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Summary: In the paper, a nonlinear analysis of RC phase‐shift oscillator with operational amplifier is done. Using Kirchhoff's laws, a nonlinear system of three differential equations that describes the behavior of the oscillator is obtained. This system is analyzed using the Poincaré–Andronov–Hopf bifurcation theory. The basic principles of the Poincaré–Andronov–Hopf bifurcation theory are presented in advance, including the Center Manifold Theory and the Theory of Normal Forms. It was found that, under certain conditions, the oscillator system has a limit cycle, which means that the oscillator under consideration generates periodic oscillations. The period, amplitude, and stability of the generated periodic oscillations are determined theoretically. The obtained analytical results are confirmed by a numerical analysis of the nonlinear system of differential equations performed by MATLAB. [ABSTRACT FROM AUTHOR]

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and 1 : 2 , 1 : 3 , 1 : 4 strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM. [ABSTRACT FROM AUTHOR]

In this paper, a simplified discrete-time SIR model with nonlinear incidence and recovery rates is discussed. Here, using the integral step size and the intervention level as control parameters, we mainly discuss three types of codimension-two bifurcations (fold-flip bifurcation, 1:3 resonance, and 1:4 resonance) of the simplified discrete-time SIR model in detail by bifurcation theory and numerical continuation techniques. Parameter conditions for the occurrence of codimension-two bifurcations are obtained by constructing the corresponding approximate normal form with translation and transformation of several parameters and variables. To further confirm the accuracy of our theoretical analysis, numerical simulations such as phase portraits, bifurcation diagrams, and maximum Lyapunov exponents diagrams are provided. In particular, the coexistence of bistability states is observed by giving local attraction basins diagrams of different fixed points under different integral step sizes. It is possible to more clearly illustrate the model's complex dynamic behavior by combining theoretical analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

Study of strain localization in geomaterials generally consists of two aspects, the pre-localization regime and the post-localization regime. The former refers to the onset and the orientation of the shear band, which corresponds to the bifurcation theory. The latter deals with the width and the evolution of the shear band. The purpose of this paper is to make a full-range analysis of the shear band in structural level. By comparing the bifurcation conditions in classical continuum an analogical bifurcation condition in micropolar theory is defined. The factors which may affect the shear band orientation in the numerical modeling process are emphasized, and a basic principle of the geometric modeling is concluded. Furthermore, the existing shear band width measurement methods are summarized, and some other more effective methods considering the singularity of the discontinuous surface is suggested. At the last, the ratio between the node distance of the element and the internal length less than 5 is suggested as reasonable choice. [ABSTRACT FROM AUTHOR]

The paper presents a new composite structure made up of a double-layer microchannel structure, an impinging jet structure, and a turbulent flow structure. In this paper, the combination of these three structures improves the overall thermal performance of the model compared to the basic impingement-jet double-layer nested microchannel heat sinks (IDN-MHS) model. At Reynolds numbers ranging from 138.2 to 580.4 (0.25≤u in ≤1.05), the thermal performance of the model within introduction of bifurcations is enhanced compared to traditional models that only contain double-layer microchannel structures and impact jet structures. Using 3D printers and conducting experiments, it was discovered that numerical simulation and experimental images were generally consistent. Additionally, IDN-MHS with streaming vertical bifurcations, n=6 (IDN-MHS-VB6) performed well in numerical simulations, with a temperature decrease of 1.24K over IDN-MHS when compared to IDN-MHS. In addition, this model performs better than IDN-MHS with streaming vertical bifurcations, n=2, IDN-MHS with streaming vertical bifurcations, n=4, and IDN-MHS with streaming vertical bifurcations, n=8 (IDN-MHS-VB2/4/8). Compared to IDN-MHS, IDN-MHS-VB6 has the outstanding advantage of reducing substrate temperature while maintaining a relatively low pressure drop. • This structure is an innovative combination of double layer microchannel structure and bifurcation structure of impact jet structure. • In the designed composite structure, the highest substrate temperature is 1.23 K lower than IDN-MHS. • The innovative structure designed has an acceptable pressure drop. • The designed bifurcation structure can effectively increase the coolant flow rate and reduce the substrate temperature. • The IDN-MHS-VB6 allows the coolant to come into contact with more high-temperature areas, achieving cooling effect. [ABSTRACT FROM AUTHOR]

