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We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system's parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient. [ABSTRACT FROM AUTHOR]

The patchy inflammatory pattern in the gastrointestinal tract is one of the typical features of Crohn's disease, and revealing the evolutionary mechanism of Crohn's disease is helpful to further understand its pathogenesis. However, the research on the pathogenesis of Crohn's disease is not comprehensive. To this end, this paper investigates the spatiotemporal dynamics of the phagocyte-bacterium model with diffusion and explores the evolutionary mechanism of Crohn's disease from the perspective of bifurcation analysis. Theoretically, we use the normal form theory of the reaction-diffusion equation and the neutral manifold theory to derive the normal form of the system near the Turing–Hopf bifurcation. Then, the results are verified by numerical simulations, and it is found that small disturbances in the control parameters led to four different types of dynamic states in the system: uniform state, spatially non-uniform steady state, spatially uniform periodic state, and spatially non-uniform periodic state. At the same time, the effects of small alterations in intestinal permeability and phagocytes diffusion rate near the Turing–Hopf bifurcation point on pattern structure and bacterial biomass of Crohn's disease were revealed. In addition, we found that there are three different types of pattern transformation in the system. These findings will help to better understand the development process of Crohn's disease, predict its pattern transformation, and provide theoretical guidance for the research and development of new drugs. Meanwhile, our research findings provide insights into the vulnerability of the mucosal immune system in Crohn's disease. [ABSTRACT FROM AUTHOR]

Nonlinear problems involving phases occur ubiquitously throughout applied mathematics andphysics, ranging from neuronal models to the search for elementary particles. The phase variables present in such models usually enter as harmonic terms and, being unbounded, pose an open challenge for studying bifurcations in these systems through standard numerical continuation techniques. Here, we propose to transform and embed the original model equations involving phases into structurally stable generalized systems that are more suitable for analysis via standard predictor–corrector numerical continuation methods. The structural stability of the generalized system is achieved by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. As an illustration of this general approach, specific details are provided for the ac-driven, Stewart–McCumber model of a single Josephson junction. It is found that the dynamics of the junction is underpinned by a two-parameter Hopf–Hopf bifurcation, detected in the generalized system. The Hopf–Hopf bifurcation gives birth to an invariant torus through Neimark–Sacker bifurcation of limit cycles. Continuation of the Neimark–Sacker bifurcation of limit cycles in the two-parameter space provides a complete picture of the overlapping Arnold tongues (regions of frequency-locked periodic solutions), which are in precise agreement with the widths of the Shapiro steps that can be measured along the current–voltage characteristics of the junction at various fixed values of the ac-drive amplitude. [ABSTRACT FROM AUTHOR]

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example. [ABSTRACT FROM AUTHOR]

