Saddle-node Bifurcation Research Articles

Equilibria and bifurcations in contact dynamics

We provide a systematic study of equilibria of contact vector fields and the bifurcations that occur generically in 1-parameter families, and express the conclusions in terms of the Hamiltonian functions that generate the vector fields. Equilibria occur at points where the zero-level set of the Hamiltonian function is either singular or is tangent to the contact structure. The eigenvalues at an equilibrium have an interesting structure: there is always one particular real eigenvalue of any equilibrium, related to the contact structure, that we call the principal coefficient, while the other eigenvalues arise in quadruplets, similar to the symplectic case except they are translated by a real number equal to half the principal coefficient. There are two types of codimension 1 equilibria, named Type I, arising where the zero-set of the Hamiltonian is singular, and Type II where it is not, but there is a degeneracy related again to the principal coefficient and the contact of the zero-level set of the Hamiltonian with the contact structure. Both give rise generically to saddle-node bifurcations. Some special features include: (i) for Type II singularities, Hopf bifurcations cannot occur in dimension 3, but they may in dimension 5 or more; (ii) for Type I singularities, a fold-Hopf bifurcation can occur with codimension 1 in any dimension, and (iii) again for Type I, and in dimension at least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass through the imaginary axis simultaneously together with one through the origin) may also occur with codimension 1.

Catastrophic and noncatastrophic population crashes in a bitrophic system with dynamic additional food provision to cooperative predators.

In this article we contemplate the dynamics of an additional food-provided prey-predator system. We assume that the behavior of cooperative predators induces fear in prey, which radically affects the prey's birth and death rates. We observe that the structural instability imposed by strong cooperative hunting among predators goes away with higher intensities of fear levels affecting the prey's reproductive output and mortality. High levels of prey refuge are not conducive to the survival of predators. In such a situation, adequate supply of high-quality additional food is favorable regarding the persistence and stability of the system. Interestingly, the system potentially exhibits two stable configurations under identical ecological conditions by allowing different bifurcation scenarios, including saddle-node and backward bifurcations, and associated hysteresis effects with prey refuge along with additional food quantity and quality. In the stochastic environment, the system experiences critical transitions through bifurcation-induced tipping events with time-varying additional food for predators. Enhanced disturbance events promote noise-induced switching and tipping events. Finally, our investigation explores whether impending population crashes resulting from the variability of additional food quantity and quality can reliably be predicted using early warning signals in the context of redshifted noise. Overall, our results may provide insights for finding control strategies in the context of community ecology.

Axle characteristics and vehicle limit cycle behaviour

ABSTRACT The paper discusses the limit cycle performance of the nonlinear 2DOF single track vehicle model. That means that periodic solutions arise under constant vehicle speed and steering angle, under excessive handling conditions, with axle characteristics playing a dominant role. These characteristics are modelled using the Magic Formula tyre model, with the model parameters varied to investigate the sensitivity of peak friction and slip stiffness on limit cycle occurrence. In the transition to limit cycling, different types of bifurcation are found, being mainly of homoclinic and Hopf type, and sometimes saddle-node bifurcation, or bifurcation involving unstable limit cycles. Dealing with models being strongly nonlinear, the research includes qualitative analysis tools being appropriate for such models, such as phase plane analysis and the application of the stability diagram. The paper discusses two conjectures on the necessary and sufficient conditions on axle characteristics for the existence of limit cycle behaviour, and the relationship between steering angle and vehicle speed at the transition to limit cycling. Limit cycle performance and the impact on global stability are elements of the dynamic vehicle performance under high acceleration conditions, and therefore very relevant for excessive handling situations and our understanding of active safety under critical circumstances.

Analysis of a Delayed Multiscale AIDS/HIV-1 Model Coupling Between-Host and Within-Host Dynamics

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h<1, the disease-free equilibrium is globally asymptotically stable; if R˜0h>1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.

First Passage Times of Long Transient Dynamics in Ecology.

Long transient dynamics in ecological models are characterized by extended periods in one state or regime before an eventual, and often abrupt, transition. One mechanism leading to long transient dynamics is the presence of ghost attractors, states where system dynamics slow down and the system lingers before eventually transitioning to the true attractor. This transition results solely from system dynamics rather than external factors. This paper investigates the dynamics of a classical herbivore-grazer model with the potential for ghost attractors or alternative stable states. We propose an intuitive threshold for first passage time analysis applicable to both bistable and ghost attractor regimes. By formulating the first passage time problem as a backward Kolmogorov equation, we examine how the mean first passage time changes as parameters are varied from the ghost attractor regime to the bistable one, through a saddle-node bifurcation. Our results reveal that the mean and variance of first passage times vary smoothly across the bifurcation threshold, eliminating the deterministic distinction between ghost attractors and bistable regimes. This work suggests that first passage time analysis can be an informative way to classify the length of a long transient. A better understanding of the duration of long transients may contribute to greater ecological understanding and more effective environmental management.

