Saddle-node Bifurcation Research Articles

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.

Dynamics of pendulum-based systems under human arm rotational movements

Wearable devices have applications in health monitoring and patient assistance devices. Harvesting energy from human motion kinetic energy by the pendulum-based electromagnetic generator is a promising way of extending the battery life of these devices. In this paper, the oscillatory and rotary dynamical response of pendulum-based systems under an open-circuit condition is investigated for two different models considering the lower arm’s rotation. The arm pitching model (in-plane excitation) represents the swing of the lower arm during walking. The arm rolling rotation model (out-of-plane excitation) represents the twisting type of motion of the lower arm movement associated with a specific kind of disorder, for example, tremors experienced by Parkinson’s patients. The governing equations of motion around stable equilibria are derived using Lagrange formulation, and their nondimensional forms are introduced for the two models. A special case of these models is developed for experimental validation and system identification, where the system parameters are identified by fitting frequency response functions on the experimental data. The system’s nonlinear dynamical response shows various bifurcations, including limit point bifurcation (i.e., saddle-node bifurcation), pitchfork bifurcation, a cascade of period-doubling bifurcations under amplitude or frequency sweeps. When the system shows a bi-stable response, the rotary response usually has a higher angular velocity amplitude than the oscillatory response. Basins of attraction plots for different operating conditions are presented to show the effect of initial conditions on the system’s type of steady-state response: oscillatory or rotary.

Modeling and Analysis of the Influence of Fear on the Harvested Modified Leslie–Gower Model Involving Nonlinear Prey Refuge

Understanding the effects of fear, quadratic fixed effort harvesting, and predator-dependent refuge are essential topics in ecology. Accordingly, a modified Leslie–Gower prey–predator model incorporating these biological factors is mathematically modeled using the Beddington–DeAngelis type of functional response to describe the predation processes. The model’s qualitative features are investigated, including local equilibria stability, permanence, and global stability. Bifurcation analysis is carried out on the temporal model to identify local bifurcations such as transcritical, saddle-node, and Hopf bifurcation. A comprehensive numerical inquiry is carried out using MATLAB to verify the obtained theoretical findings and understand the effects of varying the system’s parameters on their dynamical behavior. It is observed that the existence of these factors makes the system’s dynamic behavior richer, so that it involves bi-stable behavior.

The local bifurcation analysis of two preys stage-structured predator model with anti-predator behavior

This paper deals with two preys and stage-structured predator model with anti-predator behavior. Sufficient conditions that ensure the appearance of local and Hopf bifurcation of the system have been achieved, and it’s observed that near the free predator, the free second prey and the free first prey equilibrium points there are transcritical or pitchfork and no saddle node. While near the coexistence equilibrium point there is transcritical, pitchfork and saddle node bifurcation. For the Hopf bifurcation near the coexistence equilibrium point have been studied. Further, numerical analysis has been used to validate the main results.

Bifurcations and Bistability of an Age-Structured Viral Infection Model with a Nonmonotonic Immune Response

Recent studies have demonstrated that immune impairment is an essential factor in viral infection for disease development and treatment. In this paper, we formulate an age-structured viral infection model with a nonmonotonic immune response and perform dynamical analysis to explore the effects of both immune impairment and virus control. The basic infection reproduction number is derived for a general viral production rate, which determines the global stability of the infection-free equilibrium. For the immune intensity, we get two important thresholds, the post-treatment control threshold and the elite control threshold. The interval between the two thresholds is a bistable interval, where there are two immune-present infected equilibria. When the immune intensity is greater than the elite control threshold, only one immune-present infected equilibrium exists and it is stable. By assuming the death rate and virus production rate of infected cells to be constants, with the immune intensity as a bifurcation parameter, the system exhibits saddle-node bifurcation, transcritical bifurcation, and backward/forward bifurcation.