This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group S N [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics. [ABSTRACT FROM AUTHOR]

In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results. [ABSTRACT FROM AUTHOR]

Most researches on railway vehicle stability analysis are focused on the codimension 1 (for short, codim 1) bifurcations like subcritical and supercritical Hopf bifurcation. The analysis of codim 1 bifurcation can be completed based on one bifurcation parameter. However, two bifurcation parameters should be considered to give a general view of the motion of the system when it undergoes a degenerate Hopf bifurcation. This kind of bifurcation named the generalised Hopf bifurcation belongs to the codimension 2 (for short, codim 2) bifurcations where two bifurcation parameters need to be taken into consideration. In this paper, we give a numerical analysis of the codim 2 bifurcations of a nonlinear railway wheel-set with the QR algorithm to calculate the eigenvalues of the linearised system incorporating the Golden Cut method and the shooting method to calculate the limit cycles around the Hopf bifurcation points. Here, we found the existence of a generalised Hopf bifurcation where a subcritical Hopf bifurcation turns into a supercritical one with the increase of the bifurcation parameters, which belong to the codim 2 bifurcations, in a nonlinear railway wheel-set model. Only the nonlinear wheel/rail interactive relationship has been taken into consideration in the lateral model that is formulated in this paper. The motion of the wheel-set has been investigated when the bifurcation parameters are perturbed in the neighbourhood of their critical parameters, and the influences of different parameters on critical values of the bifurcation parameters are also given. From the results, it can be seen that the bifurcation types of the wheel-set will change with a variation of the bifurcation parameters in the neighbourhood of their critical values. [ABSTRACT FROM PUBLISHER]

In this paper, we propose an impulsive predator–prey model with communicable disease in the prey species only and investigate its interesting biological dynamics. By the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we have deduced the sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the proposed model. We also give the existences of the “infection-free” periodic solution and the “predator-free” solution. Finally, numerical results validate the effectiveness of theoretical analysis for the proposed model in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of the equilibriums, their stability and possible bifurcations in a longitudinal flight with constant forward velocity and high angle of attack. The theoretical framework is the general mathematical model describing the movements of a rigid aircraft around its center of mass [1], Necessary or sufficient conditions concerning the existence of equilibriums, their stability and possible bifurcations are given. The analysis follows and extends the results reported in the paper [2]. Numerical illustrations are given. [ABSTRACT FROM AUTHOR]

Highlights • Employing extended feedback policy to effectually control bifurcation of the fractional delayed predator–prey model. • Effects of extended feedback delay on bifurcation control are numerically explored. • Impact of the single order on the devised control strategy is thoroughly researched. • The bifurcation diagrams are nicely plotted. Abstract This paper aims at controlling bifurcation of a fractional predator–prey system by an original extended delayed feedback controller. Firstly, some sufficient conditions of delay-induced bifurcations for such uncontrolled system are captured regarding time delay as a bifurcation parameter. Secondly, a generalised delayed feedback controller is subtly designed to control Hopf bifurcation for the proposed system. It suggests that bifurcation dynamics can be controlled efficaciously for such system by carefully adjusting extended feedback delay or fractional order so long as the other parameters are established. Thirdly, the bifurcation diagrams are meticulously plotted. The obtained results consumedly popularize the previous studies concerning bifurcation control of delayed fractional-order systems. To underline the effectiveness of the proposed control scheme, some numerical simulations are ultimately addressed. [ABSTRACT FROM AUTHOR]

