Saddle-node Bifurcation Research Articles

Modeling secondary infections with temporary immunity and disease enhancement factor: Mechanisms for complex dynamics in simple epidemiological models

Modeling insights for epidemiological scenarios characterized by chaotic dynamics have been largely unexplored. A rigorous analysis of such systems are essential for a real predictive power and a more accurate disease control decision making. Motivated by dengue fever epidemiology, we study a basic SIR–SIR type model for the host population, capturing differences between primary and secondary infections. This model is the minimalistic version to previously suggested multi-strain models for dengue fever in which deterministic chaos was found in wider parameter regions. Without strain structure of pathogens, we consider temporary immunity after a primary infection and disease enhancement in a subsequent infection to identify to which extent these biological mechanisms can generate complex behavior in simple epidemiological models.Stability analysis of the system is performed using the classical linearization theory, and the qualitative behavior of the model is investigated with a detailed bifurcation analysis. Rich dynamical structures are identified, including the Bogdanov–Takens, cusp and Bautin bifurcations which has never been described in dengue fever epidemiology. Besides the conventional transcritical bifurcation, a backward bifurcation occurs for higher disease enhancement in secondary infections, exhibiting bi-stability when biological temporary immunity period is assumed. The backward bifurcation is formalized using the center manifold theory. While the Hopf and the global homoclinic bifurcation curves were computed numerically, analytical expressions for the transcritical and tangent bifurcations are obtained. The combination of temporary immunity and disease enhancement play a significant role in the complexity of the system dynamics, with chaotic behavior observed after including seasonal forcing.

Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing

We consider a single theta neuron with delayed self-feedback in the form of a Dirac delta function in time. Because the dynamics of a theta neuron on its own can be solved explicitly—it is either excitable or shows self-pulsations—we are able to derive algebraic expressions for the existence and stability of the periodic solutions that arise in the presence of feedback. These periodic solutions are characterized by one or more equally spaced pulses per delay interval, and there is an increasing amount of multistability with increasing delay time. We present a complete description of where these self-sustained oscillations can be found in parameter space; in particular, we derive explicit expressions for the loci of their saddle-node bifurcations. We conclude that the theta neuron with delayed self-feedback emerges as a prototypical model: it provides an analytical basis for understanding pulsating dynamics observed in other excitable systems subject to delayed self-coupling.

STEP-BY-STEP RESOLUTION OF SINGULARITIES AND STUDYING INTERACTIONS BETWEEN THEM

An explicit method, based on subsequent small perturbations, allowing one to study the algebraic and geometric nature of multiple isolated singularities of a polynomial vector field, is discussed. The main ingredients of the method are (i) establishing a canonical form of a singularity, (ii) explicit decomposition of a compound singularity into simpler ones, and (iii) deriving asymptotic laws of decomposition/collision of singularities. In particular, the saddle-node, pitchfork, and quadruple bifurcations of zeros of a polynomial vector field are considered from the various novel and perhaps unexpected angles. Several examples of subsequent phase portraits illustrating possible interactions between equilibrium of ODEs are also discussed.

Dynamics and Bifurcations in Filippov Type of Competitive and Symbiosis Systems

Filippov systems have found applications in various fields. This paper mainly studies five Lotka–Volterra models of Filippov type, including a competitive system with linear interaction between two species, a competitive system with Holling type II or type III functional response and a symbiosis system with Holling type II or type III functional response. We investigate the stability of all equilibria and the boundary equilibrium bifurcations of these systems, either a persistence bifurcation or a nonsmooth fold bifurcation. We present the numerical simulation results for each case. Consequently, based on both theory and simulation, we analyze the ecological aspects under the intervention of harvesting at different prescribed thresholds.

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

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.

Bifurcation and Chaos in a Phytoplankton–Zooplankton Model with Holling Type-II Response and Toxicity

Phytoplankton and zooplankton’s interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplankton should be in steady state but in previous eras, there have been a universal explosion in destructive plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So in this paper, we studied dynamical characteristics of a discrete-time phytoplankton–zooplankton model with Holling type-II predational functional response and toxic substance distribution. More specifically, we studied local stability with topological classification at equilibrium solutions, periodic points and possible bifurcation analysis of the model under consideration. It is proved that at trivial and semitrivial equilibrium solutions the phytoplankton–zooplankton model does not undergo flip bifurcation but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution. It is also proved that the interior equilibrium solution model does not undergo the fold bifurcation. We also studied chaos in the sense of feedback control strategy. Finally, extensive numerical simulations are provided to demonstrate theoretical results.

