Equilibrium Branches Research Articles

Effect of realistic distributed loads on the bi-stable behavior of a pre-stressed aileron

AbstractWe study both experimentally and numerically the aeroelastic response of a pre-stressed curved aileron in a high Reynolds number flow undisturbed at infinity. The structure is designed to have a peculiar nonlinear behavior. Specifically, the aileron has only one stable equilibrium when the external forces are vanishing, but it is bistable when distributed aerodynamic loads are applied. Hence, for sufficiently high fluid velocities, another equilibrium branch is possible. We test a prototype of such an aileron in a wind tunnel. A sudden change (snap) of the shell configuration is observed when the fluid velocity exceeds a critical threshold: the snapped configuration is characterized by sensibly lower drag. However, when the velocity is reduced to zero, the structure recovers its initial shape. A similar nonlinear behavior can have important applications for drag-reduction strategies since the transition between a bluff body-like and a slender body-like behavior is controlled by the free-stream fluid velocity and does not require any external actuation.

Nonlinear in-plane buckling of small-curved and large-curved FG porous microbeams via strain gradient-based isogeometric collocation formulations

In the current investigation, for the first time, the changes in the limit loads and equilibrium branches associated with the nonlinear in-plane stability characteristics of curved microbeams are explored in the presence of different microstructural gradient tensors. In this regard, multiple microsize-dependent equilibria are analyzed relevant to thermomechanical loaded small-curved, medium-curved, and large-curved microbeams made of functionally graded porous (FGP) metal reinforced with nanofillers possessing clamped end supports. To this purpose, based upon the strain gradient elasticity within the framework of the third-order shear flexible curved beam model, the isogeometric collocation formulations incorporating Greville abscissae are constructed resulting in higher-continuity characters as well as remarkable accuracy for higher-order approximations. It is deduced that for the small-curved FGP reinforced microbeam, no limit load can be found due to the absence of the buckling phenomenon, but after rising the temperature by an enough amount, the initial instability mode appears. However, for the medium-curved FGP reinforced microbeam, the limit instability mode of buckling occurs which results in the normalized upper limit load equal to 0.7161 based on the classical theory and 0.7606 based on the strain gradient elasticity (6.21% enhancement). Also, it results in the normalized lower limit load equal to 0.3060 based on the classical theory and 0.3521 based on the strain gradient elasticity (15.07% enhancement). On the other hand, for the large-curved FGP reinforced microbeam, the bifurcation mode of buckling occurs which results in the normalized upper limit load equal to 1.0494 based on the classical theory and 1.1358 based on the strain gradient elasticity (8.23% enhancement). Also, it results in the normalized lower limit load equal to 0.2225 based on the classical theory and 0.2513 based on the strain gradient elasticity (12.94% enhancement).

Investigations on the limit-point buckling of curved beams

Calculating the critical load that could lead to beam failure through buckling is a complex issue. This paper intends to analyze the stability of arches taking into account how the curvature, material properties, load positions and geometry affect the behavior. Unlike in many other articles, the impact of the bending moment on the membrane strain is also incorporated into the model. To derive the buckling equilibrium equation, the principle of virtual work is applied, and analytical solutions for the limit point buckling are provided. It is worth noting that the load location has a significant impact on the buckling load, and this relationship is strongly associated with the ratio of the arch length to the radius of the gyration of the structure. It is found that in the case of fixed arches, two stable equilibrium branches and one unstable branch can be observed when buckling is possible. When the load is positioned sufficiently far from the crown point, the load-carrying capacity of the structure improves.

Route to mixed-mode oscillations via step-shaped sharp transition of equilibria in a nonlinear gyroscope oscillator

Sharp transitions in relation to the variation of system parameters are frequently encountered in many multiple-timescale systems, and they have been found to be an important factor related to the generation of mixed-mode oscillations (MMOs). The present paper aims to report a novel type of sharp transition, referred to as step-shaped sharp transition, in a nonlinear gyroscope oscillator with multiple-frequency excitations, and investigate the resulting MMOs. We show that step-shaped sharp quantitative changes in relation to the variation of system parameters can be observed in the equilibrium branch, which yields the step-shaped sharp transition. In particular, with the increase of the frequency ratio between the parametric and external excitations, more step-shaped sharp transitions appear in the equilibrium branches, which evolve into the ones displaying different structures. Based on this, the rectangular-pulse-shaped explosion of equilibria is created. Furthermore, these sharp transitions can form active areas for the MMOs, leading to the alternations between large-amplitude and small-amplitude oscillations, and finally the route to MMOs is created. Our findings have significant implications for understanding the fast-slow dynamics of the nonlinear gyroscope oscillator, contributing to the exploration of new routes to MMOs. Thus, the results could provide theoretical support for the potential application of gyroscopes.

