Saddle-node Bifurcation Point Research Articles

Hysteresis behavior and generalized Hopf bifurcation in a three-degrees-of-freedom aeroelastic system with concentrated nonlinearities

A three-degrees-of-freedom aeroelastic system with concentrated structural nonlinearities is considered. The aerodynamic loads in the model are simulated by using unsteady aerodynamic forces. First, we evaluate the linear stability region of the aerodynamic system employing the Lienard–Chipart criterion so that the stability boundaries of the center of gravity position and the uncoupled pitch natural frequency are obtained. Next, we present the distribution of the first Lyapunov coefficient l1 in the parameter space by using the Poincaré projection method to determine the type of Hopf bifurcation (subcritical or supercritical). We identify three types of hysteresis loops associated with Hopf bifurcations: the first type contains one stable equilibrium point and one stable limit cycle, the second type consists of one stable equilibrium point and two stable limit cycles, and the third type comprises two stable limit cycles. Nonlinear analysis shows that the variations in hysteresis loop types are mainly caused by the nonlinear flutter critical points, namely the saddle–node bifurcation points of limit cycles. Furthermore, we determine a degenerate Hopf bifurcation type known as generalized Hopf bifurcation (Bautin bifurcation) and its corresponding second Lyapunov coefficient, l2. Subsequently, we conduct nonlinear flutter analyses for positive and negative values of l2 by using bifurcation diagrams and phase portraits. The results show that when the uncoupled pitch natural frequency is used as the control parameter, the occurring generalized Hopf bifurcations are all supercritical (l2<0). However, when the center of gravity position is used as the control parameter, the generalized Hopf bifurcations that occur have both supercritical and subcritical (l2>0) modes.

Iterative Searching Method for Closest Saddle-Node Bifurcation Point in Multi-Infeed HVDC Systems

Iterative Searching Method for Closest Saddle-Node Bifurcation Point in Multi-Infeed HVDC Systems

A Method for Stabilizing the Vibration Amplitude of a Flip-Flow Vibrating Screen Using Piecewise Linear Springs

The flip-flow vibrating screen (FFVS) is a novel multi-body screening equipment that utilizes vibrations to classify bulk materials in the field of screening machinery. The relative amplitude of FFVSs determines the tension and ejection intensity of elastic flip-flow screen panels, which is a critical operating parameter affecting the screening performance. However, FFVSs generally suffer from large variations of relative amplitude caused by the loading of materials and the changes in shear spring stiffness (the temperature changes of the shear springs lead to their stiffness changes), which significantly reduce the screening efficiency and lifespan of FFVSs. To address this problem, this paper proposes a nonlinear stiffness-based method for stabilizing the vibration amplitude of FFVSs using piecewise linear springs. By introducing these springs between the two frames, the sensitivity of the relative amplitude to shear spring stiffness is reduced, thereby achieving the stabilization of the relative amplitude of FFVSs. In this study, the variations of the vibration amplitude of the FFVS due to the loading of materials and the changes in shear spring stiffness were first demonstrated in a reasonable operating frequency range. Then the reasonable operating frequency range and dynamics of the resultant nonlinear flip-flow vibrating screen (NFFVS) with piecewise linear springs were investigated using the harmonic balance method (HBM) and the Runge–Kutta numerical method. The operating frequency region for the NFFVS lies between the critical frequency ωcs and the frequency ωlb corresponding to the saddle-node bifurcation point. Finally, a test rig was designed to validate the theoretical predictions. Theoretical and experimental results demonstrate that piecewise linear springs can effectively stabilize the relative amplitude of the FFVS.

Two-parametric unfoldings for planar invisible double-tangency singularities.

We consider the family of planar discontinuous piecewise linear systems in two regions with a straight line as boundary and one tangency point in each region. First, we recall a seven-parametric canonical form for such a family. Next, we select two parameters as bifurcation parameters. One of them represents the distance between the tangency points. It is known that this bifurcation parameter unfolds the called pseudo-Hopf bifurcation. The coefficient that determines the stability of the crossing limit cycle that emerges from this bifurcation mechanism, called the first Lyapunov coefficient, is considered our second bifurcation parameter. Finally, with these two bifurcation parameters, we give a two-parametric unfolding for the invisible fold-fold and focus-fold singularities. The bifurcation diagram for each unfolding consists of two curves of bifurcation points: a curve of pseudo-Hopf bifurcation points and a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin (pB) bifurcation because of the dynamical behavior that is same as the Bautin bifurcation for smooth dynamical systems.