Scenario-based line switching for enhancing static voltage stability with uncertainty of renewables and loads

The fast-growing renewables, such as wind power and photovoltaic, introduce a large number of uncertainties into the power grid. The uncertainty of renewables, coupled with that of load, increases the complexity of the analysis of static voltage stability of power systems. In this paper, the nonlinear impact of the uncertainties from both renewables and loads on static voltage stability (limited by two types of typical bifurcations, i.e., saddle-node bifurcation and structure-induced bifurcation) is investigated and illustrated. To model the uncertainties, a scenario-based scenario construction method is presented to obtain the representative scenarios tailored for the static voltage stability problem which can avoid the mis-elimination of the extreme scenarios. Then, a stochastic line switching problem is formulated to maximize the load margin to the static voltage stability of the look-ahead power system. To fast identify the line solutions, a three-stage methodology, including a scenario construction stage, the identification stage, and the evaluation stage, is proposed to balance the speed and accuracy. Finally, the computational results based on the IEEE 118-bus power system, 1648-bus power system, and 3120-bus power system demonstrate the efficiency of the proposed methodology.

Generalized model for steady-state bifurcations without parameters in memristor-based oscillators with lines of equilibria

We demonstrate how the pitchfork, transcritical and saddle-node bifurcations of steady states observed in dynamical systems with a finite number of isolated equilibrium points occur in systems with lines of equilibria. The exploration is carried out by using the numerical simulation and linear stability analysis applied to a model of a memristor-based circuit. All the discussed bifurcation scenarios are considered in the context of models with the piecewise-smooth memristor current-voltage characteristic (Chua’s memristor), as well as on examples of oscillators with the memristor nonlinearity that is smooth everywhere. Finally, we compare the dynamics of ideal-memristor-based oscillators with the behavior of models taking into account the memristor forgetting effect. The presented results are obtained for electronic circuit models, but the studied bifurcation phenomena can be exhibited by systems with lines of equilibria of any nature.

Nonlinear dynamic analysis of parked large wind turbine blade considering harmonic inertial excitation using continuum mathematical model

In this paper, an investigation on the nonlinear dynamics of parked large wind turbine blades considering harmonic inertial excitation is presented. The investigation is based on the continuum mathematical model which is derived using the extended Hamilton principle. Employing the Euler–Bernoulli beam model, the equations of motion are derived by considering the flap-wise and edge-wise deformations as well as the cubic nonlinearity. The proposed model is validated through comparisons with both published results and numerical simulations. The nonlinear dynamics of the NREL 5-MW wind turbine blade is investigated numerically using a bifurcation toolbox MATCONT. The steady state primary resonances of the first three modes are compared and analyzed with varying excitation frequency and excitation direction. Results show that for parked blades the aerodynamic damping increases with the increase of inflow wind speed, and the response under low wind speed becomes more unfavorable than high wind speed. The blade response can suddenly jump up or down when the excitation direction varies slightly because of the saddle node bifurcation. The 3:1 internal resonance between the 1st and 3rd modes is also observed in this study, suggesting the possible interaction of the two modes during large amplitude vibration. Quasi-periodic motions can also be induced because of the Hopf bifurcation in the case of negative aerodynamic damping.

Bifurcation and stability analysis of static equilibrium configuration of curved pipe conveying fluid