In this paper, based on an original mass transfer model, we propose a new volatile organic compound (VOC) emission model which contains a homogeneous linear parabolic partial differential equation with mixed boundary conditions and a nonlinear ordinary differential equation. First, we transfer above equations to nonlinear delay differential equations by using an integral mean value theorem. Second, we prove the global asymptotic stability of the unique positive equilibrium for ordinary differential equation. Next, we show linear stability analysis and obtain the existence of Hopf and Hopf-zero bifurcations associated with the improved nonlinear delayed VOC emission system. Moreover, we derive explicit formulae for determining the stability of Hopf-bifurcating periodic solutions and the direction of Hopf bifurcation. Finally, detailed numerical simulations, such as local asymptotically stable equilibrium, global asymptotically stable equilibrium, the effects of parameters on VOC emission, are presented to demonstrate the applications of the theoretical results, and the explanations and comparisons for numerical results are also presented. In particular, the interesting simulation results associated with the global asymptotic stability of equilibrium for delayed differential equation for some region of delay are also shown. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate the application of the conserved-mass metamaterial concept to control the flutter onset in aircraft wings and mitigate their induced vibrations. The numerical study is conducted on a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. The aeroelastic system is attached to an array of passive vibration absorbers. The mass of the vibration absorbers is deducted from the mass of the aeroelastic structure, which makes the total mass of the system conserved. The equal mass constraint is imposed so that improvements in the aeroelastic performance are due to the addition of the vibration absorbers and not the added mass attached to the main aeroelastic system. The linear analysis of the aeroelastic system reveals that a proper selection of the stiffness and position of the vibration absorber leads to 23.4% increase in the flutter speed when deploying one single absorber. Placing the vibration absorber closer to the leading edge is observed to delay the occurrence of flutter. Furthermore, considering an array of distributed vibration absorbers with a proper selection of their stiffness is found to result in an increase of the flutter speed by 84%. We also develop the normal form of the aeroelastic system equipped with the vibration absorber using the method of multiple scales to investigate the limit cycle oscillations (LCOs) beyond the flutter boundary (Hopf bifurcation). The analytical results are verified against their numerical counterparts. The normal form is used to examine the effects of the vibration absorber stiffness nonlinearities on the dynamic behavior of the aeroelastic system. Incorporating quadratic and cubic stiffness coefficients is found to degrade the aeroelastic system by amplifying the amplitude of LCOs beyond flutter. However, soft pitch spring proved to be beneficial to decrease the LCO amplitude. Moreover, the increase in the number of the vibration absorbers while imposing the equal mass constraint is observed to lead to relatively lower LCO amplitudes in the post-flutter regime. [ABSTRACT FROM AUTHOR]

We study the bifurcations of slow-fast cycles with two canard points in singularly perturbed planar systems. After analyzing the local dynamics of two canard points lying on the S-shaped critical manifolds, we give a sufficient condition under which there exist three hyperbolic limit cycles bifurcating from some slow-fast cycles. The proof is based on the geometric singular perturbation theory. Then, we apply the results to cubic Liénard equations with quadratic damping, and prove the coexistence of three large limit cycles enclosing three equilibria. This is a new dynamical configuration and has never been previously found in the existing references. [ABSTRACT FROM AUTHOR]

We examine applicability of normal forms of non-holonomic robotic systems to the problem of motion planning. A case study is analyzed of a planar, free-floating space robot consisting of a mobile base equipped with an on-board manipulator. It is assumed that during the robot's motion its conserved angular momentum is zero. The motion planning problem is first solved at velocity level, and then torques at the joints are found as a solution of an inverse dynamics problem. A novelty of this paper lies in using the chained normal form of the robot's dynamics and corresponding feedback transformations for motion planning at the velocity level. Two basic cases are studied, depending on the position of mounting point of the on-board manipulator. Comprehensive computational results are presented, and compared with the results provided by the Endogenous Configuration Space Approach. Advantages and limitations of applying normal forms for robot motion planning are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form, and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third- and fourth-order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. Then, a nonlinear stability analysis for two periodic solutions is performed in the third- and fourth-order resonance cases as well as in the nonresonance case. [ABSTRACT FROM AUTHOR]

The Bogdanov–Takens (B–T) bifurcation of a delayed predator-prey system with double Allee effect in prey are studied in this paper. According to the existence conditions of B–T bifurcation, we give the associated generic unfolding, and derive the normal forms of the B–T bifurcation of the model at its interior equilibria by generalizing and using the normal form theory and center manifold theorem for delay differential equations. By analyzing the topologically equivalent normal form system, one find that the Allee effect and delay can lead to varies dynamic behaviors, which is believed to be beneficial for understanding the potential mathematical mechanism that driving population dynamics. [ABSTRACT FROM AUTHOR]

