Saddle-node Bifurcation Point Research Articles

Snap-through of a pinned-clamped elastica with arbitrarily movable support at the clamped end

This paper studies the snap-through of a pinned-clamped elastica when the support of the clamped end can be moved arbitrarily in plane. The universal snap curve, which describes the critical boundary conditions of the pinned-clamped elasticas, is firstly obtained by determining the saddle-node bifurcation points of the moment-rotation response curves. Based on the universal snap curve, the stability of the pinned-clamped elastica can be determined when the support at the clamped end is moved. The critical boundary can also be directly obtained, where the elastica loses stability and the snap-through occurs between the non-inverted shape and the inverted shape. This study can be useful to reveal the snap-through behavior for some other complex systems where movable supports exist.

Impact of nanostructures on the superharmonic resonance characteristics of nanobeam-based capacitors: Analytical approach

In this paper, the nonlinear vibration of a size-dependent doubly clamped NEMS has been investigated based on the consistent couple-stress theory and Euler-Bernoulli beam theory. The impact of surface elasticity, dispersion Casimir force, and fringing field effect are considered in the nonlinear model. Here, the nanobeam is supposed to be under a single-side electrostatic actuation, which is a combination of DC and AC voltages. The governing differential equation of motion is derived using the extended Hamilton’s principle and discretized to a nonlinear ODE using Galerkin’s procedure. The multiple time scales method is applied to the reduced-order model in order to analytically obtain the nanobeam frequency–response curves under hard AC load. The influence of the small-scale parameter, Casimir force, and surface effect are investigated on both the static pull-in and the superharmonic resonance characteristics of the system. It is shown that the application of non-classical continuum theory effectively shifts the saddle-node bifurcation point’s loci to lower frequencies, and diminishes the maximum amplitude of the system response without changing the system stiffness. Moreover, it is concluded that the influence of surface energy on the system dynamic behavior depends on the value of DC voltage load. To validate the results obtained from perturbation analysis, the shooting method is schemed through numerical integration of the reduced-order model equation.

The Local Bifurcation of an Eco-Epidemiological Model in the Presence of Stage- Structured with Refuge

In this paper, we establish the conditions of the occurrence of the local bifurcations, such as saddle node, transcritical and pitchfork, of all equilibrium points of an eco-epidemiological model consisting of a prey-predator model with SI (susceptible-infected) epidemic diseases in prey population only and a refuge-stage structure in the predators. It is observed that there is a transcritical bifurcation near the axial and free predator equilibrium points, near disease-free equilibrium point is a saddle-node bifurcation and near positive (coexistence) equilibrium point is a saddle-node bifurcation, a transcritical bifurcation and a pitchfork bifurcation. Further investigations for Hopf bifurcation near coexistence equilibrium point are carried out. Finally, numerical simulations are used to illustrate the occurrence of the local bifurcations of this model.

Out-of-plane motion detection in encapsulated electrostatic MEMS gyroscopes: Principal parametric resonance

The impetus of this research is to numerically investigate a measurement method for out-of-plane motion detection in an encapsulated electrostatic MEMS gyroscope. This method is based on a variant of the harmonic detection of resonance (HDR) of the motion-induced current proposed by S. Park et al. [1]. The MEMS device consists of a cantilever microbeam having a paddle at its free end, and is under two simultaneous electrostatic actuations through side-wall and bottom electrodes along in-plane and out-of-plane directions, respectively. In common MEMS gyroscopes, the sense direction lays on the out-of-plane direction, and therefore, it is significant to detect the motion characteristics along this direction. To this aim, the current passing through the bottom electrode is collected. Because the resonance bandwidth for parametric resonance case is much narrower comparing to that of primary and secondary resonances, this motion characteristic is effectively applicable while measurement with ultra-sensitivity is purposed. In this research, it is shown that the out-of-plane resonance characteristics are retrieved through locking on a specific harmonic in the frequency content of the current signal, considering highest sensitivity to harmonic motions. To capture the frequency-amplitude response of the system, a combination of shooting and arc-length continuation scheme is applied to the governing dynamic motion's equations. Some intersections between the stable and unstable solutions branches as such, saddle-node, Hopf, and pitchfork (subcritical and supercritical) bifurcation points, occurring in the system behavior are addressed. Furthermore, for different levels of AC voltage amplitude, it is shown that the variations in the bifurcation points’ position can be figured out from the output current signal, so that this issue is defined as the fundamental basis for current sensing application.

