Multiple Bifurcation Points Research Articles

Modelling the prudent predation in predator–prey interactions

Prudent predators may evolve a strategy of prudent feed at a suitable rate, which is detrimental to their survival, but would not overexploit the prey thus is beneficial to the sustainability of resources. In this paper, by introducing a prey-dependent predation rate function, a two prey and one predator system with prudent predation is established, and the dynamics of the system as well as its subsystems are investigated. The existence and stability of the equilibrium are analyzed and the occurrence of Hopf bifurcation is studied. Numerical simulations are carried out to verify the analytical results and expand the theoretical analyses: (i) In the subsystems, it is possible to have multiple Hopf bifurcation points and prudence acts as a stabilizing factor; (ii) Suitable level of prudence will benefit the predator while sustaining the prey; (iii) Prudent predation can stabilize the system from chaos, which means chaotic dynamics can be controlled by the prudent predation. These results may reveal the important role of predator initiative in predator–prey interactions and enrich the dynamics of predator–prey system.

A Malware Propagation Model with Dual Delay in the Industrial Control Network

The malware attacks targeting the industrial control network are gradually increasing, and the nonlinear phenomenon makes it difficult to predict the propagation behavior of malware. Once the dynamic system becomes unstable, the propagation of malware will be out of control, which will seriously threaten the security of the industrial control network. So, it is necessary to model and study the propagation of malware in the industrial control network. In this paper, a SIDQR model with dual delay is proposed by fully considering the characteristics of the industrial control network. By analyzing the nonlinear dynamics of the model, the Hopf bifurcation is discussed in detail when the value of dual delay is greater than zero, and the expression for the threshold is also provided. The results of the experiments indicate that the system may have multiple bifurcation points. By comparing different immune and quarantine rates, it is found that the immune rate can be appropriately increased and the isolation rate can be appropriately reduced in the industrial control network, which can suppress the spread of malware without interrupting the industrial production.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

A class of periodic lattices for tuning elastic instabilities

With in mind microstructures exhibiting unconventional macroscopic mechanical behaviors, characterized by overall auxetic responses and strain localization due to local elastic instabilities, in this work we conceived a simple two-dimensional non-chiral architecture in the form of a periodic lattice, whose drawing is decided by varying the thickness ratios of the cells’ walls and in turn their slenderness. Inspired by nature, which often uses supposedly naive geometries that conceal complex functions, and focusing the attention on two limit geometrical configurations of main interest, we show how these non-chiral settings, as a function of the prescribed boundary conditions, might lead to symmetry breaking associated to a variety of non-trivial deformation modes, which in some cases also retrace and generalize chiral as well as auxetic responses already observed in literature in simpler microstructures. Interestingly, for both the above mentioned limit cases, we were able to idealize the mechanical response of the system through geometrically nonlinear beam-based models, in this way obtaining helpful analytical solutions and explicit formulas to estimate the tangent effective stiffness of the lattice as well as the critical load at the onset of instability. The post-buckling was instead analyzed and discussed in detail through parametric numerical finite element simulations and ad hoc laboratory experiments, performed by faithfully realizing 3D printed prototypes, constructed via additive manufacturing technologies and made of rubber-like material, to follow extreme deformation patterns, including multi-stable states, localization and compaction with possible self-contact/touching of the elements. At the end, we exploited the obtained closed-form solutions and some associated inequalities to derive the transition from the discrete lattice to its continuum limit, which – together with the multiple equilibrium bifurcation points exhibited by the proposed microstructure – could be used to broaden the spectrum of metamaterial geometries designed to exhibit complex tunable properties.

Workspace Boundary Estimation for Soft Manipulators Using a Continuation Approach

Considering a soft manipulator configuration, controlled via installed bounded actuators, this paper addresses the end-effector workspace estimation problem for such a soft robot. For this, the Discrete Cosserat method is adopted to deduce the mathematical model of soft manipulators, based on which a continuation method that accounts for simple and multiple bifurcation points to solution curves is developed to map its workspace boundaries. Difficulties encountered in calculating tangents at simple and multiple bifurcation points are studied, and an efficient solution is provided. Numerical simulations applied to planar and spatial soft manipulator configurations are presented to emphasize the validity of the proposed methodology.