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of the phase space. The second method is the modal derivative approach, and more specifically, the quadratic manifold defined in order to derive a second-order nonlinear change of coordinates. Both methods share a common point of view, willing to introduce a nonlinear mapping to better define a reduced-order model that could take more properly into account the nonlinear restoring forces. However, the calculation methods are different and the quadratic manifold approach has not the invariance property embedded in its definition. Modal derivatives and static modal derivatives are investigated, and their distinctive features in the treatment of the quadratic nonlinearity are underlined. Assuming a slow/fast decomposition allows understanding how the three methods tend to share equivalent properties. While they give proper estimations for flat symmetric structures having a specific shape of nonlinearities and a clear slow/fast decomposition between flexural and in-plane modes, the treatment of the quadratic nonlinearity makes the predictions different in the case of curved structures such as arches and shells. In the more general case, normal form approach appears preferable since it allows correct predictions of a number of important nonlinear features, including the hardening/softening behaviour, whatever the relationships between slave and master coordinates are. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and analyze a modified Leslie–Gower predator–prey model. Our model incorporates refuge of preys, additional fixed food for predators, harvesting of preys through a continuous threshold policy and a time delay as to account for predators maturity time. We first carry out a qualitative analysis of the model without time delay, showing existence of extinction, prey-free, predator-free and coexistence equilibria. We further study their stability conditions. Relying only on theoretical results of the model, we construct bifurcation diagrams involving refuge and harvest limit parameters. This led to summarize different scenarios for the model including elimination of one species or competition of both species that are proved possible. Furthermore, considering the time delay as bifurcation parameter, we analyze the stability of the coexistence equilibria and prove the system can undergoes a Hopf bifurcation. The direction of that Hopf bifurcation and the stability of the bifurcated periodic solution are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, based on the original friction-induced oscillation, we propose a new friction-induced mathematical model associated with moving belt of particle supply device. Two delay feedback control methods are provided to make the new model be stable. Linear stability analysis is carried out to obtain the stability of equilibrium and the stability boundary, corresponding to the critical value of Hopf bifurcation, dividing stable and unstable regimes of equilibrium. Next, we derive explicit formulae for determining the stability of Hopf bifurcating periodic solutions and the direction of Hopf bifurcation. Finally, detailed numerical simulations using MATLAB software are presented to demonstrate the application of the theoretical results, and we also compare the efficacy of the two control forces in new friction-induced models. [ABSTRACT FROM AUTHOR]

This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package. [ABSTRACT FROM AUTHOR]

It is possible to use numerical techniques to provide solutions to nonlinear dynamical systems that can be considered exact up to numerical tolerances. However, often, this does not provide the user with sufficient information to fully understand the behaviour of these systems. To address this issue, it is common practice to find an approximate solution using an analytical method, which can be used to develop a more thorough appreciation of how the parameters of a system influence its response. This paper considers three such techniques—the harmonic balance, multiple scales, and direct normal form methods—in their ability to accurately capture the forced response of nonlinear structures. Using frequency detuning as a method of comparison, it is shown that it is possible for all three methods to give identical solutions, should particular conditions be used. [ABSTRACT FROM AUTHOR]

In this paper, integrability problem and bifurcation of limit cycles for cubic Kukles systems which are assumed to have a nilpotent origin are investigated. A complete classification is given on the integrability conditions and proven to have a total of 7 cases. Bifurcation of limit cycles is also discussed; six or seven limit cycles can be obtained by two different perturbation methods. Integrability problem and bifurcation of limit cycles for the cubic Kukles systems with a nilpotent origin have been completely solved. [ABSTRACT FROM AUTHOR]

The dynamics of a nonlinear electro-optical oscillator with delayed bandpass feedback is investigated. By analyzing the distribution of the eigenvalues, the existence and stability of Hopf bifurcations are obtained. Particularly, the stability switches are found as the delay varies, where the time delay is regarded as bifurcation parameter. And then, by applying the normal form method and center manifold theory of functional differential equations, an algorithm for determining the sense of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived. For illustrating the theoretical results, some numerical simulations are performed. [ABSTRACT FROM AUTHOR]

Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical structures, due to their ability to offer insight into the system dynamics. A comparison of their accuracy has not previously been undertaken, so is addressed in this paper. This is achieved by computing the backbone curves of two systems: the single-degree-of-freedom Duffing oscillator and a non-symmetric, two-degree-of-freedom oscillator. The DNF method includes an inherent detuning, which can be physically interpreted as a series expansion about the natural frequencies of the underlying linear system and has previously been shown to increase its accuracy. In contrast, there is no such inbuilt detuning for MS, although one may be, and usually is, included. This paper investigates the use of the DNF detuning as the chosen detuning in the MS method as a way of equating the two techniques, demonstrating that the two can be made to give identical results up to ε2 order. For the examples considered here, the resulting predictions are more accurate than those provided by the standard MS technique. Wolfram Mathematica scripts implementing these methods have been provided to be used in conjunction with this paper to illustrate their practicality. [ABSTRACT FROM AUTHOR]

