practical relevance, the form of this embedding is poorly understood or entirely unknown. This paper explores how measurement of the critical phenomena that generically precede such bifurcations can be used to make inferences about some properties of their embeddings, and, conversely, how prior knowledge about the mechanism of bifurcation can robustify predictions of an oncoming tipping event. These modes of analysis are first demonstrated on a simple fluid flow system undergoing a Hopf bifurcation. The same approach is then applied to data associated with the West African monsoon shift, with results corroborated by existing models of the same system. This example highlights the effectiveness of the methodology even when applied to complex climate data, and demonstrates how a well-resolved spatial structure associated with the onset of atmospheric instability can be inferred purely from time series measurements.

Characteristics analysis of flame propagation and its coupling effect with other parameters in LPG pipeline

AbstractTo study the flame propagating characteristic and its coupling effect with other parameters in the LPG pipeline, a typical pipeline model with two equal‐length branches perpendicular to each other is designed for experiment and simulation. Then, gas explosion scenarios are experimentally tested and numerically simulated, which is followed by the analysis of flame shape changing with time and peak temperature changing with space. Results show that when passing through the bifurcation, flame propagates to vertical branch B in a sharp knife shape affected by the strong vortex, reflected airflow, and compressed pressure wave in the pipeline with a diameter of 0.125 m. At the monitoring point that is 0.4 m away from the bifurcation point, the peak temperature of the vertical branch B is 57.87% bigger than that of the horizontal branch C, and its arrival time is 80% longer than that of the horizontal branch C, due to the existence of flame in vertical branch B. What's more, in both branches, the coupling results between peak temperature and peak velocity agree very well with the growth function, while the coupling results between peak temperature and peak pressure agree well with the decay function, providing aids to the optimal layout design of industrial pipeline branches as well as to the explosion suppression measures.

Analyzing the Kuralay-II equation: bifurcation, chaos, and sensitivity insights through conformable derivative and Jacobi elliptic function expansion

This study explores the intricate dynamics of the Kuralay-II equation by employing the conformable derivative. Using the Galilean transformation, we can establish a dynamical system related to the equation. We investigate bifurcation methods in this derived system using planar dynamical systems theory. By introducing a perturbed term, we thoroughly investigate the possibility of chaotic behaviors in the Kuralay-II equation using comprehensive two-phase portraiture. Through careful analysis, we have determined that even small changes in the initial conditions have little impact on the stability of the solution, which has been confirmed by employing the Runge–Kutta method. In addition, we obtain exact solutions for the Kuralay-II equation by the Jacobi elliptic function expansion method. Graphical results of some solutions are showcased, offering a comprehensive evaluation using MATLAB across various dimensions. This study has yielded significant findings, such as the discovery of bifurcation points, the determination of conditions for chaos, and the analysis of stability under perturbations. These results have enhanced our understanding of the behavior of the Kuralay-II equation.

Some Unfamiliar Structural Stability Aspects of Unsymmetric Laminated Composite Plates.

It is widely recognized that certain structures, when subjected to static compression, may exhibit a bifurcation point, leading to the potential occurrence of a secondary equilibrium path. Also, there is a tendency of deflection increment without a bifurcation point to occur for imperfect structures. In this paper, some relatively unknown phenomena are investigated. First, it is demonstrated that in some conditions, the linear buckling mode shape may differ from the result of geometrically nonlinear analysis. Second, a mode jumping phenomenon is described as a transition from a secondary equilibrium path to an obscure one as a tertiary equilibrium path or a second bifurcation point. In this regard, some non-square plates with unsymmetric layer arrangements (in the presence of extension-bending coupling) are subjected to a uniaxial in-plane compression. By considering the geometrically linear and nonlinear problems, the bucking modes and post-buckling behaviors, e.g., the out-of-plane displacement of the plate versus the load, are obtained by ANSYS 2023 R1 software. Through a parametric analysis, the possibility of these phenomena is investigated in detail.