Nonlinear Behavior and Reduced-Order Models of Islanded Microgrid

An islanded microgrid consisting of grid-forming converters, being a high-order nonlinear system, exhibits rich nonlinear dynamical phenomena. The use of appropriate reduced-order models offers useful physical insights into the behavior of the system without the need for excessive computational resources. In this article, we derive a number of reduced-order models capable of describing the slow-scale dynamics of an islanded microgrid comprising a number of grid-forming converters. It is shown that slow-scale Hopf and homoclinic bifurcation behaviors arise from the stability of the voltage loops of grid-forming converters and are unrelated to the transmission network dynamics. Therefore, omitting the network dynamics does not affect the accuracy of reduced-order models in representing the slow-scale dynamics of the system. This is especially beneficial for modeling the microgrid with a complex transmission network. Furthermore, on this basis, all inner loops can be omitted when studying saddle-node bifurcation, leading to the development of power-flow-based reduced-order models. Finally, the stability of an islanded microgrid with a complex transmission network is evaluated.

Jump and Initial-Sensitive Excessive Motion of a Class of Relative Rotation Systems and Their Control via Delayed Feedback

Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback are investigated. The Method of Multiple Scales and the Melnikov method are applied to analyze critical conditions for bi-stability and initial-sensitive excessive motion, respectively. By introducing the fractal of basins of attraction and the erosion of the safe basin to depict jump and initial-sensitive excessive motion, respectively, the point mapping approach is used to present numerical simulations which are in agreement with the theoretical prediction, showing the validity of the analysis. It is found that jump between bistable attractors can be due to saddle–node bifurcation, while initial-sensitive excessive motion can be due to heteroclinic bifurcation. Under a positive coefficient of the gain, the types of delayed feedback can both be effective in reducing jump and initial-sensitive excessive motion. The results may provide some reference for the performance improvement of rotors and main bearings.

Dynamic behavior of self-acting gas-lubricated long journal bearing

Planar dynamics of a rotor supported by self-acting gas-lubricated journal bearing is investigated theoretically. Whirling of the shaft is one of the common problems in a gas-lubricated journal bearing system. As whirling increases, it becomes large enough to cause a destructive contact between the bearing surfaces. It is essential to predict the parameter range for the safe operation of the gas-lubricated journal bearing systems. An analytical model of the long, self-acting gas-lubricated journal bearing system is numerically integrated for analysis of steady-state solutions and limit cycle oscillations. Dynamic behavior of the system is analyzed with the variation of dimensionless bearing compressibility parameter Λ and dimensionless bearing stability parameter Λ2Ω. For a set of stability parameters Λ2Ω ranging from 0.01 to 1000, the critical value of Λ is determined which corresponds to angular speed ω of the rotor. With the decrease of parameter Λ2Ω, critical value Λ decreases. Limit cycles bifurcating from the equilibrium point at critical values are unstable for a low value of Λ2Ω=0.01 and stable for other higher values. In both the cases of stable and unstable limit cycles, limit cycles disappear beyond a compressibility parameter Λ and no further fold bifurcation of the limit cycle is detected. Results are presented as equilibrium point bifurcation and bifurcation of the limit cycle. The dynamics of the system and safe operating range are presented with diagrams. The results obtained are validated through trajectories of the journal center by numerical integration of the system of equations for different initial conditions presented in the figures.

