Saddle-node Bifurcation Point Research Articles

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

Global dynamics and integrity of a micro-plate pressure sensor

This paper investigates the complex bifurcation scenario of electrically-actuated circular micro-plates subjected to differential pressure. Our analysis deals with the primary, secondary and ultimate saddle-node bifurcation points responsible for the device snap-through and addresses the pressure range in which the robustness of the two main stable configurations is undermined by minor coexisting attractors. By making use of basins of attraction and integrity profiles, safe dynamical regions of motion are evaluated with respect to the applied DC voltage. It is found that period-doubling bifurcations are accountable for a sensible reduction in the dynamical integrity, small variations of the DC voltage largely modify the response of the pressure sensor.

Saddle Node Bifurcation Point Analysis of Voltage Stability of IEEE 30-BUS Power System for Removal of Generation through Bacteria Foraging Optimization Algorithm

This paper deals with the identification of instability nodes of IEEE 30 BUS power system to generation removal. Optimal sizing and locations of reactive power compensations are obtained. Firstly one of the generators is assumed to be removed from service and the saddle node bifurcation (SNB) point voltages are evaluated without reactive power compensation. Secondly two generators are assumed to be removed from service and the saddle node point voltage magnitudes are obtained without reactive power compensation. For both cases the study is conducted by placing optimal reactive power compensations at optimal locations using Bacterial Foraging Optimization Algorithm (BFOA).

Efficient method to identify saddle‐node and limit‐induced bifurcation points of power system

High intermittency in today's renewable-rich power systems, and the prohibitive cost of upgrading the network infrastructure along with the load growth, has rendered voltage instability an imminent threat for many power systems. This necessitates faster and more efficient ways of identifying the voltage stability (VS) limits, associated with specific bifurcation points of power system model, which are suitable for real-time applications. To date, continuation power flow (CPF) has conventionally been used to identify bifurcation points of power systems, through plotting power–voltage ( P – V ) curves. However, existing CPF methods are complex and computationally demanding. To tackle this issue, in this study, accurate identification of both saddle-node and limit-induced bifurcation points of power systems is carried out by using a new and efficient continuous power flow algorithm, in which all the complexities associated with the existing CPF methods are relaxed. Low execution time (as compared to the existing CPF methods), ease of implementation, and automated applicability, make the proposed algorithm highly suitable for fast and accurate VS assessment of renewable-rich, uncertain, power systems. Experiments, carried out on several different size power systems, verify that the proposed method can be effectively used to identify the VS limits of practical real-life power systems, despite its ease of implementation and lower computational burden.

Bifurcation and stability analysis of a nonlinear rotor system subjected to constant excitation and rub-impact

Abstract This paper focuses on the mechanism of a complex bifurcation behavior caused by flight maneuvers in an aircraft rub-impact rotor system with Duffing type nonlinearity. The maneuver load during flight maneuvers may induce a rub-impact phenomenon, accompanied by complex nonlinear behaviors such as periodic, sub-harmonic and quasi-periodic motions, but the bifurcation mechanism is not so clear. In this study, the harmonic balance method combined with an alternating frequency/time domain procedure (HB-AFT method) is formulated and used to derive the approximate periodic solutions of the system. In conjunction with the arc-length continuation, the solution branches for both periodic 1-T motion (synchronous oscillation) and periodic 2-T motion (sub-harmonic oscillation) are traced. Then with the aid of the Floquet theory, the stabilities of the obtained periodic solutions are examined. The changes in the number or the stability of the solutions lead to the identification of the bifurcation points, which can be classified qualitatively into three types, i.e. Neimark-Sacker bifurcation point (NSBP), quasi-periodic Hopf bifurcation point (QPHBP) and saddle-node bifurcation point (SNBP). In addition, the constant excitation significantly affects the instability as well as the bifurcation of the rotor system. In the case of a smaller constant excitation, the rotation region with respect to the quasi-periodic motions may disappear, accompanied with a PBP (pitchfork bifurcation point) instead of the two NSBPs and one QPHBP. The bifurcation analysis in this paper provides deep insight into the mechanism of the complex nonlinear phenomenon induced by the constant excitation. The results obtained will also contribute to a better understanding of the nonlinear dynamic behaviors of aircraft rotor systems during flight maneuvers.