The local invariant (stable, unstable, and center) manifolds of a saddle × center × center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller. [ABSTRACT FROM AUTHOR]

Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many systems exhibit non-smooth dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker-Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems. [ABSTRACT FROM AUTHOR]

The problem of determining the bounds of applicability of perturbation expansions in terms both of the system parameters and the state-space variable amplitude is a key point in the perturbation analysis of nonlinear systems. In the present paper an analysis in a finite neighborhood of a Hopf bifurcation is presented in order to analyze the conditions under which a Normal Form zero-divisors-based approach fails to describe the local dynamics and, therefore, a small divisor approach is required. The condition of “smallness” referred to the divisors is analyzed from both a qualitative and a quantitative point of view. Finally, a simple but effective analytical and numerical example is introduced to illustrate the theoretical issues along with an interpretation within a codimension-two framework. [ABSTRACT FROM AUTHOR]

In this paper, we study the mathematical model about variable frequency water supply system in the process of applying glue to particleboard. Based on the original linear ordinary differential model, the effects of time delay and the nonlinear factor are considered. Then, we obtain the delayed nonlinear differential equation associated with variable frequency water supply system. Further, we consider the existence and stability of the equilibria and the existence of several types of bifurcations in this functional differential equation. Next, we derive the normal forms of Hopf bifurcation and Bogdanov-Takens bifurcation by using the multiple time scales method and the center manifold reduction method, respectively, and analyze the classifications of local dynamics. Finally, by using Matlab software, we obtain numerical simulations with estimated values of parameters and show the existence of stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and a complex chaotic attractor from a sequence of period-doubling bifurcations. [ABSTRACT FROM AUTHOR]

This paper considers the computation of normal form associated with codimension-two Bogdanov-Takens (BT) bifurcation in delay differential equations. The main attention is focused on dynamical systems described by delay differential equations having a double-zero eigenvalue with geometric multiplicity one, which is usually called non-semisimple double-zero eigenvalue. Explicit formulas for computing the normal form of such systems with two unfolding parameters are obtained by applying center manifold reduction and the method of normal forms. In particular, the normal form associated with the flow on a center manifold up to third-order terms are derived. As an application, an Oregonator oscillator having such a BT singularity with delay is analyzed using our explicit formulas. [ABSTRACT FROM AUTHOR]

In this paper, stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of bifurcation response equations are considered. They are characterized as (1) one pair of purely imaginary eigenvalues and two pairs of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary eigenvalues in nonresonant case and one pair of conjugate complex roots with negative real parts; (3) three pairs of purely imaginary eigenvalues in nonresonant case. With the aid of Maple software and normal form theory, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are obtained, which can lead to static bifurcation and Hopf bifurcation. Under certain conditions, 2-D tori motion may occur. The complex dynamical motions are considered in this paper. Finally, the numerical solutions achieved by the fourth-order Runge-Kutta method agree with the analytic results. [ABSTRACT FROM AUTHOR]

In this paper, a damped harmonic oscillator with multiple delays is investigated. The existence conditions under which the origin of the system is a Bogdanov-Takens (B-T) singularity are derived. By utilizing the center manifold reduction and choosing suitable bifurcation parameters, the second- and the third-order normal forms of the B-T bifurcation for the system are obtained. Finally, numerical simulations are presented to illustrate the theoretical criteria. [ABSTRACT FROM AUTHOR]