Confinement effects in laminar swirling jets

This paper explores the effect of axial and radial confinement on the flow topology of laminar swirling jets. Its objective is to provide a unifying perspective toward swirling jet mechanics that connects earlier reports across a variety of confined and unconfined flow situations, and over a range of swirl ratio $S$ values. The analysis focuses separately on the influence of the jet's injection depth $L$ in a radially unconfined flow and of the chamber diameter $C$ in radially confined jets. In the former case, it shows that axial confinement influences strongly the jet's behaviour when $L$ is small, allowing bistable steady states: a central jet (CJ) solution with or without a small central recirculation zone (CRZ), and a wall jet (WJ) solution with a wide-open CRZ spreading along the reservoir's edge. Similar behaviour is identified for radially confined jets, where bistable CJ and WJ states appear over a range of moderate $C$ values, and the WJ state adopts a conical CRZ. In either case, the WJ solution appears or disappears via saddle–node bifurcations when the confinement is made sufficiently strong or weak, respectively. This dynamics is attributed to an exchange of dominance between central and outer low-pressure regions as the flow transitions from CJ to WJ, or vice versa. The findings demonstrate that the hysteresis associated widely with swirling jets is controlled not just by vortex breakdown, but also by confinement through the Coandă effect. Such confinement is found to alter significantly the state-space structure even when the walls are far from the nozzle.

Transient Turing patterns in a morphogenetic model

One of the most surprising mechanisms to explain the symmetry breaking phenomenon linked to pattern formation is known as Turing instabilities. These patterns are self-organising spatial structures resulting from the interaction of at least two diffusive species in specific conditions. The ideas of Turing have been used extensively in the specialised literature both to explain developmental patterns, as well as synthetic biology design. In the present work we study a previously proposed morphogenetic synthetic circuit consisting of two genes controlled by the same regulatory system. The spatial homogeneous version of this simple model presents a rich phase diagram, since it has a saddle-node bifurcation, spirals and limit cycle. Linear stability analysis and numerical simulations of the complete model allow us to determine the conditions for the development of Turing patterns, as well as transient patterns. We found that the parameter region where Turing patterns are found is much smaller than the region where transient patterns occur. We observed that the temporal evolution towards Turing patterns can present one or two different length scales, depending on the initial conditions. Further, we found a parameter region where the persistence time of the transient patterns depends on the distance between the parameters values on which the system is operating and the boundary of Turing patterns. This persistence time has a singularity at a critical distance that gives place to metastable patterns. To the best of our knowledge, transient and metastable patterns associated with Turing instabilities have not been previously reported in morphogenetic models.

Bifurcations of Sleep Patterns due to Homeostatic and Circadian Variation in a Sleep-Wake Flip-Flop Model

Differential equation-based physiological models of sleep-wake networks describe sleep-wake regulation by simulating the activity of wake- and sleep-promoting neuronal populations and the modulation of these populations by homeostatic and circadian ($\sim24$ h) drives. Here, we consider a sleep-wake flip-flop network model consisting of mutually inhibitory interactions between wake- and sleep-promoting neuronal populations. Motivated by changes in sleep behavior during early childhood as babies transition from napping to non-napping behavior, we vary homeostatic and circadian modulation and analyze effects on resulting sleep-wake patterns. To identify the types and sequences of bifurcations leading to changes in stable sleep-wake patterns in this piecewise-smooth model, we employ multiple mathematical methods, including fast-slow decomposition and numerical computation of circle maps. We find that the average daily number of sleeps exhibits a period adding sequence as the homeostatic time constants are reduced, and that the temporal circadian profile influences the number of observed solutions in the sequence. These solutions emerge through sequences of saddle-node and border collision bifurcations, where the particular sequence depends on parameter values. When the temporal circadian profile is steep, as is observed with long day lengths, some sleep patterns are lost and bistability of other patterns may occur. We analyze a limiting case of the temporal circadian waveform, a circadian hard switch model, to understand this loss of solutions. Generally, our holistic analysis approach provides an alternative analysis method for model systems that defy conventional numerical bifurcation analysis techniques.