Fuzzy Load-Shedding Strategy Considering Photovoltaic Output Fluctuation Characteristics and Static Voltage Stability

Based on the equilibrium point equations of a classic three-node system integrated with a large-scale photovoltaic cell (PV) power plant, the impact of PV output fluctuation on the saddle-node bifurcation (SNB) was derived and analyzed. When PV runs in a unity power factor and the PV output active power Ppv is not too large (several hundred MW and below), the PV output fluctuation has little effect on the SNB point position and load margin index, so that the load margin index can be calculated online using the SNB point at Ppv = 0 pu. On the other hand, the local reactive power compensation in the load center can effectively raise the load bus voltage and make the voltage stability problem become more concealed; the traditional under-voltage load-shedding (UVLS) strategy only carries out load shedding when the bus voltage amplitude is below the specified value and cannot effectively maintain the system static voltage stability in some occasions. In this paper, a fuzzy load-shedding strategy considering the impact of PV output fluctuations for the large-scale PV grid-connected system was designed, taking the load bus voltage amplitude and load margin index as fuzzy input variables, and the load-shedding command as a fuzzy output variable. Nine fuzzy IF-THEN rules were extracted for the fuzzy controller and the corresponding practical calculation method of load-shedding quantity was put forward. The simulation results of the classic three-node system and IEEE 14-bus system, both with a 100 MW PV power plant, verified the effectiveness of the fuzzy load-shedding controller whose input variable load margin index was calculated using the SNB point when the PV active power output was 0. The designed fuzzy load-shedding strategy can compensate for the defect—that the traditional UVLS strategy cannot effectively guarantee the system static voltage stability—and it can be widely used in power grids integrated with PV power plants whose scales are at a level of several hundred MW and below.

Remarkable similarities of two pairs of stable and saddle canards in a van der Pol oscillator under extremely weak periodic perturbation

Canards are interesting nonlinear phenomena that have generated intense research interest since their discovery in the late 20th century. We are interested here in how canard-generating dynamics are influenced by extremely weak periodic perturbations that cause the formation of saddle-node bifurcations in the fundamental harmonic entrainment region. In a previous study, we discovered that another entrainment region exists within the fundamental harmonic entrainment region surrounded by the second saddle-node bifurcation curves. We found that two pairs of stable and saddle canards coexist in this second entrainment region under such weak periodic perturbation. Moreover, the stable and saddle canards are matched pairwise; i.e., each stable canard quite closely resembles a corresponding saddle canard. Calculation of the correlation coefficients of the four canards revealed two similar solutions on the order of 0.9999⋯ between the two pairs of similar canards. In contrast, the correlation coefficients of the dissimilar canards differ from unity in proportion to the difference between the given bifurcation parameter value and the parameter values at the saddle-node bifurcation points. Approximately, they take values from 0.998 to 0.975. These contrasts are noteworthy. Similar bifurcation phenomena were observed in the 1/2-subharmonic entrainment region. We hypothesize that the two pairs of stable and saddle canards are invariant with respect to a slight shift of time at the saddle-node bifurcation points, and we numerically prove that such a property approximately holds at the bifurcation points.

Bifurcation mechanism of not increase but decrease of spike number within a neural burst induced by excitatory effect

Nonlinear dynamics is identified to play very important roles in identifying the complex phenomenon, dynamical mechanism, and physiological functions of neural electronic activities. In the present paper, a novel viewpoint that the excitatory stimulus cannot enhance but reduce the number of the spikes within a burst, the novel viewpoint which is different from the traditional viewpoint, is proposed and is explained with the nonlinear dynamics. When the impulse current or the autaptic current with suitable strength is used in the suitable phase within the quiescent state of the bursting pattern of the Rulkov model, a novel firing pattern with reduced number of spikes within a burst is evoked. The earlier the application phase of the current within the quiescent state, the higher the threshold of the current strength to evoke the novel firing pattern is and the less the number of the spikes within a burst of the novel firing pattern. Moreover, such a novel phenomenon can be explained by the intrinsic nonlinear dynamics of the bursting combined with the characteristics of the current. The nonlinear behaviors of the fast subsystem of the Rulkov model are acquired by the fast and slow variable dissection method, respectively. For the fast subsystem, there exist a stable node with lower membrane potential, a stable limit cycle with higher membrane potential, a saddle serving as the border between the stable node and limit cycle, a saddle-node bifurcation, and a homoclinic orbit bifurcation. When external simulation is not received, the bursting pattern of the Rulkov model exhibits behavior alternating between the spikes corresponding to the limit cycle of the fast subsystem and quiescent state of the fast subsystem, which is located within the parameter region between the saddle-node bifurcation point and the homoclinic orbit bifurcation point of the fast subsystem. The spikes begin with the saddle-node bifurcation and end with the homoclinic orbit bifurcation. As the bifurcation parameter turns close to the homoclinic orbit bifurcation, the disturbation or stimulus that can induce the transition from the quiescent state to the spikes becomes strong. Therefore, as the application phase of the current within the quiescent state becomes earlier, the strength threshold of the current that can induce the transition from the quiescent state to the spikes becomes stronger, and the initial phase of the spikes becomes closer to the homoclinic orbit bifurcation, which leads the parameter region of the spikes to become shorter and then leads the number of spikes within a burst to turn less. It is the dynamical mechanism of the decrease of the spike number induced by the excitatory currents. The results enrich the nonlinear phenomenon and dynamical mechanism, present a novel viewpoint for the excitatory effect, and provide a new approach to modulating the neural bursting patterns.