Multiple bifurcation paths visualized by a computational asymptotic stability theory

Multiple bifurcation (MB) is a compound stability problem of nonlinear structures, in which the singular system stiffness matrix at the stability point coincidentally undergoes two or more zero eigenvalues. The corresponding critical eigenvectors are generally coupled in the actual buckling modes, as frequently observed in symmetric structures.This paper presents a practical diagnosis to visualize all secondary paths branching from the compound stability point when the stiffness matrix is deficient in rank by two. To find post-critical equilibria around an MB point (MBP), asymptotic expansions are assumed to consist of two homogeneous and one particular solution of the singular stiffness equations. Furthermore, hyper-dual numbers are introduced to numerically evaluate the derivatives of the system stiffness with respect to the nodal degrees-of-freedom. The resulting bifurcation equations are a set of three simultaneous cubic polynomial equations with unknown perturbation parameters that can be solved by a popular graphical software. The number and location of existing equilibria above and beneath the MBP at the load level can exactly indicate each type of branching path, such as asymmetric, unstable, or stable symmetric bifurcation paths.Two numerical examples demonstrate that the proposed asymptotic theory can reliably diagnose MB. The first one is the Augusti model, i.e., a simple rigid column supported by elastic springs that shows that the numerical prediction by the proposed method is consistent with Augusti’s analytical results. The second example was computed using the plate and shell finite element (FE) program to verify that the asymptotically expanded and visually solved bifurcation equations work well and can be implemented in existing FE codes for stability analysis, including imperfection-sensitivity and optimization.

Post-buckling analysis of functionally graded multilayer graphene platelet reinforced composite cylindrical shells under axial compression.

Axially compressed composite cylindrical shells can attain multiple bifurcation points in their post-buckling procedure because of the natural transverse deformation restraint provided by their geometry. In this paper, the post-buckling analysis of functionally graded (FG) multilayer graphene platelets reinforced composite (GPLRC) cylindrical shells under axial compression is carried out to investigate the stability of such shells. Rather than the critical buckling limit, the focus of the present study is to obtain convergence post-buckling response curves of axially compressed FG multilayer GPLRC cylindrical shells. By introducing a unified shell theory, the nonlinear large deflection governing equations for post-buckling of FG multilayer GPLRC cylindrical shells with wide range of thickness are established, which can be easily changed into three widely used shell theories. Load-shortening curves for both symmetric and asymmetric post-buckling modes are obtained by Galerkin's method. Numerical results illustrate that the present solutions agree well with the existing theoretical and experimental data. The effects of geometries and material properties on the post-buckling behaviours of FG multilayer GPLRC cylindrical shells are investigated. The differences in the three shell theories and their scopes are discussed also.

E protein support of Zeb2 expression regulates development of multiple hematopoietic lineages

Abstract Zeb2 is an E-box binding transcriptional repressor that were first discovered to be important for epithelial to mesenchymal transition. We recently discovered a genetic circuit involving Zeb2, Nfil3, Id2 that governs dendritic cell (DC) specification in the multipotent common dendritic cell progenitor (CDP). Nfil3 expression is required for the transition from a Zeb2hiand Id2lo stage into a Zeb2lo and Id2hi stage, which has exclusively cDC1 potential. This circuit blocks E protein activity to exclude pDC potential within the progenitor. However how E proteins regulate pDC development is still unclear. Here, we identified an enhancer element 165kb downstream of Zeb2 transcriptional starting site that can be bound by E proteins. Deletion of this enhancer completely abrogates pDC development. Interestingly, the activity of this enhancer is also required for monocytes and B cell development, while tissue resident macrophages used a different enhancer. A circuit involving Zeb2, Id2 and E proteins, where E proteins binding at this cis-element supports Zeb2 expression, is essential for lineage specification at multiple bifurcation point in the hematopoietic tree.