The effectiveness of the nonlinear energy sink in controlling the limit cycle oscillations of a nonlinear aeroelastic system is assessed. The system consists of a rigid airfoil elastically mounted on linear and nonlinear springs. The coupled equations of the airfoil and sink are derived using Lagrange's equations. The nonlinear quasi-steady aerodynamics are used to model the aerodynamic loads. Parameters including the mass and placement of the passive controller are varied in order to test its efficiency in suppressing undesirable aeroelastic behavior under varying conditions. The nonlinear normal form governing the responses of the airfoil and the energy sink is derived. The contribution of the aerodynamic, structural and sink nonlinearities to the type of instability is quantified. The results show that the nonlinear energy sink has a limited impact on the system's response in terms of effectively delaying the onset of flutter, changing the type of instability or reducing the amplitude of the limit cycle oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a modified Lesile-Gower predator-prey system with delay is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. Furthermore, different to previous papers, a multiple time scale technique is employed to calculate the normal form on the center manifold of delay differential equations, which is much easier to implement in practice than the conventional method, center manifold reduction. Finally, to verify our theoretical predictions, some numerical simulations are also included. [ABSTRACT FROM AUTHOR]

In this paper, we study dynamics in delayed class-B laser system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations and derive the normal forms near the Hopf and double Hopf bifurcations critical points. By analyzing local dynamics near bifurcation critical points, we show how the delayed feedback control parameters effect the dynamical behaviors of the system. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameter. Namely, even for parameter values not chosen in the neighborhood of the Hopf bifurcation critical points, two families of stable periodic solutions, which are resulted from Hopf bifurcation, exist in a large region of delay, and they merge into a family of stable and globally existed periodic solutions. Finally, by choosing proper control parameters, numerical simulations, including stable equilibrium, stable periodic solutions and stable quasiperiodic solutions are presented to demonstrate the theoretical results. Therefore, in accordance with above theoretical analysis, reasonable lasers with proper control parameters can be designed in order to achieve various applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the traditional Van der Pol oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity, which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of center manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback. [ABSTRACT FROM AUTHOR]

In this paper, a finance system with time delay is considered. By linearizing the system at the unique equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the unique equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one. [ABSTRACT FROM AUTHOR]

In this paper, a ratio-dependent predator-prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction. [ABSTRACT FROM AUTHOR]

In this paper, a four-neuron BAM neural network model with multiple delays is considered. The existence conditions under which that the origin of the system is Bogdanov-Takens (B-T) or triple zero singularity are given. By choosing the connected weights as bifurcation parameters and using the center manifold reduction and the normal form theory and the formula developed by Xu and Huang (J Differ Equ 244:582-598 ) and Qiao et al. (Chinese Ann Math Ser A 31:59-70 ), the versal unfoldings and the normal forms for this singularity were given to analyze the behaviors of the system. This paper is a further study of paper Cao and Xiao (IEEE Trans Neural Netw 18:416-430 ). [ABSTRACT FROM AUTHOR]

This paper analyzes the double Neimark-Sacker bifurcation occurring in a two-DoF system, subject to PD digital position control. In the model the control force is considered piecewise constant. Introducing a nonlinearity related to the saturation of the control force, the bifurcations occurring in the system are analyzed. The system is generally losing stability through Neimark-Sacker bifurcations, with relatively simple dynamics. However, the interaction of two different Neimark-Sacker bifurcations steers the system to much more complicated behavior. Our analysis is carried out using the method proposed by Kuznetsov and Meijer. It consists of reducing the dynamics of the nonlinear map to its local center manifold, eliminating the non-internally resonant nonlinear terms and transforming the nonlinear map to an amplitude map, that describes the local dynamics of the system. The analysis of this amplitude map allows us to define regions, in the space of the control gains, with a close interaction of the two bifurcations, which generates unstable quasiperiodic motion on a 3-torus, coexisting with two stable 2-torus quasiperiodic motions. Other regions in the space of the control gains show the coexistence of 2-torus quasiperiodic solutions, one stable and the other unstable. All the results described in this work are analytical and obtained in closed form, numerical simulations illustrate and confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this manuscript, we provide a framework for zero-Hopf singularity for general delayed differential equations (DDEs). We formulate the explicit conditions such that zero-Hopf singularity occurs for general DDEs and derive the corresponding normal forms up to the third-order terms. As an application of our framework, we discuss zero-Hopf singularity for a Fitzhugh-Nagumo model with delayed feedback control. [ABSTRACT FROM AUTHOR]