Influences of generic payload and constraint force on modal analysis and dynamic responses of flexible manipulator

This article presents the dynamic modeling and nonlinear analysis of two-link flexible manipulator with generic payload whose center of gravity is different than the point of attachment with the link. The manipulator is driven by harmonic joint motions and subjected to a constraint force at the terminal end due to its interaction with working environment. Hamilton’s approach is adopted to dynamically model the multi-link manipulator for longitudinal and transverse vibrations incorporating the nonlinearities associated with axial stretching. The resulting set of eigenfrequency equations are solved numerically with subsequent generation of eigenspectrums. Further, this paper investigates the resonant behavior of manipulator links for the subharmonic and primary resonance conditions due to end follower force and revolute joint motions, respectively. The resonant responses show the jump phenomenon and existence of multi-valued solutions at pitchfork and saddle-node bifurcation points. The numerical and graphical results thus obtained, clearly indicate the significant influence of variation of system configurations and generic payload on eigencharacteristics and nonlinear responses, and hence must be given consideration during dynamic modeling of a manipulator in order to attenuate undesirable vibrations.

CRNT4SBML: a Python package for the detection of bistability in biochemical reaction networks.

Signaling pathways capable of switching between two states are ubiquitous within living organisms. They provide the cells with the means to produce reversible or irreversible decisions. Switch-like behavior of biological systems is realized through biochemical reaction networks capable of having two or more distinct steady states, which are dependent on initial conditions. Investigation of whether a certain signaling pathway can confer bistability involves a substantial amount of hypothesis testing. The cost of direct experimental testing can be prohibitive. Therefore, constraining the hypothesis space is highly beneficial. One such methodology is based on chemical reaction network theory (CRNT), which uses computational techniques to rule out pathways that are not capable of bistability regardless of kinetic constant values and molecule concentrations. Although useful, these methods are complicated from both pure and computational mathematics perspectives. Thus, their adoption is very limited amongst biologists. We brought CRNT approaches closer to experimental biologists by automating all the necessary steps in CRNT4SMBL. The input is based on systems biology markup language (SBML) format, which is the community standard for biological pathway communication. The tool parses SBML and derives C-graph representations of the biological pathway with mass action kinetics. Next steps involve an efficient search for potential saddle-node bifurcation points using an optimization technique. This type of bifurcation is important as it has the potential of acting as a switching point between two steady states. Finally, if any bifurcation points are present, continuation analysis with respect to a user-defined parameter extends the steady state branches and generates a bifurcation diagram. Presence of an S-shaped bifurcation diagram indicates that the pathway acts as a bistable switch for the given optimization parameters. CRNT4SBML is available via the Python Package Index. The documentation can be found at https://crnt4sbml.readthedocs.io. CRNT4SBML is licensed under the Apache Software License 2.0.

Bifurcations of Negative Responses to Positive Feedback Current Mediated by Memristor in a Neuron Model with Bursting Patterns

In contrast to traditional viewpoint that positive feedback current always enhances neural firing activities, in the present paper, we identify that the excitatory feedback current mediated by memristor can induce negative responses of bursting patterns, which can be well interpreted with bifurcations. For the Hindmarsh–Rose neuron model without memristor, the period-adding bifurcations of bursting patterns and increase of firing frequency can be induced by increasing the excitatory effect of the background current. After introducing a memristor to simulate the biological synapse or electromagnetic induction effect, inverse period-adding or complex bifurcations of bursting patterns are induced by the excitatory feedback current mediated by the memristor. The number of spikes per burst becomes smaller and the firing frequency becomes lower when increasing the positive feedback gain. Such negative responses of bursting patterns to the positive feedback current are demonstrated in a circuit designed with Digital Signal Processor systems of the MatLab software. Furthermore, the underlying bifurcation mechanism of the negative responses to the positive feedback is acquired with fast–slow variable dissection method. With increasing feedback gain, the initial phase of the burst, which corresponds to a saddle-node bifurcation point of the fast subsystem, delays, while the termination phase of the burst, which corresponds to a saddle-homoclinic bifurcation point, remains unchanged. Therefore, the burst becomes narrower with increasing feedback gain, which leads to decrease in the number of spikes within a burst and decrease in firing frequency. The results present a paradoxical nonlinear phenomenon and the dynamical mechanism, which is helpful for understanding the functions of memristor and roles of the electromagnetic induction current.