The Parameter Design of Nonlinear Energy Sink Installed on the Jacket Pipe by Using the Nonlinear Dynamical Theory

The aim of this paper is to investigate the vibration suppression effect of a nonlinear energy sink (NES) on the wind-vortex-induced vibration of deep-water jacket pipes. Based on nonlinear dynamical theories, the coupled dynamic equation of a deep-water jacket pipe with NES was established, and the nonlinear response characteristics and motion stability of the system were studied by using the multi-scale mixed harmonic balance method and Routh–Hurwitz criterion. The paper also analyzed the influence of the nonlinear stiffness and damping parameters of an NES on the dynamic response of the pipe. It is concluded that when the nonlinear stiffness of an NES increases to a certain extent, the motion response of the pipe appeared as a saddle-node and Hopf bifurcation, and the vibration amplitude of the pipe is greatly suppressed. For a system with nonlinear motion, NES damping mainly affects the range of the unstable region. For the system without nonlinear motion, the increases in NES damping will reduce the amplitude near the resonance region. Selecting proper parameters of an NES nonlinear stiffness and damping can effectively suppress the vortex-induced vibration of pipes.

Modal analysis of viscoelastic three-dimensional rotating beam with generic tip mass

In this paper, the nonlinear dynamic modelling, free vibration and nonlinear analysis of three-dimensional viscoelastic Euler-Bernoulli beam undergoing hub motion incorporating substantial tip mass is accomplished. The kinetic, and potential energy of the system are derived in terms of velocities of general point on the link and center of gravity of the tip mass expressed in multi-floating co-ordinate systems. Hamilton's principle is used to obtain the governing equations of motion and linearly coupled boundary conditions. The material of the beam is considered as viscoelastic constituted of Kelvin-Voigt rheological model. The mass attached at the terminal end of the beam is assumed to have eccentricity in axial, lateral, as well as transverse directions. Free vibration analysis is performed on the linearized system model to obtain the transcendental eigenfrequency equation. Further, the dynamic equations of motion of beam are discretized using the obtained mode shapes and the response of system under rotary motion of hub is investigated. The steady state solutions and frequency response curves exhibiting bi-stable and tri-stable regions are obtained using method of multiple scales. The bifurcation diagrams of the system are studied for resonance conditions when the frequency of the rotary motion becomes equal or nearly equal to the normalized beam frequencies. The saddle node and pitchfork bifurcations exhibiting multiple solutions and jump phenomena are observed and investigated to avoid catastrophic failure of the system. The numerical simulations of modal parameters of the system, nonlinear characteristics, and their parametric dependency is discussed thoroughly.

Consensus, Polarization and Hysteresis in the Three-State Noisy q-Voter Model with Bounded Confidence.

In this work, we address the question of the role of the influence of group size on the emergence of various collective social phenomena, such as consensus, polarization and social hysteresis. To answer this question, we study the three-state noisy q-voter model with bounded confidence, in which agents can be in one of three states: two extremes (leftist and rightist) and centrist. We study the model on a complete graph within the mean-field approach and show that, depending on the size q of the influence group, saddle-node bifurcation cascades of different length appear and different collective phenomena are possible. In particular, for all values of , social hysteresis is observed. Furthermore, for small values of , disagreement, polarization and domination of centrists (a consensus understood as the general agreement, not unanimity) can be achieved but not the domination of extremists. The latter is possible only for larger groups of influence. Finally, by comparing our model to others, we discuss how a small change in the rules at the microscopic level can dramatically change the macroscopic behavior of the model.

Temperature elevations can induce switches to homoclinic action potentials that alter neural encoding and synchronization

Almost seventy years after the discovery of the mechanisms of action potential generation, some aspects of their computational consequences are still not fully understood. Based on mathematical modeling, we here explore a type of action potential dynamics – arising from a saddle-node homoclinic orbit bifurcation - that so far has received little attention. We show that this type of dynamics is to be expected by specific changes in common physiological parameters, like an elevation of temperature. Moreover, we demonstrate that it favours synchronization patterns in networks – a feature that becomes particularly prominent when system parameters change such that homoclinic spiking is induced. Supported by in-vitro hallmarks for homoclinic spikes in the rodent brain, we hypothesize that the prevalence of homoclinic spikes in the brain may be underestimated and provide a missing link between the impact of biophysical parameters on abrupt transitions between asynchronous and synchronous states of electrical activity in the brain.