Multiple modes of bursting phenomena in a vector field of triple Hopf bifurcation with two time scales

For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations.

Slow–Fast Dynamics of a Coupled Oscillator with Periodic Excitation

We study the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced. The model under consideration is a coupled system of Bonhöffer–van der Pol (BVP) equations with a slow-varying periodic excitation. We apply the modified slow–fast analysis method to perform a detailed study on all the equilibrium branches and their bifurcations of the generalized autonomous system. According to different dynamical behaviors, we explore the dynamical evolution of existing attractors, which reveals the coexistence of a quasi-periodic attractor with diverse types of bursting attractors. Further investigation shows that the coexisting steady states may cause spiking oscillations to behave in combination of a 2D torus and a limit cycle. We also identify a period-2 cycle bursting attractor as well as a quasi-periodic attractor according to the period-2 limit cycle.

Hopf bifurcation in a delayed prey–predator model with prey refuge involving fear effect

This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points.

Experimental characterization and phenomenological modeling of nonlinear viscoelasticity, plasticity and damage of continuous carbon fiber-reinforced thermoplastics

The presented work addresses the experimental characterization and phenomenological modeling of the complex rate dependent mechanical response of continuous fiber-reinforced thermoplastics (cFRTP). The theoretical basis is given by the concept of an equilibrium curve of the material which provides a comprehensive framework for separating the mechanical response into a rate dependent overstress and a rate independent equilibrium stress. In long-term creep and relaxation tests, it is shown that relaxation tests provide several advantages over creep tests when determining the equilibrium response. A stepwise loading–unloading relaxation (SR) test is proposed which makes it possible to extract nonlinear viscoelasticity, damage and plasticity present in the material from a single test. The application to cFRTP is exemplified by using the proposed method for the experimental characterization of a continuous carbon fiber-reinforced polycarbonate (CF-PC). Based on the obtained experimental data, a phenomenological material model is developed which describes the mechanical response of a unidirectional ply. The model consists of an overstress branch addressing the nonlinear viscoelastic material response and an equilibrium branch that utilizes a coupled damage-plasticity approach. After calibrating the model using the experimental data, it is shown that the proposed model is able to accurately reproduce the mechanical response of cFRTP.

Phase-field viscoelastic fracture modeling of polymer composites using strain tensor spectral decomposition

The mechanical behaviors of polymer composites exhibit strong time-dependent nonlinearities, and the tensile and compressive strengths can be vastly different. Such a strength differential effect poses a great challenge in fracture modeling using existing phase-field models. To address this problem, this study develops a new viscoelastic fracture phase-field model, in which the strain tensor spectral decomposition is proposed to decompose the elastic energy of the equilibrium and non-equilibrium branches into tensile and compressive components. The non-negativity of elastic energy and the thermodynamic consistency are theoretically verified. Various case studies, spanning from material uniaxial tension–compression and compression under isostatic pressure to fracture processes of realistic structural components, are used to validate the proposed method. Results of the proposed method are compared with those of existing methods and the actual testing data of polymer composites to demonstrate the effectiveness of the method. It is shown that the proposed method can accurately model not only the strong strength differential effect but also the isostatic pressure dependence of viscoelastic materials. Furthermore, the proposed method offers a more precise prediction of the fracture path and can deal with the loading rate effect.

All possible bursting oscillations in neighborhood of a Hopf bifurcation point induced by low-frequency excitation

Hopf bifurcation may be one of most common bifurcations, which results in the transitions between quiescence and spiking states in a bursting attractor, since it causes the alternation between an equilibrium point and a limit cycle. Though a lot of bursting attractors related to Hopf bifurcation have been obtained, while how many types of bursting attractors may appear in the neighborhood of a Hopf bifurcation point still remains an open problem. Here we try to answer the question based on the normal form up to the seventh order of a vector field with Hopf bifurcation at the origin. By introducing a low-frequency external excitation to the normal form, we derive the equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying excitation, based on which four typical cases corresponding to qualitatively different evolution of the behaviors are presented. Accordingly, four types of quasi-periodic bursting oscillations, i.e. Hopf/Hopf bursting attractors with (1) and (2) without saddle on the limit cycle bifurcation in the spiking oscillations, respectively, (3) fold/Hopf/fold/Hopf bursting attractors, and (4) bursting attractor with saddle on limit cycle bifurcation at the transition, are obtained with the increase of the exciting amplitude, the mechanism of which is obtained by overlapping equilibrium branches together with bifurcations in generalized autonomous system and transformed phase portrait. It is found that when two coexisted stable attractors are separated by an unstable attractor on the phase plane in generalized autonomous system, the influence of one of the stable attractors on the dynamics of full system may disappear. Furthermore, when the window for a stable attractor between two bifurcation points along slow-varying parameter axis is relatively short, the effect of the attractor as well as the bifurcation can only appear when the external excitation changes slowly enough. The amplitude of the repetitive spiking oscillations may vary according to two different limit cycles separated by saddle on limit cycle bifurcation, at which a sudden change of the oscillating amplitude can be observed.