We present bifurcation and stability analysis of static equilibrium configurations of a clamped-clamped curved pipe conveying fluid in this paper. To capture large deformations of the pipe, this pipe is modeled with an absolute nodal coordinate formulation (ANCF). A technique of parameter continuation is used to solve the equilibria of the governing equation of this pipe system. Effects of external force, flow velocity and arc angle on the nonlinear responses of the curved pipe conveying fluid are discussed in detail. In the case of the curved pipe without any external loadings, the system admits two stable equilibria and an unstable equilibrium when the flow velocity exceeds a critical value on which a saddle-node bifurcation occurs. This critical flow velocity that characterizes the onset of the multistability is increased as the arc angle of the pipe increases. In addition, a pitchfork bifurcation occurs along the branch of the unstable equilibria. When the curved pipe is subjected to a concentrated force at its midpoint, the pipe deforms only in symmetrical mode shapes and undergoes a snap-through buckling under variations in the concentrated force. In contrast, when the curved pipe is subject to a gravity and the arc angle of the pipe exceeds a critical value, a pitchfork bifurcation buckling is observed under variations in the gravity. Along the secondary branch of this pitchfork bifurcation, equilibria in asymmetric mode shapes are found. Finally, effects of system parameters on these critical buckling loads are carefully explored.

Stability and Bifurcation Analysis of Two-Species Prey–Predator Model Incorporating External Factors

Constant prey refuge with immigration and harvesting in two species would result in significant diversity in the dynamics of a prey–predator population. The phrase “refuge” increases the likelihood of prey population survival in the face of a predator population. Based on these findings, we created and examined a two-species prey–predator system with immigration and harvesting factors, including refuge to only prey population. All ecologically possible equilibrium points are studied for the proposed system. Routh–Hurwitz stability criterion is used for local stability analysis. Global stability of the interior equilibrium point is examined with a suitable Lyapunov function. Local bifurcation of the proposed system, such as saddle-node bifurcation, is analyzed. The conditions for the emergence of this bifurcation at the critical threshold near the nonhyperbolic equilibrium point are established by utilizing Sotomayor’s theorem. The transversality condition is validated for the occurrence of Hopf-bifurcation. The first Lyapunov number is exploited for determining the nature of Hopf bifurcating periodic solution. Finally, numerical simulations are illustrated to validate our theoretical predictions.

Bifurcation analysis of a modified continuum traffic flow model considering driver’s reaction time and distance

A modified continuum traffic flow model is established in this paper based on an extended car-following model considering driver’s reaction time and distance. The linear stability of the model and the Korteweg–de Vries (KdV) equation describing the density wave of traffic flow in the metastable region are obtained. In the new model, the relaxation term and the dissipation term exist at the same time, thus the type and stability of equilibrium solution of the model can be analyzed on the phase plane. Based on the equilibrium point, the bifurcation analysis of the model is carried out, and the existence of Hopf bifurcation and saddle-node bifurcation is proved. Numerical simulations show that the model can describe the complex nonlinear dynamic phenomena observed in freeway traffic, such as local cluster effect. Various bifurcations of the model, such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles, Cusp bifurcation and Bogdanov–Takens bifurcation, are also obtained by numerical simulations, and the traffic behaviors of some bifurcations are studied. The results show that the numerical solution is consistent with the analytical solution. Consequently, some nonlinear traffic phenomena can be analyzed and predicted from the perspective of global stability.

Bifurcation analysis based on new macro two-velocity difference model

Based on the new macroscopic two-velocity difference model, this paper analyzes the linear stability of the new model and studies the nonlinear bifurcation theory. First, the linear stability analysis method is used to study the stability conditions of the shock wave in the model. Then, considering the long wave model in the coarse-grained scale, the reduced perturbation method is used to analyze the characteristics of the traffic flow in the metastable region, and the solitary wave solution of the Korteweg-de Vries (KdV) equation in the metastable region is derived. In addition, by using the bifurcation analysis method, the type, and stability of the equilibrium solution are discussed and the existing conditions of the saddle-node bifurcation are proven. Then, taking the saddle-node bifurcation as the starting point, we draw the density space-time diagram and phase plane diagram of the system. It is proven that the newly proposed model can describe complex traffic phenomena such as stop-and-go and sudden changes in stability, which is of great help to solve traffic congestion.

Bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic’s future course when the public healthcare facilities, specifically the number of hospital beds, are limited. The feasibility and stability of the obtained equilibria are analyzed, and the basic reproduction number ([Formula: see text]) is obtained. We show that the system exhibits transcritical bifurcation. To show the existence of Bogdanov–Takens bifurcation, we have derived the normal form. We have also discussed a generalized Hopf (or Bautin) bifurcation at which the first Lyapunov coefficient evanescences. To show the existence of saddle-node bifurcation, we used Sotomayor’s theorem. Furthermore, we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure. An optimal control setting is studied analytically using optimal control theory, and numerical simulations of the optimal regimen are presented as well.

