Pitchfork Bifurcation Research Articles

The Bifurcation Analysis and Persistence of the Food Chain Ecological Model with Toxicant

In this work, the occurrence conditions of both local Bifurcation and persistence were studied, Saddle-node bifurcation appears near fourth point, near the first point, the second point and the third point a transcritical bifurcation occurred but no pitchfork bifurcation happened near any of the four equilibrium points. In addition to study conditions for Hopf-bifurcation near positive stable point that is the fourth point. Besides discuss persistence occurrence as globally property of the food chain of three species include prey, first predator and top predator with impact of toxin in all species and harvesting effect on the predator’s only. Numerical results for the set of hypothetical of parameters come to confirm our analytical results about illustration the occurrence of local bifurcation and persistence of this model.

PEMODELAN DINAMIKA VIRUS RUBELLA DENGAN RISIKO SINDROM RUBELLA BAWAAN

Rubella is a common cause of childhood rash and fever. It is typically a mild infection but it can be serious condition in pregnant women, as it may cause Congenital Rubella Syndrome (CRS) in the fetus. The study aimed to model and analyze it in order to get picture about the dynamics of the rubella virus transmission. The paper discussed two models of rubella transmission involving child-bearing age women and newly born infants of infected mothers. The models are SEIR-IR without seasonality and SEIR-IR with seasonality. The results from the first model showed that a pitchfork bifurcation occurred in the dynamics of the solution of the system with constant infection rate and the increase of the infection rate value do not make impact to the incidence of rubella among infants. The second model involving seasonality gave interesting dynamics where big seasonality may lead to a more complex dynamics of the solution.

Analysis of supercritical pitchfork bifurcation in active magnetic bearing-rotor system with current saturation

The bifurcation characteristics of the active magnetic bearing-rotor system subjected to the external excitation were investigated analytically when it was operating at a speed far away from its natural frequencies. During operation of the system, some nonlinear factors may be prominent, for example, the nonlinearity of bearing force and current saturation. Nonlinear factors can lead to some complicated behaviors, which have negative effects on the operating performance and stability. To analyze the bifurcations happening at the speed far away from harmonic resonances, an approximate analytical method that can be applicable to the bifurcation analysis of the forced vibration system was proposed. By applying it to the active magnetic bearing-rotor system, multiple static equilibriums and periodic solutions were obtained, and then, the stability analysis was conducted based on Floquet theory. The validity and accuracy of the approximate analytical method were verified by the numerical integration method and generalized cell mapping digraph method. It was found that there was supercritical pitchfork bifurcation of static equilibrium in the active magnetic bearing-rotor system. The influences of external excitation and controller parameters on dynamical characteristics were discussed. Based on analysis results, controller parameters were also improved to prevent nonlinear behaviors and improve system performance.

Onset of temporal dynamics within a low reynolds-number laminar fluidic oscillator

We investigate the onset of temporal dynamics of a fluidic oscillator (FO) in the laminar regime at very low Reynolds number (Re=Uhi/ν, where U and hi are the velocity and channel height at inlet, and ν is the kinematic viscosity of the fluid). Both two- and spanwise-periodic-three-dimensional simulations are performed using high-order spectral element methods to characterise the flow inside the FO cavity and the outcoming jet. While the flow remains steady and symmetric for sufficiently low Re, the two-dimensional FO undergoes a symmetry-breaking Hopf bifurcation at ReH2=75.7 that results in a space-time symmetric periodic solution. The remnant space-time symmetry is later broken in a pitchfork bifurcation of limit cycles at ReP2=294.2. Taking three-dimensionality into consideration results in an early spanwise-invariance disruption at about Re3≃68 of wavelength 13hi that retards the onset of the oscillation to ReH3=82.1. This effect is little representative of actual FO implementations, which are typically much narrower than the wavelength of the spanwise instability observed at these low Re. Contrary to what happens with FOs in the turbulent oscillatory regime, the oscillation frequency of the output jet in the laminar regime is found to decrease with Re. The mechanism that drives the oscillation, however, remains the same: as the high speed flow inside the cavity attaches to one of the internal walls through the Coandă effect, the feedback channels divert part of the momentum back, which pushes the incoming flow against the opposite wall. The alternate deviation of the flow inside the cavity to one or the other side results in the periodic flipping of the output jet. The pressure contribution to net momentum at the end of the feedback channels is found to be double that of the advective momentum flux at the early time-periodic regime but both become comparable as Re is increased. The jet sweeping angle amplitude is more pronounced for the two-dimensional FO as compared to three-dimensional at a fixed given Re, the Coandă effect being only partially fulfilled in the latter case. In both cases the sweep amplitude increases with Re. The instability of the output jet, which becomes slightly chaotic already at very low Re, is responsible for triggering the cavity instability that drives the oscillation slightly earlier than it would, should outside noise be suppressed altogether.

Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

In this paper, complex dynamical behaviors of a predator-prey system with the Beddington–DeAngelis functional response and the Allee-like effect on predator were studied by qualitative analysis and numerical simulations. Theoretical derivations have given some sufficient and threshold conditions to guarantee the occurrence of transcritical, saddle-node, pitchfork, and nondegenerate Hopf bifurcations. Computer simulations have verified the feasibility and effectiveness of the theoretical results. In short, we hope that these works could provide a theoretical basis for future research of complexity in more predator-prey ecosystems.

Amplitude death, oscillation death, and periodic regimes in dynamically coupled Landau–Stuart oscillators with and without distributed delay

The effects of a distributed ’weak generic kernel’ delay on dynamically coupled Landau–Stuart limit cycle oscillators are considered. The system is closed via the ’linear chain trick’ and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. After reformulation, the effect of the weak generic kernel delay turns out to be mathematically similar to the linear augmentation scheme considered earlier for coupled oscillator systems. For our coupled Landau–Stuart oscillators the distributed delay produces transitions from amplitude death (AD) or oscillation death (OD) to periodic behavior via Hopf Bifurcations, with the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The transition from AD to OD occurs here via a supercritical pitchfork bifurcation, as also seen previously for some other couplings. In addition, the dynamically coupled Landau–Stuart system has a secondary limit cycle emerging within the OD parameter regime, in contrast to other couplings and systems studied earlier.The various transitions among AD, OD and periodic domains that we observe are more intricate than the simple AD states, and the rough boundaries of the parameter regimes where they occur, predicted by linear stability analysis and experimentally verified in earlier work on dynamically coupled systems. Also, although our delayed system mathematically resembles linear augmentation schemes used earlier, the numerical searches here reveal more complex transitions among various dynamical regimes than the direct AD and OD found in those studies.

Universality of photon counting below a local bifurcation threshold

At a bifurcation point, a small change of a parameter causes a qualitative change in the system. Quantum fluctuations wash out this abrupt transition and enable the emission of quantized energy, which we term photons, below the classical bifurcation threshold. Close to the bifurcation point, the resulting photon counting statistics is determined by the instability. We propose a generic method to derive a characteristic function of photon counting close to a bifurcation threshold that only depends on the dynamics and the type of bifurcation, based on the universality of the Martin-Siggia-Rose action. We provide explicit expressions for the cusp catastrophe without conservation laws. Moreover, we propose an experimental setup using driven Josephson junctions that exhibits both a fold and a pitchfork bifurcation behavior close to a cusp catastrophe.

The influence of a pitchfork bifurcation of the critical points of a symmetric caldera potential energy surface on dynamical matching

Many organic chemical reactions are governed by potential energy surfaces that have a region with the topographical features of a caldera. If the caldera has a symmetry then trajectories transiting the caldera region are observed to exhibit a phenomenon that is referred to as dynamical matching. Dynamical matching is a constraint that restricts the way in which a trajectory can exit the caldera based solely on how it enters the caldera. In this paper we show that bifurcations of the critical points of the caldera potential energy surface can destroy dynamical matching even when the symmetry of the caldera is not affected by the bifurcation.