Mini-Workshop: Mathematics of Differential Growth, Morphogenesis, and Pattern Selection

Living structures are highly heterogeneous systems that consist of distinct regions made up of characteristic cell types with a specific structural organization. During evolution, development, disease, or environmental adaptation each region may grow at its own characteristic rate. Differential growth creates a balanced interplay between tension and compression and plays a critical role in biological function. In plant physiology, typical every-day examples include the petioles of celery, caladium, or rhubarb with a slower growing compressive outer surface and a faster growing tensile inner core. In developmental biology, differential growth is critical to the organogenesis of various structures including the gut, the heart, and the brain. From a structural point of view, these phenomena are close associated with instabilities, of twisting, looping, folding, and wrinkling. From a mathematical point of view, the governing equations of organogenesis are highly nonlinear and often characterized through multiple bifurcation points. Bifurcation is critical in symmetry breaking, pattern formation, and selection of shape. While biologists are studying differential growth, morphogenesis, and pattern selection merely by observation, our goal in this workshop is to explore, discuss, and advance the fundamental theory of differential growth to characterize morphogenesis and pattern selection by mathematical modeling. This workshop will bring together scientists with similar interests and complementary backgrounds in applied mathematics, mathematical biology, developmental biology, plant biology, dynamical systems, biophysics, biomechanics, and clinical sciences. We will identify common features of growth phenomena in living systems with the overall objectives to establish a unified mathematical theory for growing systems and to identify the necessary mathematical tools to address challenging questions in biology and medicine.

Competition between drift and spatial defects leads to oscillatory and excitable dynamics of dissipative solitons.

We have reported in Phys. Rev. Lett. 110, 064103 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064103 that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-2 bifurcation points. Moreover, we demonstrate that the mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons.

Elastic postbuckling response of axially-loaded cylindrical shells with seeded geometric imperfection design

Elastic instabilities (such as buckling) have been recognized as a promising phenomenon to design smart materials and mechanical systems. Thin-walled cylindrical shells under axial compression can attain multiple bifurcation points in their postbuckling regime due to the natural transverse deformation restraint provided by their geometry; but harnessing such behavior for smart purposes is lacking extensive study due to its notoriously high imperfection sensitivity. In this paper, the concept of seeded geometric imperfection (SGI) design is proposed to modify and control the elastic postbuckling behavior of cylindrical shells. Eigenvalue-based mode shapes were used as basic geometric forms to generate a seeded imperfection. Prototyped SGI cylindrical shells were fabricated through 3D printing and tested under loading–unloading cycles. Numerical and experimental results suggest that the SGI cylindrical shells are less sensitive to initial imperfections and load variation than uniform ones. Cylindrical shells with seeded geometry can be potentially used in the design of smart devices and mechanical systems such as energy harvesters and self-powered sensors.

A hybrid method for airway segmentation and automated measurement of bronchial wall thickness on CT.

Inflammatory and infectious lung diseases commonly involve bronchial airway structures and morphology, and these abnormalities are often analyzed non-invasively through high resolution computed tomography (CT) scans. Assessing airway wall surfaces and the lumen are of great importance for diagnosing pulmonary diseases. However, obtaining high accuracy from a complete 3-D airway tree structure can be quite challenging. The airway tree structure has spiculated shapes with multiple branches and bifurcation points as opposed to solid single organ or tumor segmentation tasks in other applications, hence, it is complex for manual segmentation as compared with other tasks. For computerized methods, a fundamental challenge in airway tree segmentation is the highly variable intensity levels in the lumen area, which often causes a segmentation method to leak into adjacent lung parenchyma through blurred airway walls or soft boundaries. Moreover, outer wall definition can be difficult due to similar intensities of the airway walls and nearby structures such as vessels. In this paper, we propose a computational framework to accurately quantify airways through (i) a novel hybrid approach for precise segmentation of the lumen, and (ii) two novel methods (a spatially constrained Markov random walk method (pseudo 3-D) and a relative fuzzy connectedness method (3-D)) to estimate the airway wall thickness. We evaluate the performance of our proposed methods in comparison with mostly used algorithms using human chest CT images. Our results demonstrate that, on publicly available data sets and using standard evaluation criteria, the proposed airway segmentation method is accurate and efficient as compared with the state-of-the-art methods, and the airway wall estimation algorithms identified the inner and outer airway surfaces more accurately than the most widely applied methods, namely full width at half maximum and phase congruency.