A finding of the maximal saddle-node bifurcation for systems of differential equations

A variational method is presented for directly finding the bifurcation point of nonlinear equations as the saddle-node point of the extended nonlinear Rayleigh quotient. In the main result, this method is justified for finding the maximum saddle-node bifurcation point of the set of stable positive solutions to a system of equations with nonlinearities of convex-concave type.

Periodic solutions and frequency lock-in of vortex-induced vibration energy harvesters with nonlinear stiffness

The phenomenon of amplitude jump usually means the presence of multi-solution ranges and unstable periodic solutions, which has not been fully explored in the field of vortex-induced vibration energy harvesting. In this study, semi-analytical periodic solutions of vortex-induced vibration energy harvesters (VIVEHs) are obtained using the incremental harmonic balance method, and their stabilities are determined via the Floquet theory. It is found that the VIVEH with linear stiffness and those with three-order nonlinear stiffness have four saddle-node bifurcation points and two multi-solution ranges, which leads to the occurrence of amplitude jumps. The hardening effect can shift frequency lock-in regions to higher wind speeds, thus increasing the aerodynamic forces, and realizing wide wind speed bandwidths and high voltage outputs. The wind tunnel experiments verify the accuracy of theoretical calculations, the occurrence of amplitude jump phenomena, and the existence of multi-solution ranges. The VIVEH with nonlinear stiffness experimentally reached a maximum voltage of 9.87 V, and a frequency lock-in region of 2.27–5.36 m/s.

Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential equations

Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor.

Modeling the Dynamics of a Ratio-Dependent Leslie–Gower Predator–Prey System with Strong Allee Effect

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Numerical simulation of the bifurcation-remerging process and intermittency in an undriven direct current glow discharge.

As a complex nonlinear medium, gas discharge plasma can exhibit various nonlinear discharge behaviors. In this study, in order to investigate the chaos phenomenon in the subnormal glow region of an undriven direct current glow discharge, a two-dimensional plasma fluid model is established coupled with a circuit model as a boundary condition. Using the applied voltage as control parameter in the simulation, the complete period-doubling bifurcation and inverse period-doubling bifurcation processes in the oscillation region are found, and the influence of the applied voltage on the spatiotemporal distribution of plasma parameters during the bifurcation-remerging process is examined. In addition, the spatial distribution of the plasma parameters of the bifurcation-remerging process is also examined. Also, a series of periodic windows are present in the chaotic region, where the positions and relative order are generally consistent with the universal sequence. Additionally, this study showed that the intermittent chaos appears near the period-3 window, and the bursts appearing in the approximate periodic motion becomes more and more frequent as the control parameters move away from the saddle-node bifurcation point, which shows the typical type-I intermittent chaos characteristics.

Nonlinear super-harmonic and sub-harmonic resonance analysis of the horizontal-axis wind turbine blades using multiple scale method including the pre-twist effects

The focus of this paper is on the nonlinear secondary resonance of HAWT blades under the super-harmonic and sub-harmonic excitations. To pursue this goal, a 3-DOF dynamic model of the blade is presented including the pre-twist effects and then the method of multiple scale (MS) method is employed to investigate the nonlinear secondary resonance. Nine different cases of secondary resonances including 3 (2) super-harmonic resonance cases of order 2 (3) as well as 2 (2) sub-harmonic resonance cases of order 1/2 (1/3) are detected. For each case, the amplitude-frequency and phase-frequency responses are derived analytically. The stability of each branch of the solution is evaluated through the Hartman-Grobman theorem and several saddle-node bifurcation points are revealed, considering the NREL 5 MW HAWT blade as the case study. Moreover, a parametric study is carried out to study the effects of damping, pre-twist, angular velocity, and excitation force amplitude. It is found that these parameters have a great influence on amplitude-frequency response (AFR) curves. Eventually, some comparisons are made to demonstrate the validity of the formulation and simulation results.

Pseudo-Bautin Bifurcation in 3D Filippov Systems

Consider a piecewise linear dynamical system in [Formula: see text], divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems.

Viscous froth model applied to the motion and topological transformations of two-dimensional bubbles in a channel: three-bubble case.