Abstract The impact of multiple time delays on the dynamics of two disc dynamos with viscous friction is studied in this paper. We consider the stability of equilibrium states for different delay values, and determine the location of relevant Hopf bifurcations using the normal form method and the center manifold theory. By performing numerical calculations and analysis, we verify the validity of our analytically obtained results. Our research results reveal a classical period-doubling route towards deterministic chaos in the studied system, and play an important role for the better understanding of the complex dynamics of two disc dynamos with viscous friction subject to multiple time delays. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the dynamics of a predator-prey model with Beddington-DeAngelis functional response and nonselective harvesting. By using the Lyapunov-Schmidt reduction, we obtain the existence of spatially nonhomogeneous steady-state solution. The stability and existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the change of a specific parameter are investigated by analyzing the distribution of the eigenvalues. We also get an algorithm for determining the bifurcation direction of the Hopf bifurcating periodic solutions near the nonhomogeneous steady-state solution. Finally, we show some numerical simulations to verify our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved method for analyzing the stability of the nonlinear hybrid dynamical systems (NHDSs) using a monodromy matrix. We define an n‐dimensional NHDS and focus on an orbit whose initial values exist near the periodic orbit. Because the system has nonlinearity, the circuit equation cannot be expressed in the form of a linear ordinary differential equation, meaning that the existing monodromy‐matrix‐based stability analysis method cannot be used because it requires a matrix exponential. Therefore, we propose a general theory for calculating orbital perturbations during Poincaré observation. Perturbation through switching events is expressed by a state‐transition matrix, which we refer to as the saltation matrix. By computing the perturbations, we obtain the monodromy matrix, whose characteristic multipliers denote the stability of the periodic orbit. We apply the proposed algorithm to an interrupted electric circuit with a nonlinear characteristic to confirm its validity. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

Abstract In Pelton turbine, two-phase homogeneous model is the fundamental base to investigate the flow inside the turbine. Meanwhile, flow pattern in section of the distributor and bifurcation pipe produces the secondary flow which influences the shape of the jet downstream 6 nozzles. This paper numerically investigated the dynamic process using CFX commercial code under different operating conditions. Volume of Fluid(VOF) model was used to capture the air/water interface. This work focuses on the jet bifurcation problem which can seriously influence the Pelton turbine performance and furthermore it is not yet investigated in hydraulic research. Results indicate that the shape of the distributor induces the secondary flow and Dean vortex structure which are symmetric with the central horizontal plane of the pipe and nozzles. On the other hand, this sequence of vortex develops and transits along the flow and finally affects the shape of jet. Effect of secondary flow in the distributor stretches the jet flow in vertical direction and compress it in horizontal direction. Meanwhile, bifurcating phenomenon is observed outside the nozzle which is also obtained in classical experimentations. This study reveals that the secondary flow is important to be taken into consideration to optimize the performance of Pelton turbine. Highlights • Six nozzles of a Pelton turbine are numerically simulated with unsteady two-phase model. • Secondary flow is detected in the flowing parts and Dean vortex are observed clearly. • Jet bifurcation is visualized and quantitatively confirmed by the former experimentation in the literature. [ABSTRACT FROM AUTHOR]

In this paper we define the index at infinity of an asymptotically linear autonomous Hamiltonian system. We use this index to prove the existence and bifurcation from infinity of periodic solutions of the system. We apply the degree for G-invariant strongly indefinite functionals defined by Gołȩbiewska and Rybicki in (Nonlinear Anal 74:1823-1834, 2011). [ABSTRACT FROM AUTHOR]

A field observation on a terrestrial vertebrate has shown that the fear of predators can affect the behavior of prey populations and it can greatly reduce their reproduction. On the other hand, it has been observed that providing additional food to the predator decreases the predatory attack rate and increases the growth rate of the predator. In this paper, we have investigated the dynamical behavior of a predator–prey model incorporating both the effects of fear and additional food. Positivity, uniform boundedness and extinction criteria of the system are studied. Equilibrium points and their stability behaviors are also discussed here. Existence of a Hopf-bifurcation is established by considering the level of fear as bifurcation parameter. The effect of time-delay is discussed, where the delay may be considered as gestation time of the predator. Numerical simulations are performed using MATLAB to verify our analytical findings. [ABSTRACT FROM AUTHOR]

This paper describes a prey–predator type ecological model with infection in the prey populations. We also consider here a nonlinear functional response for disease transmission and a constant amount of refuge for the sound prey populations. The dynamical behavior of the mathematical model is described from the point of view of stability and bifurcation. A geometric method is also applied to establish the global asymptotic stability at the co-existence equilibrium point. Some computer simulation works have been presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents the calculation of Lyapunov exponents in a beam subjected to distributed impact. In this case, discontinuity sporadically occurs over the contacted area. The impact area is spatially discretized and the transition condition at impact instance is applied to all impact nodes in the finite element manner. The solution of the continuous vibration in the impacted beam is expressed in the modal expansion form. Then, the spectra of Lyapunov exponents in the periodic and chaotic motion are estimated for the truncated number of modes. Numerical results show that the calculated largest Lyapunov exponent agrees with the bifurcation plot. [ABSTRACT FROM AUTHOR]