Controlling mode-coupling instability in friction-induced vibration by acceleration feedback

Self-excited friction-induced oscillation, such as brake squeal, railway wheel squeal, chattering in machine tools, etc., is harmful in many mechanical systems. In the present study, the efficacy of acceleration feedback control with a second order filter is theoretically studied to mitigate friction-induced mode-coupling instability. Two different control strategies based on acceleration feedback are employed to control friction-induced mode-coupling instability of a two degrees-of-freedom linear system. Linear stability analysis near the equilibrium point of the system shows the efficacy of control laws to stabilize the system. The optimum filter parameters are determined by the pole crossover optimization method to attenuate the transient response as well as to achieve greater relative stability of the system. The Robustness study confirms the ability of the controller to stabilize the system in the presence of uncertainties in system parameters. Analytical results match closely with simulation results. Finally, nonlinearity in the form of stiffness is incorporated into the system for which the dynamics of the system show the presence of subcritical, supercritical Hopf bifurcation, and cyclic fold bifurcation. In the presence of nonlinearity, the feedback control is also able to reduce the range of instability of static equilibrium as well as the amplitude of the self-excited oscillation. The method of averaging is used to determine the stability and amplitude of the limit cycle. For the nonlinear system, the bifurcation diagram is constructed in AUTO and validated with results obtained from MATLAB SIMULINK and the method of averaging.

Distinct bursting oscillations in parametrically excited Liénard system

In this paper, we report the occurrence of two distinct types of bursting oscillations (BOs) in a parametrically excited Liénard system. When the system is linearly damped, it undergoes saddle–node bifurcations leading to asymmetric bursting oscillations as the control parameter is varied. On the other hand, if it is damped with a nonlinear force, then it exhibits supercritical pitchfork and supercritical pitchfork/fold bifurcations resulting in symmetric bursting oscillations. We have identified the equilibrium points of the system for both the linear and nonlinear damping cases and have analyzed their stability. We have constructed the bifurcation and two-parameter stability plots to characterize the two distinct bursting oscillations and their transitions. Also, we have proposed an analog electronic model of our system and made real time experimental observations to validate our numerical results.

Dynamics of periodic solution to a electrostatic Micro-Electro-Mechanical system

A canonical mass–spring model of electrostatically actuated Micro-Electro-Mechanical System (MEMS) is studied. We investigate the existence and bifurcations of the periodic solution by means of the continuation theorem of Mańasevich and Mawhin with techniques of a priori estimate and bifurcation theory. The main results answer the open problem proposed by P. Torres in the known literature. It also reveals that the system undergoes saddle node and period doubling bifurcation leading to the multiplicity of periodic solution. Moreover, bistability of periodic solution generated by the combination of these bifurcations is detected for the first time, which provides some new insight into the so called pull-in instability in the system. At last, periodic orbits with different stability and corresponding time series are given to illustrate the results.

Bifurcation, stability, and critical slowing down in a simple mass–spring system

We analyse a simple mass–spring system as an accessible context for showcasing how continuous changes to system parameters can lead to critical transitions (‘tipping points’). Two kinds of transition are explored in particular: saddle–node bifurcations, due to changes in a mass forcing parameter a; and pitchfork bifurcations, due to changes in a spring separation parameter X. Both types of bifurcation arise as features of a cusp catastrophe characterised in X−a parameter space by the critical curve X2/3+a2/3=1, leading to hysteresis cycles, as described by C. Ong (2021), and non-reversible pitchfork catastrophes, which are discussed here for the first time. In each case we demonstrate critical slowing down of the oscillation period τ→∞ as the system approaches bifurcation.

Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect

Abstract We propose and analyze a Lotka-Volterra commensal model with an additive Allee effect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle-node bifurcations and transcritical bifurcations are derived by using Sotomayor’s theorem. Third, we give sufficient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee effect case, the additive Allee effect has a negative effect on the final density of both species, with increasing Allee effect, the densities of both species are decreasing. For the strong Allee effect case, the additive Allee effect is one of the most important factors that leads to the extinction of the second species. The additive Allee effect leads to the complex dynamic behaviors of the system.