Bifurcation and chaotic behavior in the discrete BVP oscillator

In this work, a new class of discrete Bonhoeffer–van der Pol (BVP) system with an odd function is proposed and investigated. At first, the necessary and sufficient conditions on the existence and stability of the fixed points for this system are given. We then show the system passes through various bifurcations of codimension one, including pitchfork bifurcation, saddle–node bifurcation, flip bifurcation and Neimark–Sacker bifurcation under some certain parameter conditions. The center manifold theorem and bifurcation theory are the main tools in the analysis of the local bifurcations. Furthermore, we prove rigorously there exists Marotto’s chaos in this discrete BVP system, which means the fixed point eventually evolves into a snap-back repeller. Finally, numerical simulation evidences are provided not only to further demonstrate our theoretical analysis, but also to exhibit the complex dynamical phenomena, such as the period-9, -17, -18 orbits, attracting invariant cycles, quasi-periodic orbits, ten-coexisting chaotic attractors, etc. These phenomena illustrate relatively rich dynamical behaviors of the discrete BVP oscillator.

Bifurcation analysis of Friedkin–Johnsen and Hegselmann–Krause models with a nonlinear interaction potential

Opinion dynamics of a group of individuals is the change in the members’ opinions through mutual interaction with each other. The related literature contains works in which the dynamics is modeled as a continuous system, of which behavioral patterns are analyzed in regard to the parameters contained in the system. These models are constructed by the assumption that the individuals are interdependent. Besides, the decisions of the individuals are only affected by two forces: self-bias force and group influence force. In this work, a nonlinear dynamical system which models the evolution of the decision of a group under the existence of a leader is considered. The two well-known opinion evolutions Friedkin–Johnsen and Hegselmann–Krause models are incorporated with a nonlinear exponentially decaying interaction potential to analyze the dynamics. The model considered in this work, which reflects simultaneously the effect of a leader and a nonlinear potential in the group dynamics, is analyzed the first time in the literature. Through this mechanism, the aim of the study is to understand the transitions between the final stable situations of the system which can be agreement, majority rule or disagreement. Bifurcation analysis of the system is performed to obtain stability results on the system. It is shown that one of these transition mechanisms is an imperfect pitchfork bifurcation. The study successfully produces boundary curves of the different regions in the parameter space that separates the stable states. The results of the work well present the distinctions between these final situations analytically, in case of N=3 agents plus a leader. The group mechanism considered is promising in the sense of having possible further modeling opportunities. For a large N number of people, clusters of opinions can be studied numerically, for analyzing how a leader can affect a big group of agents. The same can be tried for a big community with several leaders, which can represent the dynamics of electoral processes. The basic construction in this work will be a starting point for such further analyses.

Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington DeAngelis functional response

In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings.

Propagating vortices in ferrofluidic Couette flow under magnetic fields – Part I: Axial and symmetry breaking transversal orientated fields

Numerical investigations of propagating vortex states (pVs) for ferrofluidic Couette flow with small aspect ratio and fixed non-rotating end-walls are presented. We study structural modifications and changes in spatial and temporal behavior for pV solutions. The system is subjected to either pure axial or pure transverse magnetic field, with the latter already breaking the basic system symmetries. While under axial magnetic field pVs remain basically the same, they are found to appear in various configurations with different symmetry variation in flow structure and flow dynamics under symmetry breaking transversal magnetic field. pVs appearing in a pitchfork bifurcation are found either symmetric or asymmetric/alternating, regarding the full cycle of vortex generation, propagation and annihilation, in upper and lower system half.

Transition from steady to chaotic flow of natural convection on a section-triangular roof

Flow phenomena on a roof are investigated by employing direct numerical simulation. A sequence of pitchfork bifurcations of steady plumes on the roof occurs for small Rayleigh numbers, which is analyzed using a topological method. Furthermore, a Hopf bifurcation followed by periodic doubling, quasiperiod bifurcations, and a transition to chaos appear as the Rayleigh number is increased.