The stability and bifurcation of delayed feedback spin stabilization of a rigid spacecraft is investigated in this paper. The spin is stabilized about the principal axis of the intermediate moment of inertia using a simple delayed feedback control law. In particular, linear stability is analyzed via the exponential-polynomial characteristic equations and then the method of multiple scales is used to obtain the normal form of the Hopf bifurcation. Bifurcation diagrams and the dynamics of the delayed closed-loop system are verified using continuation software and with numerical simulations. [ABSTRACT FROM AUTHOR]

A brief review is stated on a hypernormalization method of autonomous differential equations in the vicinity of an equilibrium. The formalization is obtained from the method of cohomology spectral sequences. We skip the technical concepts such that it can be accessible for a broad audience. Three normal form styles are demonstrated by computing three different and known first-level normal forms of Bogdanov-Takens singularity. [ABSTRACT FROM AUTHOR]

In this paper, we consider a classical van der Pol equation with a general delayed feedback. Firstly, by analyzing the associated characteristic equation, we derive a set of parameter values where the Hopf bifurcation occurs. Secondly, in the case of the standard Hopf bifurcation, the stability of bifurcating periodic solutions and bifurcation direction are determined by applying the normal form theorem and the center manifold theorem. Finally, a generalized Hopf bifurcation corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance) is analyzed by using a normal form approach. [ABSTRACT FROM AUTHOR]

We perform an analytical and experimental investigation into the dynamics of an aeroelastic system consisting of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The experimental results show that the onset of flutter takes place at a speed smaller than the one predicted by a quasi-steady aerodynamic approximation. On the other hand, the unsteady representation of the aerodynamic loads accurately predicts the experimental value. The linear analysis details the difference in both formulation and provides an explanation for this difference. Nonlinear analysis is then performed to identify the nonlinear coefficients of the pitch spring. The normal form of the Hopf bifurcation is then derived to characterize the type of instability. It is demonstrated that the instability of the considered aeroelastic system is supercritical as observed in the experiments. [ABSTRACT FROM AUTHOR]

A symbolic computation scheme and its corresponding Maple program are developed to compute the normal form for the Hopf bifurcation in a neutral delay differential equation. In the symbolic computation scheme, the neutral delay differential equation is considered as an ordinary differential equation in an appropriate infinite-dimensional phase space so that both center manifold reduction and normal form computation can be simultaneously conducted without computing center manifold beforehand. The Maple program is proved to provide an easy way to compute the normal form of the neutral delay differential equation automatically by only inputting some basic information of the equation. As an illustrative example, the application of the Maple program to a container crane with a delayed position feedback control is given. The results reveal that the normal form obtained by using the center manifold reduction and the normal form computation is in a full agreement with the result derived by applying the method of multiple scales. Moreover, numerical analysis is presented to validate the analytical results. [ABSTRACT FROM AUTHOR]

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system's instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed. [ABSTRACT FROM AUTHOR]

This paper investigates the Hopf bifurcation of a four-dimensional hyperchaotic system with only one equilibrium. A detailed set of conditions is derived which guarantees the existence of the Hopf bifurcation. Furthermore, the standard normal form theory is applied to determine the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods. In addition, numerical simulations are used to justify theoretical results. [ABSTRACT FROM AUTHOR]

We follow two approaches to derive the normal form that represents the aeroelastic response of the Goland wing. Such a form constitutes an effective tool to model the main physical behaviors of aeroelastic systems and, as such, can be used for developing a phenomenological reduced-order model. In the first approach, an approximation of the wing's response near the Hopf bifurcation is constructed by directly applying the method of multiple scales to the two coupled partial-differential equations of motion. In the second approach, we apply the same method to a Galerkin discretized model that is based on the mode shapes of a cantilever beam. The perturbation results from both approaches are verified by comparison with results from numerical integration of the discretized equations. [ABSTRACT FROM AUTHOR]