The viscous froth model is used to predict rheological behaviour of a two-dimensional (2D) liquid-foam system. The model incorporates three physical phenomena: the viscous drag force, the pressure difference across foam films and the surface tension acting along them with curvature. In the so-called infinite staircase structure, the system does not undergo topological bubble neighbour-exchange transformations for any imposed driving back pressure. Bubbles then flow out of the channel of transport in the same order in which they entered it. By contrast, in a simple single bubble staircase or so-called lens system, topological transformations do occur for high enough imposed back pressures. The three-bubble case interpolates between the infinite staircase and simple staircase/lens. To determine at which driving pressures and at which velocities topological transformations might occur, and how the bubble areas influence their occurrence, steady-state propagating three-bubble solutions are obtained for a range of bubble sizes and imposed back pressures. As an imposed back pressure increases quasi-statically from equilibrium, complex dynamics are exhibited as the systems undergo either topological transformations, reach saddle-node bifurcation points, or asymptote to a geometrically invariant structure which ceases to change as the back pressure is further increased.

Static Voltage Stability Analysis Based on Multi-Dimensional Holomorphic Embedding Method

The importance of voltage stability problems has progressively increased in tandem with continuously increasing electric power demand. To fast calculate the V-Q sensitivity values of multiple operating conditions which characterize generations/loads scaled in different proportions and avoid calculating power flow repeatedly, we propose a multi-variable V-Q sensitivity method based on the multi-dimensional holomorphic embedding method (MDHEM) for static voltage stability analysis. First, the MDHEM is utilized to calculate the power flow analytical expressions. Second, based on recursion theory, bus voltage expressions are used to derive voltage magnitude expressions, and multi-variable V-Q sensitivity expressions are further derived offline. Third, the V-Q sensitivity values are calculated online to identify the weak buses under multiple operating conditions and judge the saddle-node bifurcation point (SNBP). Numerical results are presented to better explain the high computational efficiency and accuracy of the proposed method.

Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Quorum-sensing induced transitions between bistable steady-states for a cell-bulk ODE-PDE model with lux intracellular kinetics.

Intercellular signaling and communication are used by bacteria to regulate a variety of behaviors. In a type of cell-cell communication known as quorum sensing (QS), which is mediated by a diffusible signaling molecule called an autoinducer, bacteria can undergo sudden changes in their behavior at a colony-wide level when the density of cells exceeds a critical threshold. In mathematical models of QS behavior, these changes can include the switch-like emergence of intracellular oscillations through a Hopf bifurcation, or sudden transitions between bistable steady-states as a result of a saddle-node bifurcation of equilibria. As an example of this latter type of QS transition, we formulate and analyze a cell-bulk ODE-PDE model in a 2-D spatial domain that incorporates the prototypical LuxI/LuxR QS system for a collection of stationary bacterial cells, as modeled by small circular disks of a common radius with a cell membrane that is permeable only to the autoinducer. By using the method of matched asymptotic expansions, it is shown that the steady-state solutions for the cell-bulk model exhibit a saddle-node bifurcation structure. The linear stability of these branches of equilibria are determined from the analysis of a nonlinear matrix eigenvalue problem, called the globally coupled eigenvalue problem. The key role on QS behavior of a bulk degradation of the autoinducer field, which arises from either a Robin boundary condition on the domain boundary or from a constant bulk decay, is highlighted. With bulk degradation, it is shown analytically that the effect of coupling identical bacterial cells to the bulk autoinducer diffusion field is to create an effective bifurcation parameter that depends on the population of the colony, the bulk diffusivity, the membrane permeabilities, and the cell radius. QS transitions occur when this effective parameter passes through a saddle-node bifurcation point of the Lux ODE kinetics for an isolated cell. In the limit of a large but finite bulk diffusivity, it is shown that the cell-bulk system is well-approximated by a simpler ODE-DAE system. This reduced system, which is used to study the effect of cell location on QS behavior, is easily implemented for a large number of cells. Predictions from the asymptotic theory for QS transitions between bistable states are favorably compared with full numerical solutions of the cell-bulk ODE-PDE system.