Dynamics of an SIRWS model with waning of immunity and varying immune boosting period

SIRS models capture transmission dynamics of infectious diseases for which immunity is not lifelong. Extending these models by a W compartment for individuals with waning immunity, the boosting of the immune system upon repeated exposure may be incorporated. Previous analyses assumed identical waning rates from R to W and from W to S. This implicitly assumes equal length for the period of full immunity and of waned immunity. We relax this restriction, and allow an asymmetric partitioning of the total immune period. Stability switches of the endemic equilibrium are investigated with a combination of analytic and numerical tools. Then, continuation methods are applied to track bifurcations along the equilibrium branch. We find rich dynamics: Hopf bifurcations, endemic double bubbles, and regions of bistability. Our results highlight that the length of the period in which waning immunity can be boosted is a crucial parameter significantly influencing long term epidemiological dynamics.

All Possible Bursting Attractors in the Neighborhood of Hopf Bifurcation Point Under Periodic Excitation

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors.

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

Classification of Codimension-1 Singular Bifurcations in Low-Dimensional DAEs

The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise smooth resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

Stochastic resonance in stochastic PDEs

We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to 1 for supercritical noise intensity. Concentration estimates of solutions are given in the $$H^s$$ Sobolev norm for any $$s<\frac{1}{2}$$ . The results generalise to an infinite-dimensional setting those obtained for 1-dimensional SDEs in [5].

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.

Individual and combined static stabilities in electrostatically actuated initially curved coupled micro beams

A stability study of electrostatically actuated micro-meta-structure (MMS), composed of two double clamped, initially curved micro beams and coupled via a rigid truss at their respective centres, is studied via a reduced order (RO) model. The study is based on the energy method and Lyapunov stability analyses, carried out in conjunction with the implicit arc-length “Riks” continuation method, to characterise the stability properties of several key configurations. By sweeping through each of the configurations, we show that it is possible to ascertain the stability of each beam individually, as well as of the entire structure, globally. The analysis has brought forth interesting characteristics, hitherto unknown, which complement any static analysis, by foretelling locations at which the MMS loses stability, causing a shift in limit point locations. It is shown that due to stability mismatch between the two beams, tristable properties, which could be present in one of them, will be nullified due to instability present in the other one. The Lyapunov analysis further discloses the location of a new equilibrium branch, located at an offset from the predicted equilibrium curve. The study begins with an analysis of the structure under displacement-independent “mechanical” load to facilitate its understanding, while allowing validation of the RO model against a finite element (FE) model, thus laying a foundation for the electrostatic loading analysis. It is concluded that a metastructure, composed of two sub-structures, is stable if an only if (iff) the two sub-structures are stable. Otherwise, new equilibrium branches will present themselves when conducting an experimental quasi-static loading. The study also shows that as far as stability analysis and quasi-static loadings are concerned, the newly formalised dynamic RO model bears merit.

Effect of density dependence on coinfection dynamics: part 2

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate {bar{gamma }} of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

A Form-Finding Method for Branching Structures Based on Dynamic Relaxation

Branching structure is often used as a supporting structure of the grid shell due to its geometrical and force-transferring features, and the rationality of its shape is very important. The “physical” and “numerical” hanging models can be used for the joint form-finding of the branching structure and free-form grid shell. However, slack elements may exist in the equilibrium model which corresponds to the inefficient members in the form-found branching structure. To solve this problem, a form-finding method of branching structure based on dynamic relaxation is proposed in this study. The proposed method clusters the elements of the branching model and equalizes the axial forces of the elements in the same cluster, in other words, there are no slack elements in the equilibrium branching model. This method overcomes the defect that the equilibrium branching model may have slack elements and needs many manual adjustments during the procedure of determining the rational shape of a branching structure, and effectively prevents the inefficient members existing in the form-found structure. Numerical examples are provided to demonstrate the characteristics of the proposed method and its effectiveness is verified as well.