This work investigates the influence of structural and aerodynamic nonlinearities on the dynamic behavior of a piezoaeroelastic system. The system is composed of a rigid airfoil supported by nonlinear torsional and flexural springs in the pitch and plunge motions, respectively, with a piezoelectric coupling attached to the plunge degree of freedom. The analysis shows that the effect of the electrical load resistance on the flutter speed is negligible in comparison to the effects of the linear spring coefficients. The effects of aerodynamic nonlinearities and nonlinear plunge and pitch spring coefficients on the system's stability near the bifurcation are determined from the nonlinear normal form. This is useful to characterize the effects of different parameters on the system's output and ensure that subcritical or 'catastrophic' bifurcation does not take place. Numerical solutions of the coupled equations for two different configurations are then performed to determine the effects of varying the load resistance and the nonlinear spring coefficients on the limit-cycle oscillations (LCO) in the pitch and plunge motions, the voltage output and the harvested power. [ABSTRACT FROM AUTHOR]

In the neural system, action potentials play a crucial role in many mechanisms of information communication. The quiescent state, spiking and bursting activities are important biological behaviors with the different neurocomputational properties. In this paper, based on the bifurcation mechanisms involved in the generation of action potentials, an interesting mathematical study of bursting behavior is obtained. The transition between the bursting and quiescence state is investigated,which shows that the time delay must be large enough for bursting behavior to occur in a delayed system. Two types of the codimension-two bifurcation, i.e., Bogdanov-Takens (BT) bifurcation and saddle-node homoclinic (SNH) bifurcation are investigated also. The bifurcation curves of the parameters and the phase portraits for the different regions are shown. The local existence of the homoclinic curve is achieved by using the center manifold reduction and normal form method. For occurrence of a periodic stimulation in the neighborhood of the SNH bifurcation, the system can switch over from an equilibrium state to an oscillatory state either through saddle-node on an invariant circle bifurcation (called circle bifurcation for simplicity) or saddle-node (SN) bifurcation, and back from the oscillatory state to the equilibrium state through the circle or homoclinic bifurcation. Complex bursting phenomena are displayed for the different values of delay couplings and stimulation intensities. Some types of bursting behaviors, such as Circle/Circle (Type II or parabolic bursting), Circle/Homoclinic, SN/Circle (triangular bursting), SN/Homoclinic (Type I or square-wave bursting), and Fold/Hopf bursting are obtained in the firing area. The results show that the different burstings are related to the delay coupling and external inputs. [ABSTRACT FROM AUTHOR]

In this paper, Hopf bifurcation and Bogdanov-Takens bifurcation with codimension 2 in a coupled FitzHugh-Nagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov-Takens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov-Takens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov-Takens bifurcation indicates that the neuron signals can transit between resting and spiking. [ABSTRACT FROM AUTHOR]

We use a systematic approach combining a path-following scheme, the method of multiple scales, the method of harmonic balance, Floquet theory, and numerical simulations to investigate the local and global dynamics and stability of cutting tool on a lathe due to the regenerative mechanism. First, we use the method of multiple scales to determine the normal form of the Hopf bifurcation at all of the stability boundaries and calculate the limit cycles generated by the bifurcation. Then, we use a combination of a path-following scheme and the method of harmonic balance to continue the branch of generated limit cycles. Thus, we calculate small- and large-amplitude limit cycles and ascertain their stability using Floquet theory. We validate these results using numerical simulations. Then, we search for isolated branches of large-amplitude solutions coexisting with the linearly stable trivial solution. We use all of the results to generate bifurcation diagrams consisting of multiple large-amplitude stable and unstable branches of limit cycles coexisting with the trivial response, indicating three regions of operation, as in the experimental observations. Then, we investigate bifurcation control using cubic-velocity feedback and show that the unconditionally stable region can be expanded at the expense of the conditionally stable region. [ABSTRACT FROM AUTHOR]