We consider the family of dehomogenized Loud's centers X μ = y ( x − 1 ) ∂ x + ( x + D x 2 + F y 2 ) ∂ y , where μ = ( D , F ) ∈ R 2 , and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family { X μ , μ ∈ R 2 } distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set Γ B of codimension 1 in R 2 . In the present paper we succeed in proving that a subset of Γ B has criticality equal to one. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the (2+1) dimensional sine-Gordon equation analysed earlier [R. Radha and M. Lakshmanan, J. Phys. A Math. Gen. 29, 1551 (1996)] employing the Truncated Painlevé Approach. We then generate the solutions in terms of lower dimensional arbitrary functions of space and time. By suitably harnessing the arbitrary functions present in the closed form of the solution, we have constructed dromion solutions and studied their collisional dynamics. We have also constructed dromion pairs and shown that the dynamics of the dromion pairs can be turned ON or OFF desirably. In addition, we have also shown that the orientation of the dromion pairs can be changed. Apart from the above classes of solutions, we have also generated compactons, rogue waves and lumps and studied their dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates the effect of low frequency parametric excitations on the coupled dynamics of a spinning disc bounded in a compressible fluid filled enclosure. Specifically, the effect of principal parametric resonance due to disc accelerations is studied. The angular acceleration of the disc results in the governing equations being represented as nonlinearly coupled non-autonomous gyroscopic system. Investigations into the bifurcation behaviour is first studied using Floquet theory on the linearised equations; the stable and unstable regions of the trivial solution are obtained in the amplitude–frequency parameter space corresponding to the acceleration, for various damping configurations. A codimension-two bifurcation is observed when two control parameters are considered in the bifurcation analysis. The bifurcation behaviour subsequently investigated through direct numerical integration of the governing equations, reveals that the coupled system exhibits a two-period quasi periodic oscillation in the post bifurcation regime. [ABSTRACT FROM AUTHOR]

In this paper, we study dynamics in delayed micro-electromechanical nonlinear coupling system, with particular attention focused on Hopf and Hopf-pitchfork bifurcations. Based on the distribution of eigenvalues, we prove that a sequence of Hopf and Hopf-pitchfork bifurcations occur at the trivial equilibrium as the delay increases and obtain the critical values of two types of bifurcations. Next, by applying the multiple time scales method, the normal forms near the Hopf and Hopf-pitchfork bifurcations critical points are derived. Finally, bifurcation analysis and numerical simulations are presented to demonstrate the application of the theoretical results. We show the regions near above bifurcation critical points in which the micro-electromechanical nonlinear coupling system exists stable fixed point or stable periodic solution. Detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameters. Namely, the stable fixed point and stable periodic solution can exist in large regions of unfolding parameters as the unfolding parameters increase away from the critical value. [ABSTRACT FROM AUTHOR]

In the current paper, we continue the mathematical analysis studied in Li and Wu (2017) [15] and Oeda (2011) [22] , and further study the effect of cross-diffusion for the predator on the stationary problem. The existence of positive solutions is first established by the bifurcation theory. We next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity. Moreover, as the prevention coefficient tends to infinity, we obtain two shadow systems and give the complete limiting characterization of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases. [ABSTRACT FROM AUTHOR]

2-D numerical models have been used to analyze flow distribution performance in consecutive bifurcating fluid flow distributors according to constructal theory. The quantitative analysis of flow distribution uniformity performance for Y shape distributors with different bifurcation angles as well as inlet velocities are evaluated with water as working fluid, a hypothesis on the mechanism for the maldistribution is also presented. It is found that the flow distribution uniformity deteriorates with the increase of bifurcation angles from 15° to 90°, and the worst performance occurs when bifurcation angle is 90°, where the largest pressure drop also appears. Furthermore, the flow distribution performance also decreases with increasing inlet velocity for the same distributor. Finally, an improvement to obtain more uniform flow uniformity for the T shape distributor is proposed by ‘shrinking’ the vertical channel width of each bifurcation. The conclusions obtained in this paper are potentially helpful for the design of fluid distributors for multi channel devices. [ABSTRACT FROM AUTHOR]