Pressure Drop Oscillations With Symmetry Breakdown in Two-Phase Flow Parallel Channels

Abstract Two-phase flow in parallel heated channels is prone to symmetry breakdown resulting in mass flow maldistribution. Moreover, in the presence of compressible volume (CV), such systems also undergo pressure drop oscillations (PDOs). The performances of such systems depend on the effect of these flow instabilities. However, the simultaneous occurrence of these two-phenomena has been rarely reported in the literature. In this work, an approach is applied in a two-channel system to demarcate the parameter space of mass flow rate and inlet temperature into several areas, where these two phenomena take place. The loss in the symmetry in the flow rate is observed as the mass flow rate is varied, which leads to flow maldistribution. The PDO are also observed for specific values of mass flow rate in the system. One unique feature of the parallel channel system is the existence of the oscillatory and stable (albeit asymmetric) states at the same parameter values. For these parameter values, the final state of the system is dependent on the type of initial disturbance. The flow maldistribution due to symmetry breakdown is identified by the pitchfork bifurcation, and oscillations of mass flow rate are identified by the presence of Hopf bifurcation. Moreover, the physical interpretation of the different phenomena in the system is carried out using internal and external pressure drop characteristics curves.

Nonlinear Network Dynamics with Consensus\u2013Dissensus Bifurcation

We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.

A new perspective on static bifurcations in the presence of viscoelasticity

This manuscript explores the effect of viscoelasticity on static bifurcations: such as pitchfork, saddle-node, and transcritical bifurcations, of a single-degree-of-freedom mechanical oscillator. The viscoelastic behavior is modeled via a differential form, where the extra degree of freedom represents the internal force provided by the viscoelastic element. The governing equations are derived from a simplified lumped parameter model consisting of a rigid rod incorporating a viscoelastic element and subjected to axial and transverse forces at the free end, in addition to an external time-varying moment applied to the rod. In order to study the effect of viscoelasticity on bifurcation diagrams, the equations of motion are non-dimensionalized. Next, a review of static bifurcations in the absence of viscoelasticity is conducted, followed by an examination of the effect of viscoelasticity on the bifurcation diagrams. Finally, this paper investigates the effects of viscoelasticity on the transient behavior of the oscillator. Results show that the Deborah number, which measures the ratio of the viscoelastic time constant to the natural periodic time of the system, controls the duration of time needed to maintain oscillations around an unstable point before jumping to a stable equilibrium point.

Chua’s oscillator: an introductory approach to chaos theory

The purpose of this article is to build an introductory approach to chaos theory from the Chua’ oscillator for students of mathematics, physics and engineering. The Chua’ circuit is one of the simplest dynamic systems that produce irregular behavior. Despite its simplicity, this oscillating circuit is a very useful device for studying basic principles of chaos theory. This study begins with the definition of equilibrium points, this is done in a graphic and analytical way. The analysis of the stationary solution of differential equations system allows to show the occurrence of pitchfork bifurcation. The eigenvalues are calculated and the trajectories of the Chua’ oscillator are analyzed, then the occurrence of Hopf bifurcation is demonstrated. Period-doubling cascade are observed using phase portrait and orbit diagram. Finally, sensitivity to initial conditions is studied by calculating Lyapunov exponents.

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions

This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion. Several criteria are compared and it is found that a criterion based on a regression artificial neural network (ANN) provides the most accurate results for a channel flow problem exhibiting a supercritical pitchfork bifurcation. The same benchmark test is then used to compare the local ROM approach with the regression ANN selection criterion to an established global projection-based ROM and a recently proposed ANN based method called POD-NN. We show that our local ROM approach gains more than an order of magnitude in accuracy over the global projection-based ROM. However, the POD-NN provides consistently more accurate approximations than the local projection-based ROM.

The interplay between symmetry-breaking and symmetry-preserving bifurcations in soft dielectric films and the emergence of giant electro-actuation

Soft elastomers that can exhibit extremely large deformations under the action of an electric field are essential for applications such as soft robotics, stretchable and flexible electronics, energy harvesting among others. The critical limiting factor in conventional electro-actuation of such materials is the occurrence of the so-called pull-in instability. In this work, we demonstrate an extraordinarily simple way to coax a dielectric thin film towards a symmetry-breaking pitchfork bifurcation state while avoiding pull-in instability. Through the nonlinear interplay between the two bifurcation modes, we predict electro-actuation strains that exceed what is conventionally possible by 200%, and at significantly lower applied electric fields.

Dynamics of inverted flags: Experiments and comparison with theory

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow – also known as inverted flags – has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small- and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.