Bifurcation analysis of a modular model of embryonic kidney development

Many biological processes show switching behaviors in response to parameter changes. Although numerous surveys have been conducted on bifurcations in biological systems, they commonly focus on over-represented parts of signaling cascades, known as motifs, ignoring the multi-motif structure of biological systems and the communication links between these building blocks. In this paper, a method is proposed which partitions molecular interactions to modules based on a control theory point of view. The modules are defined so that downstream effect of one module is a regulator for its neighboring modules. Communication links between these modules are then considered as bifurcation parameters to reveal change in steady state status of each module. As a case-study, we generated a molecular interaction map of signaling molecules during the development of mammalian embryonic kidneys. The whole system was divided to modules, where each module is defined as a group of interacting molecules that result in expression of a vital downstream regulator. Bifurcation analysis was then performed on these modules by considering the communication signals as bifurcation parameters. Two-parameter bifurcation analysis was then performed to assess the effects of simultaneous input signals on each module behavior. In the case where a module had more than two inputs, a series of two parameter bifurcation diagrams were calculated each corresponding to different values of the third parameter. We detected multi-stability for RET protein as a key regulator for fate determination. This finding is in agreement with experimental data indicating that ureteric bud cells are bi-potential, able to form tip or trunk of the bud based on their RET activity level. Our findings also indicate that Glial cell-derived neurotrophic factor (GDNF), a known potent regulator of kidney development, exerts its fate-determination function on cell placement through destruction of saddle node bifurcation points in RET steady states and confining RET activity level to high activity in ureteric bud tip. In conclusion, embryonic cells usually show a huge decision making potential; the proposed modular modeling of the system in association with bifurcation analysis provides a quantitative holistic view of organ development.

Saddle-Node Bifurcations of Power Systems in the Context of Variational Theory and Nonsmooth Optimization

This paper presents a new concept for finding saddle-node bifurcation (SNB) points in voltage stability analysis of power systems by applying the extended functional method (EFM). This method enables the finding of the SNB point of power systems by directly calculating the extreme values of a nonsmooth variational functional, which is obtained in its turn by the so-called nonlinear generalized Collatz-Wilandt formula. The main theoretical result establishes the EFM applicability for finding the maximum loading capacity of power systems. The maximum loading capacity of the power system is shown to be located at the maximizing point of the nonsmooth function of bifurcations. The subgradient method for nonsmooth functions was applied. The EFM was performed on various IEEE test systems to find SNB points, and the results were compared with those obtained with the Continuation Power Flow (CPF) and the Point of Collapse (PoC) method. The simulation results show that the proposed method is robust and, unlike the PoC method, finds the SNB point even when good guessing of the turning point is not available. Tasks such as tracking an SNB point displaced by a contingency and infeasible power flow were performed successfully by using the EFM.

Analysis of noise-induced phenomena in the nonlinear tumor–immune system

The effect of random disturbances on the corporate dynamics of immune and tumor cells is studied on the basis of the mathematical model proposed by Kuznetsov et al. This model possesses two equilibria corresponding to opposite regimes of ”tumor dormancy” and ”tumor explosion”. We study the system dynamics under the variation of the parameter of inactivation of immune cells by tumor cells. For deterministic model, parametric mono- and bistable zones and the corresponding changes in the basins of attraction are described. We investigate how the multiplicative random noise can change the dynamics of this tumor–immune system. Mechanisms of the noise-induced suppression and growth of tumor cells are investigated in detail. Furthermore, we reveal that near the saddle–node bifurcation points, the random disturbances can destroy the bistability and generate the intermittent oscillations between high and low levels of tumor cells. We perform an analysis of these phenomena by a novel approach based on the stochastic sensitivity functions and confidence domains.