In this paper, a comprehensive analysis is presented to investigate a codimension two bifurcation that exists in a nonlinear railway bogie dynamic system combining theoretical analysis with numerical investigation. By using the running velocity V and the primary longitudinal stiffness K1x as bifurcation parameters the first and second Lyapunov coefficients are calculated to determine which kind of Hopf bifurcation can happen and how the system states change with the variance of the bifurcation parameters. It is found that multiple solution branches both stable and unstable coexist in a range of the bifurcation parameters which can lead to jumps in the lateral oscillation amplitude of the railway bogie system. Furthermore, reduce the values of the bifurcation parameters gradually. Firstly, the supercritical Hopf bifurcation turns into a subcritical one with multiple limit cycles both stable and unstable near the Hopf bifurcation point. With a further reduction in the bifurcation parameters two saddle-node bifurcation points emerge, resulting in the loss of the stable limit cycle between these two bifurcation points. [ABSTRACT FROM AUTHOR]

In this paper, we investigated deformation in rotating thick-walled circular cylinders of incompressible, isotropic, neo- Hookean material. We apply the WKB method to the bifurcation analysis of cylinder shell which is subjected an external pressure. In the all mode numbers find that present compression inner layer. Amongst other results, we obtained present compression outer layer. Finally, we compared the asymptotic results with different mode numbers. [ABSTRACT FROM AUTHOR]

Taking a herringbone star gear transmission (HSGT) with floating sun gear as an example, the system bifurcation characteristics with the changing of the eccentric error of star gear and the working frequencies are analyzed. For this analysis, a generalized dynamic model of HSGT considering the manufacturing eccentric errors, time-varying mesh stiffness, and load balancing mechanism is established and solved by numerical method. The floating process of sun gear is explained. In this paper, there are seven cases about the eccentric errors of star gears which are calculated, respectively. To study the effect of the working frequencies (including meshing frequency and rotation frequency), the calculation is done at three kinds of input speed in which the working frequencies are close to the system natural frequencies. The results are demonstrated in detail by the bifurcation diagrams, phase plane plots, and Poincare maps. The system bifurcation characteristics are particularly analyzed and compared in every case. This work provides important guidance to the engineering of HSGT. [ABSTRACT FROM AUTHOR]

Local bifurcations occur in power systems, causing changes in power system dynamic behaviors. These local bifurcations include the saddle-node bifurcation, Hopf bifurcation, and structureinduced bifurcation. This paper presents a new type of bifurcation that can occur in the optimal power flow (OPF) problem. This new type of bifurcation, termed pseudo-pitchfork bifurcation, involves bifurcations of the feasible components of the OPF problem and the disappearance of local optimal power flow solutions. The main features of this special type of bifurcation are demonstrated on several power systems with different loading condition parameters and different constraint parameters. Then the computation consideration and physical meaning of the pseudo-pitchfork bifurcation are roughly discussed. It is also demonstrated that a pseudopitchfork bifurcation occurring between feasible components can help us interpret the loss or birth of optimal power flow solutions and can lead to powerful numerical methods for computing high-quality optimal power flow solutions. [ABSTRACT FROM AUTHOR]

In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields, where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, the authors construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. By means of the bifurcation equations, the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles. Moreover, an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system. [ABSTRACT FROM AUTHOR]

We study convection in a two-dimensional container of porous material saturated with fluid and heated from below. This problem belongs to the class of dynamical systems with nontrivial cosymmetry. The cosymmetry gives rise to a hidden parameter in the system and continuous families of infinitely many equilibria, and leads to non-trivial bifurcations. In this article we present our numerical studies that demonstrate nonlinear phenomena resulting from the existence of cosymmetry. We give a comprehensive picture of different bifurcations which occur in cosymmetric dynamical systems and in the convection problem. It includes internal and external (as an invariant set) bifurcations of one-parameter families of equilibria, as well as bifurcations leading to periodic, quasiperiodic and chaotic behaviour. The existence of infinite number of stable steady-state regimes begs the important question as to which of them can realize in physical experiments. In this paper, this question (known as the selection problem) is studied in detail. In particular, we show that the selection scenarios strongly depend on the initial temperature distribution of the fluid. The calculations are carried out by the global cosymmetry-preserving Galerkin method, and numerical methods used to analyse cosymmetric systems are also described. [ABSTRACT FROM AUTHOR]