Dynamic instability of dielectric elastomer actuators subjected to unequal biaxial prestress

The paper presents a Hamiltonian approach for extracting the dynamic instability parameters of homogeneously deforming dielectric elastomer actuators subjected to an unequal biaxial prestress, and driven by a suddenly applied electric load. The approach relies on setting up the balance between the kinetic, strain, and electrostatic energy at the point of maximum overshoot in an oscillation cycle. The equation of the stagnation curve, obtained by invoking aforestated statement of energy-balance, is operated upon by the condition of instability to determine the instability parameters. The underlying principles of the approach are elucidated by considering the Ogden family of hyperelastic material models. The approach is however portrayed generically, and hence, can be extended to the other hyperelastic material models of interest. The estimates of the dynamic instability parameters are corroborated by examining the saddle-node bifurcation points in the time-history response obtained by integrating the equation of motion. A parametric study is conducted to bring out the effect of unequal biaxial prestress, and the trends of variation of the critical electric field and the thickness-stretch on the onset of dynamic instability are presented. A quantitative comparison with the static instability parameters reveals that the dynamic instability gets triggered for electric fields that are lower than those corresponding to the static instability. In contrast, the maximum stretch experienced by the actuator at the dynamic instability is significantly higher than that at the static instability. The crucial inferences can find their potential use in the design of DEAs subjected to a transient motion.

A codimension two bifurcation in a railway bogie system

In this paper, a comprehensive analysis is presented to investigate a codimension two bifurcation that exists in a nonlinear railway bogie dynamic system combining theoretical analysis with numerical investigation. By using the running velocity V and the primary longitudinal stiffness $$K_{1x}$$ as bifurcation parameters the first and second Lyapunov coefficients are calculated to determine which kind of Hopf bifurcation can happen and how the system states change with the variance of the bifurcation parameters. It is found that multiple solution branches both stable and unstable coexist in a range of the bifurcation parameters which can lead to jumps in the lateral oscillation amplitude of the railway bogie system. Furthermore, reduce the values of the bifurcation parameters gradually. Firstly, the supercritical Hopf bifurcation turns into a subcritical one with multiple limit cycles both stable and unstable near the Hopf bifurcation point. With a further reduction in the bifurcation parameters two saddle-node bifurcation points emerge, resulting in the loss of the stable limit cycle between these two bifurcation points.

Analysis of the Steady-State Multiplicity Behavior for Polystyrene Production in the CSTR

Abstract The work proposes two different approaches where the first one is based on the tools of the system theory and the other is strongly related to the principle of heat balance, in order to analyze the abnormal phenomena of the continuous styrene polymerization reactors, i. e. the multiplicity behavior in the wide range of operating conditions. More precisely, the multiplicity behavior of polystyrene production in a continuous stirred tank reactor (CSTR) is carried out by the numerical simulations through the Van Heerden diagram and the phase plane. Furthermore, the bifurcation diagrams in terms of two different inputs including jacket temperature and volumetric flow rate of initiator predict the appearance of multiplicity behavior as well as the saddle-node bifurcation points. The results, firstly, verify that the multiplicity behavior of the system appears under considered operating conditions. Secondly, the analysis of bifurcation behavior gives the theoretical prediction of multiplicity behavior once the operating conditions vary due to the soft constraints or the effect of noise and disturbance.

Theoretical convergence guarantees versus numerical convergence behavior of the holomorphically embedded power flow method

The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to nonlinear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Padé approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Padé approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl’s theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl’s theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically.

Fast Determination of Saddle-Node Bifurcations via Parabolic Approximations in the Infeasible Region

This letter presents a simple methodology to quickly obtain the saddle-node bifurcation points of power systems by first solving a sequence of power flows that ends in the infeasible region and then performing a parabolic adjustment with the results of two infeasible cases. The proposed method exploits the ability of the factored load flow procedure to converge to complex solutions well beyond the maximum loading point. Test results on benchmark cases show that the new method yields a significant reduction in the computational cost, when compared to the previous approach based on exhaustive bisection search.