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

BIFURCATION ANALYSIS AND CHAOS CONTROL OF THE DISSOLVED OXYGEN-PHYTOPLANKTON DYNAMICAL MODEL

The production of oxygen through phytoplankton photosynthesis is a crucial phenomenon in the dynamics of marine ecosystems. A generic oxygen-phytoplankton interaction model is considered to comprehend its underlying mechanism. This paper investigates the discrete-time dynamics of oxygen and phytoplankton in aquatic ecosystems, incorporating factors that cause phytoplankton mortality due to external influences. We explore the conditions for the local stability of steady states concerning the oxygen content in dissolved water and phytoplankton density. The analysis reveals that the model undergoes a co-dimension one bifurcation, encompassing flip and Neimark–Sacker bifurcations, utilizing the center manifold theorem and bifurcation theory. To manage the chaos resulting from the Neimark–Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we present numerical simulations to validate the theoretical discussion.

Dynamics and bifurcations in genetic circuits with fibration symmetries.

Circuit building blocks of gene regulatory networks (GRN) have been identified through the fibration symmetries of the underlying biological graph. Here, we analyse analytically six of these circuits that occur as functional and synchronous building blocks in these networks. Of these, the lock-on, toggle switch, Smolen oscillator, feed-forward fibre and Fibonacci fibre circuits occur in living organisms, notably Escherichia coli; the sixth, the repressilator, is a synthetic GRN. We consider synchronous steady states determined by a fibration symmetry (or balanced colouring) and determine analytic conditions for local bifurcation from such states, which can in principle be either steady-state or Hopf bifurcations. We identify conditions that characterize the first bifurcation, the only one that can be stable near the bifurcation point. We model the state of each gene in terms of two variables: mRNA and protein concentration. We consider all possible 'admissible' models-those compatible with the network structure-and then specialize these general results to simple models based on Hill functions and linear degradation. The results systematically classify using graph symmetries the complexity and dynamics of these circuits, which are relevant to understand the functionality of natural and synthetic cells.

Torsional behaviors of motor rotor of permanent magnet semi-direct drive system for aerial passenger device

Considering the influence of motor shaft torsional damping, the torsional dynamics equations of a permanent magnet semi-direct drive rotor system in an aerial passenger device are derived. The derivation relies on the coupling between the mathematical model and the gear dynamics model of the permanent magnet motor, and the Lagrange–Maxwell equations are used for streamlined analysis. To protect the motor under extreme working conditions, the influence of supercritical Hopf bifurcation caused by different torsional dampings of the motor shaft on the torsional vibration of the motor rotor is investigated. Based on the Hurwitz criterion, the influence of the linear control gain and nonlinear gain of the washout filter on the system bifurcation point and the limit cycle amplitude of the system is analyzed. Then, from the perspective of the system instability and oscillation caused by friction damping change of the drive motor and the complex electromechanical coupling behavior in the permanent magnet semi-direct drive cutting drive system of the aerial passenger device, the system stability domain is defined. The results can be applied to effectively suppress the torsional vibration of the shaft system in the permanent magnet semi-direct drive system of the aerial passenger device, providing a theoretical guidance for stable operation of the system.

Partial tipping in bistable ecological systems under periodic environmental variability.

Periodic environmental variability is a common source affecting ecosystems and regulating their dynamics. This paper investigates the effects of periodic variation in species growth rate on the population dynamics of three bistable ecological systems. The first is a one-dimensional insect population model with coexisting outbreak and refuge equilibrium states, the second one describes two-species predator-prey interactions with extinction and coexistence states, and the third one is a three-species food chain model where chaotic and limit cycle states may coexist. We demonstrate with numerical simulations that a periodic variation in species growth rate may cause switching between two coexisting attractors without crossing any bifurcation point. Such a switchover occurs only for a specific initial population density close to the basin boundary, leading to partial tipping if the frozen system is non-chaotic. Partial tipping may also occur for some initial points far from the basin boundary if the frozen system is chaotic. Interestingly, the probability of tipping shows a frequency response with a maximum for a specific frequency of periodic forcing, as noticed for equilibrium and non-equilibrium limit cycle systems. The findings suggest that unexpected outbreaks or abrupt declines in population density may occur due to time-dependent variations in species growth parameters. Depending on the selective frequency of the periodic environmental variation, this may lead to species extinction or help the species to survive.

Coupling dynamics of locally active memristor based neurons.