Quantitative models may exhibit sophisticated behaviour that includes having multiple steady states, bistability, limit cycles, and period-doubling bifurcation. Such behaviour is typically driven by the numerical dynamics of the model, where the values of various numerical parameters play the crucial role. We introduce in this paper natural correspondents of these concepts to reaction systems modelling, a framework based on elementary set theoretical, forbidding/enforcing-based mechanisms. We construct several reaction systems models exhibiting these properties. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of global, blow-up and bifurcation solutions for parametrized families of elliptic and parabolic equations with p-Laplacian and concave-convex nonlinearity. The main results are obtained by means of a generalised Collatz-Wielandt formula. [ABSTRACT FROM AUTHOR]

In this paper a harmonically excited linear oscillator with a play is investigated. Direct numerical simulation and numerical continuation techniques were employed to study the system behaviour. To conduct the numerical analysis, the system differential equations were transformed into the autonomous form and were then solved using our newly developed in-house Matlab-based computational suite ABESPOL [1] . The results are presented in form of trajectories and Poincaré maps on the phase plane, bifurcation diagrams and basins of attraction. The bifurcation analysis was supported by a path following method. The influence of each system parameter (except gap) on the system dynamics was studied in detail. The bifurcations known as interior crisis and boundary crisis were observed and discussed in this work. Notably, the parameter regions where various types of grazing induced bifurcations occurred were detected and investigated. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical study of the pulsatile transitional flow crossing a severe real stenosis located right in front of the bifurcation between the right subclavian and right common carotid arteries. The simulation allows one to determine relevant features of this subject-specific flow, such as the pressure waves in the right subclavian and right common carotid arteries. We explain the subclavian steal syndrome suffered by the patient in terms of the drastic pressure drop in the right subclavian artery. This pressure drop is caused by both the diverging part of the analyzed stenosis and the reverse flow in the bifurcation induced by another stenosis in the right internal carotid artery. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses the dynamics of particles orbiting planetary rings under a general central potential. Starting with the mathematical description of the dynamical system, we analyze the motion of a particle with infinitesimal mass as attracted by a central body surrounded by a homogeneous circular annular disk. Throughout the paper we carry out an analytic search of the most relevant equilibria solutions and, based on that, we investigate numerically the stability matrix of the system to find stability inequalities. In this way, we describe the in-plane and out-of-plane motion by means of the numerical continuation of a wide number of uni-parametric families of planar and spatial periodic orbits. We present a description of the main families of periodic orbits encountered, their bifurcations and linear stability. With the aim of reproducing a more realistic scenario, we analyse different mass proportions between the annulus and the central body, we consider an oblate planet and we also include a composition of rings in the dynamical model. [Copyright &y& Elsevier]

We seek discrete approximations to solutions u : Ω → ℝ of semilinear elliptic PDE of the form Δu + fs (u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in ℝd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0,π)³ ⊆ ℝ³. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464]. [ABSTRACT FROM AUTHOR]

In this paper we discuss the harvesting of the prey species making a fraction of them to be accessed by the predator while both the prey and predator are being subjected to Beddington-DeAngelis functional response. It is observed that a Hopf-bifurcation may occur around the interior equilibrium taking the environmental carrying capacity of the prey species as the parameter. Some numerical examples and the corresponding curves are studied using Maple to explain the results of the proposed model. [ABSTRACT FROM AUTHOR]

A stage-structure predator prey model is proposed and analyzed in this paper in which predators are divided into juvenile and mature predators using Monod–Haldane-type response function. The dynamical behavior of this system both analytically and numerically is investigated from the view point of stability and bifurcation. We investigate global stability around the interior equilibrium point E * by constructing suitable Lyapunov function. Our model simulation indicates that the conversion of prey population to juvenile predators can destabilize the model system which lead to limit cycle oscillations. We also investigate that the rate of juvenile predators becoming mature predators play an important role to destabilize the model system for the stable coexistence of both the populations. We carried out extensive numerical simulations of the model to confirm the analytical findings. [ABSTRACT FROM AUTHOR]

The present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elasto-plasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly non-linear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the Marciniak–Kuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme. [ABSTRACT FROM AUTHOR]