Bifurcations to bursting oscillations in memristor-based FitzHugh-Nagumo circuit

To characterize neuronal firing activities and elucidate its bifurcation mechanisms, a memristor-based FitzHugh-Nagumo (FHN) circuit is designed based on the FHN circuit architecture combined with a first-order memristive simulator, and its normalized system with periodic and quasi-periodic bursting oscillations is established. With the change of externally applied excitation, the memristor-based FHN system has the time-varying equilibrium point where the number, position and stability evolve slowly over time. In an evolution period of the time-domain waveform, different fold and/or Hopf bifurcations are triggered, resulting in periodic or quasi-periodic bursting oscillations. To explain the intrinsic bifurcation mechanisms, the fold and Hopf bifurcation sets are built and the transitions between the resting and spiking states are demonstrated, thus identifying the Hopf/fold and Hopf/Hopf bursting oscillations. Finally, based on the circuit simulation model, analog circuit simulations and hardware circuit measurements are developed for the memristor-based FHN circuit to confirm MATLAB numerical simulations. In addition, it is worth noting that the proposed circuit is a simple non-autonomous memristive neuron circuit that is particularly easy to physically implement.

Nonlinear dynamic morphing of conical bistable dielectric elastomer actuator

The bistable dielectric elastomer actuator (BDEA) possesses two stable positions which offers notable advantages of stable-state self-maintenance, fast response, and threshold snap-through characteristic in comparison with conventional dielectric elastomers. However, the strong nonlinearity induced by the coupling among materials, structure, and electrostatic fields greatly affect the dynamic response and gives rise to stability issues. Hence, a novel BDEA is proposed by introducing DEA film centrally connected with one mass block and linear spring, and the bistability can be adjusted by applying external voltage. A nonlinear dynamical model considering the electro-mechanical coupling effects is established using the Euler-Lagrange method, with which the snap-through procedure is theoretically analyzed and validated through the analytic method and finite element method. The influences of the electric actuation and structural parameters on the number of stable states and natural frequency are analyzed. Additionally, the supercritical pitchfork bifurcation and saddle-node bifurcation are investigated through dynamic analysis under forced vibration. Furthermore, the ranges of electrical actuation parameters can be determined for preventing the bifurcation phenomena under parametric excitations. Moreover, an active morphing strategy for achieving nonlinear dynamic morphing between steady states of BDEA using drive voltage is obtained, thereby enhancing the versatility of conical BDEA.

Swirling electrolyte. Part 1. Hierarchy of catastrophic transitions in steady annular flows

This study extends our previous work (McCloughan & Suslov, J. Fluid Mech., vol. 887, 2020, A23), where the existence of a saddle-node bifurcation of steady axisymmetric electrolyte flows driven by the Lorentz force in a shallow annular domain was first reported. Here we perform further weakly nonlinear analysis over a wider range of the governing parameters to demonstrate that the previously reported saddle-node bifurcation is a local feature of a global fold catastrophe, which, in turn, is a part of cusp catastrophe occurring as the thickness of the fluid layer increases. The amplitude equation characterising multiple flow solutions in the finite vicinity of catastrophe points is derived. The sensitivity of its coefficients and solutions to the distance from the catastrophe points is assessed demonstrating the robustness of the used analytical procedure. The asymptotic flow solution past the catastrophe point is subsequently obtained and its topology is explored confirming the existence of the secondary circulation in the bulk of flow (two-tori background flow structure). The latter is argued to lead to the appearance of experimentally observable vortices on the fluid surface. The rigorous justification of this conjecture is to be given in Part 2 of the study.

Stabilization of Laminars in Chaos Intermittency

Chaos intermittency is composed of a laminar regime, which exhibits almost periodic motion, and a burst regime, which exhibits chaotic motion; it is known that in chaos intermittency, switching between these regimes occurs irregularly. In the laminar regime of chaos intermittency, the periodic solution before the saddle node bifurcation is closely related to its generation, and its behavior becomes periodic in a short time; the laminar is not, however, a periodic solution, and there are no unstable periodic solutions nearby. Most chaos control methods cannot be applied to the problem of stabilizing a laminar response to a periodic solution since they refer to information about unstable periodic orbits. In this paper, we demonstrate a control method that can be applied to the control target with laminar phase of a dynamical system exhibiting chaos intermittency. This method records the time series of a periodic solution prior to the saddle node bifurcation as a pseudo-periodic orbit and feeds it back to the control target. We report that when this control method is applied to a circuit model, laminar motion can be stabilized to a periodic solution via control inputs of very small magnitude, for which robust control can be obtained.