Brain-like dynamics require third-order or higher-order complexity. In order to investigate the coupling neuromorphic behaviors of identical third-order memristive neurons, this paper begins with the aim of exploring two identical neuron based dynamics under distinct operating regimes and coupling strengths. Without coupling, the single neuron can exhibit resting states, periodic spikes, or chaos depending on the bias condition. The uncoupled resting neurons can be activated by resistive coupling, inducing inhomogeneous resting states (static Smale paradox) and inhomogeneous spikes (dynamic Smale paradox) due to the edge of chaos regime. Considering the single neuron at the periodic spikes or chaotic states, the coupled neurons can mimic shocking oscillation death, non-periodic asynchronization, and periodic synchronization via the Hopf bifurcation theory. From the above analyses, an artificial ring neural network is constructed using 100 memristive neurons and resistive synapses to further study the coupled mechanism, generating exotic spatiotemporal patterns such as chimera death, amplitude chimera, solitary states, and asynchronization because of symmetry breaking. This sheds new light on exploring exotic spatiotemporal patterns of networks based on memristive neurons from the perspective of the nonlinear circuit theory.

Complex Dynamics of a Discretized Predator–Prey System with Prey Refuge Using a Piecewise Constant Argument Method

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system’s bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system.

Birth of Catastrophe and Strange Attractors through Generalized Hopf Bifurcations in Covid-19 Transmission Mathematical Model

Coronavirus can be transmitted through the things that people carry or the things where it sticks to after being spread by the sufferer. Instead, various preventive measures have been carried out. We create a new mathematical model that represents Coronavirus that exists in non-living objects, susceptible, and infected subpopulations interaction by considering the Coronavirus transmission through non-living objects caused by susceptible and infected subpopulations along with its prevention to characterize the dynamics of Coronavirus transmission in the population under those conditions. One disease-free and two infection equilibrium points along with their local stability and coexistence are identified. Global stability of the disease-free equilibria and basic reproduction number are also investigated. Changes in susceptible-Coronavirus interaction rate generate Fold and Hopf bifurcations which represent the emergence of a cycle and the collision of two infection equilibrium points respectively. Catastrophe generated by the collision between an attractor and a repeller is found around a Generalized Hopf bifurcation point by changing susceptible-Coronavirus interaction rate and increasing rate of Coronavirus originating from infected subpopulation. It represents a momentary unpredictable dynamics as the effect of Coronavirus addition and infection. Non-chaotic strange attractors that represent complex but still predictable dynamics are also triggered by Generalized Hopf bifurcation when the susceptible-Coronavirus interaction rate and one of the following parameters, i.e. increasing rate of Coronavirus originating from infected subpopulation or infected subpopulation recovery rate vary.

A light-fueled self-rolling unicycle with a liquid crystal elastomer rod engine

Active materials hold substantial potential for self-sustaining motion, however, the dependency on movable light or extensive area illumination in current structures often restricts the design and practical deployment of such active machines. In this study, we report a novel zero-energy-mode, self-rolling unicycle equipped with a liquid crystal elastomer rod engine, which fueled by a fixed, small-area light source. By employing an elaborate dynamic model for liquid crystal elastomer, we derive the lateral curvature of the liquid crystal elastomer rod and the driving moment necessary for the unicycle's self-rolling. The study results demonstrate that the self-rolling of the unicycle is driven by a moment induced by the shift in the center of gravity of the curved liquid crystal elastomer rod. Our numerical simulations indicate the presence of a supercritical Hopf bifurcation point delineating the transition between the self-rolling mode and the static mode within the unicycle system. Additionally, the rolling velocity of the unicycle relies on a handful of key system parameters, notably including light intensity, light penetration depth, length of the liquid crystal elastomer rod, rolling friction coefficient, total mass of the unicycle, and wheel radius of the unicycle. The experimental validation of a self-rolling unicycle with zero-energy-mode has been conducted. The self-rolling unicycle constructed in this paper has excellent characteristics of simple structure, zero-energy-mode, horizontal constant light source, and small area light source, providing valuable insights for the utilization of photoresponsive liquid crystal elastomer rods in soft robotics, medical devices, energy harvesting systems, and actuators.