Dynamics and Physiological Roles of Stochastic Firing Patterns Near Bifurcation Points

Different stochastic neural firing patterns or rhythms that appeared near polarization or depolarization resting states were observed in biological experiments on three nervous systems, and closely matched those simulated near bifurcation points between stable equilibrium point and limit cycle in a theoretical model with noise. The distinct dynamics of spike trains and interspike interval histogram (ISIH) of these stochastic rhythms were identified and found to build a relationship to the coexisting behaviors or fixed firing frequency of four different types of bifurcations. Furthermore, noise evokes coherence resonances near bifurcation points and plays important roles in enhancing information. The stochastic rhythms corresponding to Hopf bifurcation points with fixed firing frequency exhibited stronger coherence degree and a sharper peak in the power spectrum of the spike trains than those corresponding to saddle-node bifurcation points without fixed firing frequency. Moreover, the stochastic firing patterns changed to a depolarization resting state as the extracellular potassium concentration increased for the injured nerve fiber related to pathological pain or static blood pressure level increased for aortic depressor nerve fiber, and firing frequency decreased, which were different from the physiological viewpoint that firing frequency increased with increasing pressure level or potassium concentration. This shows that rhythms or firing patterns can reflect pressure or ion concentration information related to pathological pain information. Our results present the dynamics of stochastic firing patterns near bifurcation points, which are helpful for the identification of both dynamics and physiological roles of complex neural firing patterns or rhythms, and the roles of noise.

Effects of thermo hydrodynamic (THD) floating ring bearing model on rotordynamic bifurcation

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

A detailed study of excursive (or Ledinegg) instability and density wave oscillations (DWOs) is carried out for two-phase flow in a natural circulation loop. The maps in the parametric space have been obtained, which indicate excursive instability (Ledinegg instability) as well as density wave oscillations (DWOs). The dynamic (DWOs) and static (Ledinegg) stability boundary on these maps have been drawn. These diagrams dealing with the interaction of Ledinegg instability and DWOs have rarely been reported, and detailed mathematical analysis of these maps is lacking. In the present work, a detailed study of such interactions is carried out. It is also noted that the Ledinegg instability is a manifestation of saddle node bifurcation or limit point (LP) for the dynamical system of natural circulation loop. Density wave oscillations in the dynamical system of natural circulation loop can be observed with the Hopf bifurcation. The detailed bifurcation analysis of these instabilities shows that stability maps can be divided into three broad regions, the first region can only have DWOs (Hopf bifurcation), and the second region can have Ledinegg as well as DWOs (Hopf bifurcation and saddle node bifurcation co-exists), while the third region can have only Ledinegg instability (saddle node bifurcation). The first and second region are separated by a Cusp point (CP) and it is found that between CP and Bogdanov-Takens (BT) bifurcation point both Hopf and LP exist, due to presence of both DWOs and Ledinegg instability. The region beyond BT bifurcation point only has Ledinegg or excursive instability. Moreover, DWOs in the first region have two types of Hopf bifurcation depending on the nature of limit cycles. These subcritical and supercritical Hopf bifurcations, are separated by the Generalized Hopf (GH) bifurcation. Abovementioned three regions are observed only for the Type II DWOs stability boundary, whereas Type I DWOs have only subcritical and supercritical Hopf bifurcations. The impact of design parameters on the saddle-node curve along with the shifting of interaction point (Bogdanov-Takens bifurcation) have been investigated as well.

Linear and non-linear vibrations of fluid-filled hollow microcantilevers interacting with small particles

Linear and non-linear vibrations of a U-shaped hollow microcantilever beam filled with fluid and interacting with a small particle are investigated. The microfluidic device is assumed to be subjected to internal flowing fluid carrying a buoyant mass. The equations of motion are derived via extended Hamilton's principle and by using Euler-Bernoulli beam theory retaining geometric and inertial non-linearities. A reduced-order model is obtained applying Galerkin's method and solved by using a pseudo arc-length continuation and collocation scheme to perform bifurcation analysis and obtain frequency response curves. Direct time integration of the equations of motion has also been performed by using Adams-Moulton method to obtain time histories and analyze transient cantilever-particle interactions in depth. It is shown that exploiting near resonant non-linear behavior of the microcantilever could potentially yield enhanced sensor metrics. This is found to be due to the transitions that occur as a matter of particle movement near the saddle-node bifurcation points of the coupled system that lead to jumps between coexisting stable attractors.

An extended Landau–Levich model for the dragging of a thin liquid film with a propagating surface acoustic wave

In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.

Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

How additive noise generates a phantom attractor in a model with cubic nonlinearity

Two-dimensional nonlinear system forced by the additive noise is studied. We show that an increasing noise shifts random states and localizes them in a zone far from deterministic attractors. This phenomenon of the generation of the new “phantom” attractor is investigated on the base of probability density functions, mean values and variances of random states. We show that increasing noise results in the qualitative changes of the form of pdf, sharp shifts of mean values, and spikes of the variance. To clarify this phenomenon mathematically, we use the fast–slow decomposition and averaging over the fast variable. For the dynamics of the mean value of the slow variable, a deterministic equation is derived. It is shown that equilibria and the saddle-node bifurcation point of this deterministic equation well describe the stochastic phenomenon of “phantom” attractor in the initial two-dimensional stochastic system.