Bistability and Bifurcations of Tumor Dynamics with Immune Escape and the Chimeric Antigen Receptor T-Cell Therapy

Tumor immune escape refers to the inability of the immune system to clear tumor cells, which is one of the major obstacles in designing effective treatment schemes for cancer diseases. Although clinical studies have led to promising treatment outcomes, it is imperative to design theoretical models to investigate the long-term treatment effects. In this paper, we develop a mathematical model to study the interactions among tumor cells, immune escape tumor cells, and T lymphocyte. The chimeric antigen receptor (CAR) T-cell therapy is also described by the mathematical model. Bifurcation analysis shows that there exists backward bifurcation and saddle-node bifurcation when the immune intensity is used as the bifurcation parameter. The proposed model also exhibits bistability when its parameters are located between the saddle-node threshold and backward bifurcation threshold. Sensitivity analysis is performed to illustrate the effects of different mechanisms on the backward bifurcation threshold and basic immune reproduction number. Simulation studies confirm the bifurcation analysis results and predict various types of treatment outcomes using different CAR T-cell therapy strengths. Analysis and simulation results show that the immune intensity can be used to control the tumor size, but it has no effect on the control of the immune escape tumor size. The introduction of the CAR T-cell therapy will reduce the immune escape tumor size and the treatment effect depends on the CAR T-cell therapy strength.

Bifurcation Analysis of a Discrete Amensalism Model

By using model discretization of the piecewise constant argument method, a discrete amensalism model with nonselective harvesting and Allee effect is formulated. The dynamic analysis of the model is studied and the existence and stability of the equilibrium point are discussed. The fold bifurcation and flip bifurcation at the equilibrium point of the system are proved by using the bifurcation theory and the center manifold theorem. In order to control flip bifurcation and restore the system to a stable state, a hybrid control strategy of parameter perturbation and state feedback is adopted. Finally, the effectiveness of the theoretical results and the control strategy is verified by numerical simulations.

Bifurcation analysis of autonomous and nonautonomous modified Leslie-Gower models.

In ecological systems, the predator-induced fear dampens the prey's birth rate; yet, it fails to extinguish their population, as they endure and survive even under significant fear-induced costs. In this study, we unveil a modified Leslie-Gower predator-prey model by incorporating the fear of predators, cooperative hunting, and predator-taxis sensitivity. We embark upon an exploration of the positivity and boundedness of solutions, unearthing ecologically viable equilibrium points and their stability conditions governed by the model parameters. Delving deeper, we unravel the scenario of transcritical, saddle-node, Hopf, Bogdanov-Takens, and generalized-Hopf bifurcations within the system's intricate dynamics. Additionally, we observe the bistable nature of the system under some parametric conditions. Further, the nonautonomous extension of our model introduces the intriguing interplay of seasonality in some crucial parameters. We establish a set of sufficient conditions that guarantee the permanence of the seasonally driven system. By conducting a numerical study on the seasonally forced model, we observe a myriad of phenomena manifesting the predator-prey dynamics. Notably, periodic solutions, higher periodic solutions, and bursting patterns emerge, alongside intriguing chaotic dynamics. Specifically, seasonal variations of the predator-taxis sensitivity and hunting cooperation can lead to the extinction of prey species and even the control of chaotic (higher periodic) solutions through the generation of a simple periodic solution. Remarkably, the seasonal forcing has the capacity to govern the chaotic behavior, leading to an exceptionally quasi-periodic arrangement in both prey and predator populations.

Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators.

We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.

The impact of water storage capacity on plant dynamics in arid environments: A stoichiometric modeling approach

Plants in arid environments have evolved many strategies to resist drought. Among them, the developed water storage tissue is an essential characteristic of xerophytes. To clarify the role of water storage capacity in plant performance, we originally formulate a stoichiometric model to describe the interaction between plants and water with explicit water storage. Via an ecological reproductive index, we explore the effects of precipitation and water storage capacity on plant dynamics. The model possesses saddle–node bifurcation and forward or backward bifurcation, and the latter may lead to the emergence of alternative stable states between a stable survival state and a stable extinction state. Numerical simulations illustrate the persistence and resilience of plants regulated by soil conditions, precipitation and water storage capacity. Our findings contribute to the botanical theory in the perspectives of environmental change and plant water storage traits.