A concept for energy harvesting from quasi-static structural deformations through axially loaded bilaterally constrained columns with multiple bifurcation points

A major obstacle limiting the development of deployable sensing and actuation solutions is the scarcity of power. Converted energy from ambient loading using piezoelectric scavengers is a possible solution. Most of the previously developed research focused on vibration-based piezoelectric harvesters which are typically characterized by a response with a narrow natural frequency range. Several techniques were used to improve their effectiveness. These methods focus only on the transducer’s properties and configurations, but do little to improve the stimuli from the source. In contrast, this work proposes to focus on the input deformations generated within the structure, and the induction of an amplified amplitude and up-converted frequency toward the harvesters’ natural spectrum. This paper introduces the concept of using mechanically-equivalent energy converters and frequency modulators that can transform low-amplitude and low-rate service deformations into an amplified vibration input to the piezoelectric transducer. The introduced concept allows energy conversion within the unexplored quasi-static frequency range (≪1 Hz). The post-buckling behavior of bilaterally constrained columns is used as the mechanism for frequency up-conversion. A bimorph cantilever polyvinylidene fluoride (PVDF) piezoelectric beam is used for energy conversion. Experimental prototypes were built and tested to validate the introduced concept and the levels of extractable power were evaluated for different cases under varying input frequencies. Finally, finite element simulations are reported to provide insight into the scalability and performance of the developed concept.

Quantum fidelity for degenerate ground states in quantum phase transitions

Spontaneous symmetry breaking in quantum phase transitions leads to a system having degenerate ground states in its broken-symmetry phase. In order to detect all possible degenerate ground states for a broken-symmetry phase, we introduce a quantum fidelity defined as an overlap measurement between a system ground state and an arbitrary reference state. If a system has N-fold degenerate ground states in a broken-symmetry phase, the quantum fidelity is shown to have N different values with respect to an arbitrarily chosen reference state. The quantum fidelity then exhibits an N-multiple bifurcation as an indicator of a quantum phase transition without knowing any detailed broken symmetry between a broken-symmetry phase and a symmetry phase as a system parameter crosses its critical value (i.e., a multiple bifurcation point). Each order parameter, characterizing a broken-symmetry phase from each degenerate ground state reveals an N-multiple bifurcation. Furthermore, it is shown that it is possible to specify how each order parameter calculated from degenerate ground states transforms under a subgroup of a symmetry group of the Hamiltonian. Examples are given through study of the quantum q-state Potts models with a transverse magnetic field by employing tensor network algorithms based on infinite-size lattices. For any q, a general relation between the local order parameters is found to clearly show the subgroup of the Z_{q} symmetry group. In addition, we systematically discuss criticality in the q-state Potts model.

Slow passage through multiple bifurcation points

The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations. The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp--a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly. We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter. These delay and memory effects are independent of the ramp rate, for small enough ramp rates. The slow passage through the torus break-up bifurcation is qualitatively different. It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate. This is typical of transient behavior. We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used. The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.

Binding features by relaying modulator group of neurons

Simultaneous oscillation of neuron groups could provide solution to the binding problem [1] and simultaneously firing neurons could bind together different features of the same object. The corticothalamic system plays a key role in synchronizing the activity of thalamic and cortical neurons [2] and synchronized oscillations have important function in brain activities [3]. It is shown that a relaying neuron group could cause zero time lag synchronization among distant neuron groups [4]. As there are many fields representing different features then they compete with each other to become dominant; thus, Jorg Lucke [5] proposed that there might be multiple bifurcation points where the network tends to move towards a certain stable state. Such selection is achieved by the decision making processes [6], where a common group of inhibitory neurons inhibits activity of the to-be-loosing excitatory neuron group. Based on this we have built up an experimental setup where we used two pairs of exclusive features from two dimensions and tried to bind together one from each dimension as a unified object and cause activity of the higher level group of neurons. In our experiments the Brian simulator [7] featuring a leaky integrate-and-fire neuron model is used and three main groups of excitatory neurons are formed. The first group is the modulator group (see Figure ​Figure1A1A Tc) which synchronizes activity of the next feature group (Py1) and the latter sends synaptic effects to the higher consolidating group (Py2). All the excitatory neurons are connected to a local inhibitory group (Re, In1, In2) and the feature neurons and consolidating neurons get pairwise additional inhibitory input (In3, In4). All the neurons receive Poissonian background input noise that causes averaged network activity in the frequency of 1 Hz. The neurons from modulator and feature groups get additional Poissonian input in a certain time frame and, during the simulations, activity switches from one pair of features to the other. As a result we could see the well synchronized activity of feature neurons could cause oscillating activity of neurons in the higher group (Figure ​(Figure11). Figure 1 A) Network structure. B) Raster plot of spikes of excitatory neurons. C) Averaged firing histogram of subpopulations. Upper figure Py12 and Py14 subpopulations and their common firing (blue, green and red respectively) and lower figure Py11 and Py13 (the ... There might be some number of basic mechanisms that are responsible for processing and forwarding information in the brain, including a variety of synchronizing mechanisms. In this experiment we have shown how synchronized oscillations of neurons and decision making processes can be used for binding and forwarding information between the neuron groups.