Fast–slow variable dissection with two slow variables related to calcium concentrations: a case study to bursting in a neural pacemaker model

Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear behavior alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast–slow variable dissection method with two slow variables is proposed to analyze the complex bursting simulated in a four-dimensional neuronal model to describe firing associated with pathological pain. The lumenal (\(C_\mathrm{lum}\)) and intracellular (\(C_\mathrm{in}\)) calcium concentrations are the slowest variables, respectively, in the quiescent state and burst duration. Questions encountered when the traditional method with one slow variable is used. When \(C_\mathrm{lum}\) is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With \(C_\mathrm{in}\) chosen as slow variable, \(C_\mathrm{lum}\) value of the initiation point of burst is far from the saddle-node bifurcation point, due to \(C_\mathrm{lum}\) not contained in the equation of the membrane potential. To overcome this problem, both \(C_\mathrm{in}\) and \(C_\mathrm{lum}\) are regarded as slow variables; the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing the \(C_\mathrm{lum}\) dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast–slow variable dissection and a deep understanding of the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

Numerical computations for bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations

We numerically study solitary waves in the coupled nonlinear Schr\"odinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.

On the free streamline solutions of the Falkner–Skan equation

The problem of the free streamline solutions of the Falkner–Skan equation is revisited in this paper. Until now, such solutions were found for negative values of the pressure gradient parameter β only. All of them are associated with slip velocities −1<f′0<∞ and emerge from the trivial solution f=η of the problem. The present paper shows, however, that in the positive range 1<β<∞, just below the interval −1<f′0<∞, a further branch of free streamline solutions of slip velocities −2≤f′0<−1 exists. These new solutions emerge from an exact solution of the Falkner–Skan equation which describes the flow in a converging channel with moving boundaries at the saddle–node bifurcation point f′0=−2,f′′0=0. For the large-β asymptotics of this solution branch a new algorithm is presented. The occurrence of further free streamline solutions in the range β<0, as well as the existence of free streamlines of vanishing slip velocities, f′′0=f′0=0, both for positive and negative values of β is also addressed in the paper. The flow inside a cone is also considered shortly and the occurrence of free streamline solutions is pointed out also in this case.

Influence of Allee effect in prey and hunting cooperation in predator with Holling type-III functional response

Cooperative hunting is a widespread phenomenon in the predator population which promotes the predation and the coexistence of the prey-predator system. On the other hand, the Allee effect among prey may drive the system to instability. In this work, we consider a prey-predator model with Type-III functional response involving the hunting cooperation in predator and Allee effect in the growth rate of the prey population. Here our aim mainly is to demonstrate the impact of both the Allee effect and hunting cooperation on the system dynamics. Mathematically our analysis primarily focuses on the stability of coexisting equilibrium points and all possible bifurcations that the system may exhibit. We have observed transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and SN-TC bifurcation point respectively in the course of studying the global dynamics.

Dynamical systems analysis as an additional tool to inform treatment outcomes: The case study of a quantitative systems pharmacology model of immuno-oncology.

Quantitative systems pharmacology (QSP) proved to be a powerful tool to elucidate the underlying pathophysiological complexity that is intensified by the biological variability and overlapped by the level of sophistication of drug dosing regimens. Therapies combining immunotherapy with more traditional therapeutic approaches, including chemotherapy and radiation, are increasingly being used. These combinations are purposed to amplify the immune response against the tumor cells and modulate the suppressive tumor microenvironment (TME). In order to get the best performance from these combinatorial approaches and derive rational regimen strategies, a better understanding of the interaction of the tumor with the host immune system is needed. The objective of the current work is to provide new insights into the dynamics of immune-mediated TME and immune-oncology treatment. As a case study, we will use a recent QSP model by Kosinsky et al. [J. Immunother. Cancer 6, 17 (2018)] that aimed to reproduce the dynamics of interaction between tumor and immune system upon administration of radiation therapy and immunotherapy. Adopting a dynamical systems approach, we here investigate the qualitative behavior of the representative components of this QSP model around its key parameters. The ability of T cells to infiltrate tumor tissue, originally identified as responsible for individual therapeutic inter-variability [Y. Kosinsky et al., J. Immunother. Cancer 6, 17 (2018)], is shown here to be a saddle-node bifurcation point for which the dynamical system oscillates between two states: tumor-free or maximum tumor volume. By performing a bifurcation analysis of the physiological system, we identified equilibrium points and assessed their nature. We then used the traditional concept of basin of attraction to assess the performance of therapy. We showed that considering the therapy as input to the dynamical system translates into the changes of the trajectory shapes of the solutions when approaching equilibrium points and thus providing information on the issue of therapy.