Enhancement of ECCD by the current condensation effect for stabilizing large magnetic islands caused by neoclassical tearing modes in tokamak plasmas

The radio frequency current condensation effect reported in Reiman and Fisch (2018 Phys. Rev. Lett. 121 225001) is modeled in the nonlinear resistive magnetohydrodynamic code. A series of numerical investigations have been performed to investigate the enhancement of electron cyclotron current drive (ECCD) by the current condensation effect during the control of neoclassical tearing mode (NTM) in tokamak plasmas. In the numerical model, both the parallel transport and the perpendicular transport of electron temperature are considered. The EC driven current and driven perturbed electron temperature can nonlinearly evolve within the given magnetic configuration and eventually reach saturation states. The input power threshold of ECCD and the fold bifurcation phenomenon are numerically verified via nonlinear simulations. The numerical results show good agreements with the analytical results. Moreover, spatial distributions of EC current for the two solutions at different condensed level are displayed. The control effectiveness of ECCD for large NTM islands has been evaluated while considering the current condensation effect. While taking into account current condensation effect, for a sufficiently large input power, a larger island can be more effectively stabilized than a smaller one, which suggests a reassessment of the previous idea that the ECCD should always be turned on as early as possible. The potential physics mechanism behind the ECCD control have all been discussed in detail. Furthermore, the condensation effect is found to have favorable effects on the radial misalignment of ECCD. In the consideration of the situation for extremely localized control needs, a highly peaked heating profile is adopted to verify that the fold bifurcation phenomenon still exists and the current condensation effect can still take effect in this extreme condition.

Effect of motor suspension parameters on bifurcations for a nonlinear bogie system

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie.

Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

Nonlinear shimmy and dynamic bifurcation in vehicle system with driver steering input

Based on the dynamic coupling analysis of the front wheel shimmy and vehicle plane motion, a three degrees of freedom (DOF) nonlinear shimmy model considering driver steering input is established. Firstly, when the driver steering input is not considered, the modal properties and dynamic stability of the system are investigated by solving the Jacobian matrix. Then, the nonlinear shimmy behavior considering the driver steering input is discussed by means of numerical calculations. The slow flow equations of the vehicle system considering the driver steering input are derived based on the complexification-averaging (CA) method. According to the nonlinear dynamics theory, the saddle node (SN) and Hopf bifurcation characteristics of the system are analyzed, the analytical solution of the front wheel shimmy angle is also obtained. Finally, with the help of the understeer gradient, the influence of the linear cornering stiffness of the front wheel on the nonlinear shimmy motion is discussed. It is summarized that the SN and Hopf bifurcation are more likely to occur in the shimmy system for an oversteered vehicle. The relevant conclusions are useful for the early anti-shimmy design of vehicles.

Bifurcation and hysteresis in a nonlinear transport model on network motifs

Nonlinear transport has been observed in many real systems, such as international trade, business transactions, human movement, and urban traffic. To understand the fundamental properties of transport in such systems, several models have been proposed in the past few years. These models exhibit complex behavior due to the interplay between nonlinearity and the underlying network topology. We studied a transport equation on three-body network motifs both numerically and analytically, and we found bifurcation on two motifs and hysteresis of a model on one motif. These three motifs show different bifurcation points. The first two motifs have a branching node, i.e., a node with outdegree 2, and the third motif is a fully connected bidirectional triangle. In particular, the hysteresis on the bidirectional triangle is not predictable from the linear stability analysis on the uniform stationary solution. The analysis based on the self-consistency equation of the parameter u [see Eq. (6) for the precise definition] measuring the ratio between sizes on two nodes reveals the following: The bifurcation on the first two motifs is due to supercritical pitchfork bifurcation. The difference between these two motifs comes from the existence of bidirectional edges. The hysteresis on the third motif is due to the combination of the saddle-node and transcritical bifurcation. Our results imply that the nonlinear transport model exhibits nontrivial behavior even in minimal settings and that the bifurcation on large-scale networks becomes far more complex and unpredictable from the underlying network structure. Published by the American Physical Society 2024

Existence and bifurcation of periodic solutions to the Lp-Minkowski problem with indefinite weight

This paper is devoted to the existence and bifurcation of positive periodic solutions for Lp-Minkowski problem with indefinite weight. We provide new sufficient conditions for the existence of at least one positive periodic solution. The main tools are Leray-Schauder alternative principle and a global continuation theorem by Manásevich-Mawhin. Using the numerical bifurcation theory, we study the dynamic behaviors of periodic solutions in the cases of indefinite and positive weight, where the weight term is a simple sinusoidal function. In the positive weight case, the multiplicity of positive periodic solutions is detected for the first time in the Lp-Minkowski problem, which is generated by a saddle-node bifurcation.