An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

SUMMARYRecently, a numerical method was proposed to compute a Hopf bifurcation point in fluid mechanics. This numerical method associates a bifurcation indicator and a Newton method. The former gives initial guesses to the iterative method. These initial values are the minima of the bifurcation indicator. However, sometimes, these minima do not lead to the convergence of the Newton method. Moreover, as only a single initial guess is obtained for each computation of the indicator, the computational time to obtain a Hopf bifurcation point can be quite long. The present algorithm is an enhancement of the previous one. It consists in automatically computing several initial guesses for each indicator curve. The majority of these initial values leads to the convergence of the Newton method. This method is evaluated through the problem of the lid‐driven cavity with several aspect ratios in the framework of the finite element analysis of the 2D Navier–Stokes equations. The results prove the efficiency and the robustness of the proposed algorithm. Copyright © 2011 John Wiley & Sons, Ltd.

複数の部材座屈と極限点が重複するアーチ状トラスの最悪不整

Imperfection sensitivity properties are investigated for an arch that has multiple member buckling at a limit point, which is classified as hilltop branching with multiple symmetric bifurcation points. The critical loads of imperfect structures are shown to be governed by a piecewise linear law of the imperfection parameter. Antioptimization problems are formulated to obtain the worst imperfection mode that most drastically reduces the critical load. A formula for the worst nodal imperfection as a linear combination of the critical modes is also presented. The validity of the formula is ensured by path-following analysis of arches with randomly generated imperfections.

Computation of multiple bifurcation point

PurposeThe aim of this paper is to develop a new method for finding multiple bifurcation points in structures.Design/methodology/approachA brief review of nonlinear analysis is presented. A powerful method (called arc‐length) for tracing nonlinear equilibrium path is described. Techniques for monitoring critical points are discussed to find the rank deficiency of the stiffness matrix. Finally, by using eigenvalue perturbation of tangent stiffness matrix, load parameter associated with multiple bifurcation points is obtained.FindingsSince other methods of finding simple bifurcation points diverge in the neighborhood of critical points, this paper introduces a new method to find multiple bifurcation points. It should be remembered that a simple bifurcation point is a multiple bifurcation point with rank deficiency equal to one. Therefore, the method is applicable to simple critical points as well.Practical implicationsGlobal buckling of the structures should be considered in design. Many structures (specially symmetric space structures) have multiple bifurcation points, therefore, analyst and designer should be aware of these points and should control them (for example, by changing the geometry or other related factors) for obtaining a safe and optimum design.Originality/valueIn this paper a robust method to find multiple bifurcation points is introduced. By using this method, engineers can be aware of critical load of multiple bifurcation points to control global buckling of related structures.

$\sqrt{5}:2$ MODE INTERACTIONS ON A SQUARE LATTICE

In many diverse pattern-forming systems, the observed arrangements are square-symmetric or hexagonal. However, while many investigations have been devoted separately to the competition between these patterns and rolls, important questions still remain regarding their competition with one another. This paper demonstrates that both square and hexagonal patterns can be accounted for by the resonant interaction of two sets of modes with wave numbers in the ratio [Formula: see text], aligned to lie on a square lattice. The behavior of this system is investigated by expanding about the multiple bifurcation point where the stability of both sets of modes is marginal. The corresponding amplitude equations, constructed through simple symmetry arguments, are shown to encompass various forms of hexagonal and square-symmetric patterns within a single framework. This framework is then applied to the Swift–Hohenberg partial differential equation, where both hexagons and squares with up–down-asymmetry are found to occur subcritically.