Combined impacts of the Allee effect, delay and stochasticity: Persistence analysis

We study a combined influence of the Allee effect, delay and stochasticity on the base of the phenomenological Hassell mathematical model of population dynamics. This bistable dynamical model possesses a wide variety of regimes, both regular and chaotic. In the persistence zone, these regimes coexist with the trivial equilibrium that corresponds to the extinction of the population. It is shown that borders of the persistence zone are defined by the crisis and saddle-node bifurcation points. Noise-induced transitions from the persistence to the extinction are studied both numerically and analytically. Using numerical modeling, we have found that the persistence zone can decrease and even disappear under the influence of random noise. For the theoretical study of this phenomenon, we apply the stochastic sensitivity analysis and Mahalanobis metrics.

Study on Ledinegg instability of two-phase boiling flow with bifurcation analysis and experimental verification

Some new phenomena of Ledinegg instability are observed by portraying a complete flow excursion process via bifurcation analysis and experimental verification. A mathematical model consisting of two first-order nonlinear ordinary differential equations, is constructed via the lumped parameter method and verified through linear stability analysis. The results of bifurcation analysis show that Ledinegg instability corresponds to saddle-node bifurcation, which is consistent with previous literatures. The minimum point of the internal curve is not necessarily the onset of flow instability (OFI). It is equivalent, for the saddle-node bifurcation point, OFI and the tangent point between two characteristic curves, to indicate flow excursion. The flow excursion process is unidirectional and irreversible at OFI, which decides that hysteresis phenomenon can be triggered by Ledinegg instability within the unstable bifurcation interval. Conversely, hysteresis phenomenon determines there exist two types of flow excursion and OFIs, namely, Type I and Type II which respectively correspond to decrease and increase in mass flux. Hysteresis phenomenon indicates not only that the system cannot autonomously returns to the original operating state once flow excursion occurs, but also that the sufficient and necessary condition of flow excursion is that current equilibrium state crosses over the corresponding OFI in the same equilibrium branch. To avoid the occurrence of Ledinegg instability, a versatile quantitative criterion is developed by considering two characteristic forces simultaneously. Finally, experimental results verify the correctness of bifurcation analysis.

Bifurcation analysis of a forced delay equation for machine tool vibrations

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance, misalignment of the workpiece or ultrasonic excitation), or it can arise from the cutting process itself (e.g., periodic chip formation). We investigate the classical one-degree-of-freedom tool vibration model, a delay-differential equation with quadratic and cubic nonlinearity, and periodic forcing. The method of multiple scales is used to derive the slow flow equations. Stability and bifurcation analysis of equilibria of the slow flow equations is presented. Analytical expressions are obtained for the saddle-node and Hopf bifurcation points. Bifurcation analysis is also carried out numerically. Sub- and supercritical Hopf, cusp, fold, generalized Hopf (Bautin), Bogdanov–Takens bifurcations are found. Limit cycle continuation is performed using MatCont. Local and global bifurcations are studied and illustrated with phase portraits and direct numerical integration of the original equation.

Boundary-derivative direct method for computing saddle node bifurcation points in voltage stability analysis

The Saddle Node Bifurcation (SNB) is the location where equilibrium solutions of one parametric system distinguish when its parameter changes. Load distance between the SNB and the present operation point can reflect the voltage stability of the studied system. At present, Continuous Power Flow (CPF) and Point of Collapse (POC)/Direct Method are main techniques for tracking the point. However, CPF needs multiple power flow calculations, which is not suitable for fast and accurate applications, while POC has problems of large coefficient matrix and initialization of eigenvector. A novel direct method is proposed to calculate the SNB point, titled the Boundary-Derivative Direct Method (BDDM). A one-dimensional boundary condition, which reflects the slope of the equilibrium manifold, is configured to replace the multidimensional conditions of POC. As a result, the dimension of the coefficient matrix is drastically cut and the initialization of eigenvector is avoided. Moreover, we extend the algorithm, considering multi-load growth and general limitations. Several instances on IEEE 9-bus system show the convergence of the proposed algorithm and the cases in IEEE 118-bus system and 2000-Bus Texas System, which contain multi-load increments and reactive power limitations of generators, are used to analyze the effectiveness of the proposed algorithm.

