Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先基于哈密顿变分原理，引入与损伤程度、范围和位置相关的三个无量纲参数，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

Cyclic motions near a Hopf bifurcation of a four-dimensional system

We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Oscillatory dynamics in a gene regulatory network mediated by small RNA with time delay

A genetic regulatory network mediated by small RNA with two time delays is investigated. We show by mathematical analysis and simulation that time delays can provide a mechanism for the intracellular oscillator. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive 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.     

Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations

Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions

Nonlinear dynamics of elastic structures with two-mode interactions have been extensively studied in the literature. In this work, nonlinear forced response of elastic structures with essential inertial nonlinearities undergoing three-mode interactions is studied. More specifically, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied. Linear modal characteristics of the structure with uniform beams depend on the length ratios of the three beams, the mass of the particle relative to that of the structure, and the location of the mass particle along the beams. The discretized model is studied for both external and parametric resonances for parameter combinations resulting in three-mode interactions. For the external excitation case, focus is on the system with 1:2:3 internal resonances with the external excitation frequency near the middle natural frequency. For the case of the structure with 1:2:5 internal resonances, the problem involving simultaneous principal parametric resonance of the middle mode and a combination resonance between the lowest and the highest modal frequencies is investigated. This case requires a higher-order approximation in the method of multiple time scales. For both cases, equilibrium and bifurcating solutions of the slow-flow equations are studied in detail. Many pitchfork, saddle-node, and Hopf bifurcations appear in the amplitude response of the three-beam structure, thus resulting in complex multimode responses in different parameter regions.     

The new result on delayed finance system

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.     

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.     

Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations

The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Bifurcation analysis of milling process with tool wear and process damping: regenerative chatter with primary resonance

In this paper, the occurrence of various types of bifurcation including symmetry breaking, period-doubling (flip) and secondary Hopf (Neimark) bifurcations in milling process with tool-wear and process damping effects are investigated. An extended dynamic model of the milling process with tool flank wear, process damping and nonlinearities in regenerative chatter terms is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. Non-autonomous parametrically excited equations of the system with time delay terms are developed. The multiple-scale approach is used to construct analytical approximate solutions under primary resonance. Periodic, quasi-periodic and chaotic behavior of the limit cycles is predicted in the presence of regenerative chatter. Detuning parameter (deviation of the tooth passing frequency from the chatter frequency), damping ratio (affected by process damping) and tool-wear width are the bifurcation parameters. Multiple period-doubling and Hopf bifurcations occur when the detuning parameter is varied. As the damping ratio changes, symmetry breaking bifurcation is observed whereas the variation of tool wear width causes both symmetry breaking and Hopf bifurcations. Also, under special damping specifications, chaotic behavior is seen following the Hopf bifurcation.     

Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion

A set of nonlinear differential equations is established by using Kane‘s method for the planar oscillation of flexible beams undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base, the second natural frequency of the beam may approximate the tripled first one so that the condition of 3 : 1 internal resonance of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to derive a set of nonlinear modulation equations for the principal parametric resonance of the first mode combined with 3 : 1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation,the 4-dimensional modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.     

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays

This paper presents an investigation of stability and Hopf bifurcation of a synaptically coupled nonidentical HR model with two time delays. By regarding the half of the sum of two delays as a parameter, we first consider the existence of local Hopf bifurcations, and then derive explicit formulas for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions, using the normal form method and center manifold theory. Finally, numerical simulations are carried out for supporting theoretical analysis results.     

Bifurcations in a predator-prey model with discrete and distributed time delay

In this paper, a class of predator-prey model with discrete and distributed time delay is considered. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By using the normal form theory and center manifold theory, we derive some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Hopf bifurcation in a delayed food-limited model with feedback control

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.     

Bifurcation analysis in a delayed Lokta�CVolterra predator�Cprey model with two delays

In this paper, a class of delayed Lokta?CVolterra predator?Cprey model with two delays is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also provided. Finally, main conclusions are given.