Estimating the margin to voltage collapse in a dwindling electric power network

For more than a decade, Nigeria National Grid (NNG) has encountered an average of 24 times per year of total incidences of voltage collapse in the system. Nigerian manufacturers, small scale businesses, families including private households have been unable to break even in their businesses owing to this ugly trend. The main focus of the work is to estimate the limit and margin of active power transmission across a certain defined boundary prio to voltage collapse. This will help to alert the power systems operators and planners about the dangers that lie ahead in case of any voltage collapse scenario. The paper made an exploit of Continuation Power Flow to determine the Margin to Voltage Collapse (MVC) of NNG. The effectiveness of the approach was tested in the 10Bus NNG and 30Bus NNG. For the 10Bus NNG, the results show that at 15 th iteration the corrector solution (4.3698pu) was found to be too far from the predictor value (1.7454pu). However, the maximum loading Parameter 4.3698pu with a mismatch of 0.00324pu converged at 40 th iteration search in 1.5016 seconds. To enable quicker convergence of the solutions and the computation speed, the step size was reduced to 0.04. The MVC for the 10Bus NNG and 30Bus NNG are estimated to be 4.1698 p.u and 3.5 p.u respectively. Keywords: CPF, MVC, Voltage Collapse, Voltage Stability, Saddle Node Bifurcation Point, Current Operating Point

On the convergence of the implicit Z-Bus power flow method for distribution systems

In this paper, the convergence properties of the implicit Z-Bus power flow method for distribution systems are analyzed. First, the formulation of implicit Z-Bus method is simplified to obtain its expressions in real vector space. Second, two fundamental convergence problems are respectively discussed and derived. And the results obtained with regard to the implicit Z-Bus method include: it is inevitably convergent in the entire power flow solvability region (excluding the boundary which is comprised of all the saddle node bifurcation points), and it has reliable convergence property in the aspect of the basin of attraction. Finally, an accelerating strategy based on the Aitken–Steffensen formula is proposed to accelerate the implicit Z-Bus method. Numerical tests are implemented to illustrate the convergence characteristics.

Voltage–Amplitude Response of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Cantilever Resonators

This paper investigates the voltage–amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage–amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.

Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions.

In this paper, we present a new method based on dynamical system theory to study certain type of slow-fast motions in dynamical systems, for which geometric singular perturbation theory may not be applicable. The method is then applied to consider recurrence behavior in an oscillating network model which is biologically related to organic reactions. We analyze the stability and bifurcation of the equilibrium of the system, and find the conditions for the existence of recurrence, i.e., there exists a "window" in bifurcation diagram between a saddle-node bifurcation point and a Hopf bifurcation point, where the equilibrium is unstable. Simulations are given to show a very good agreement with analytical predictions.

Improvement of the line losses, weaker buses and saddle-node-bifurcation points using reconfigurations of the identified suitable lines

For the planning, operation and design of the modern power systems, the effect of the network reconfiguration of the identified weaker lines are nowadays investigated. Gradually increasing the load, saddle-node bifurcation (SNB) is determined using modified continuation power flow (MCPF) algorithm. Judging three different sets of the security limits, the weaker buses are detected from the power flow solution at the SNB point. The three different reconfigurations are implemented on the identified weaker lines for the standard IEEE-14, 30 and 57-bus test systems separately. The optimal location and the suitable lines for the reconfiguration are demonstrated with the results. Using the network reconfiguration, the significant improvement of the SNB points or load margin, weaker buses, voltage profile, line losses for the different cases are proved showing results.

Nested Layer Particle Swarm Optimization for Detection of Saddle-Node Bifurcation Point in One-Dimensional Discrete Time Dynamical Systems

Nested Layer Particle Swarm Optimization for Detection of Saddle-Node Bifurcation Point in One-Dimensional Discrete Time Dynamical Systems