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.     

Bifurcations in a parametrically excited non-linear oscillator

A non-linear parametrically excited oscillator, that includes van der Pol as well as Duffing type non-linearities, is studied for its small non-linear motions using the method of averaging. The averaged equations, which form a dynamical system on the plane and depend on the linear damping and the detuning, are analyzed for their constant and periodic solutions. Bendixon's criterion is used to deduce the existence and the non-existence of limit cycle solutions for various values of the parameters. Then, using local bifurcation theory for “saddle-node”, pitchfork and “Hopf” bifurcations and some results from one and two parameter unfoldings of degenerate singularities, a partial bifurcation set is constructed. Since constant and periodic solutions of the averaged system correspond, respectively, to the periodic solutions and almost periodic or amplitude modulated motions of the original oscillator, the bifurcation set indicates some ways in which periodic solutions can become “entrained” or can break the entrainment for almost periodic oscillations.     

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.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. 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, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

Numerical Continuation of Solutions and Bifurcation Analysis in Multibody Systems Applied to Motorcycle Dynamics

It is shown how the equations of motion for a multibody system can be generated in a symbolic form and the resulting equations can be used in a program for the analysis of nonlinear dynamical systems. Stationary and periodic solutions are continued when a parameter is allowed to vary and bifurcations are found. The variational or linearized equations and derivatives with respect to parameters are also provided to the analysis program, which enhances the efficiency and accuracy of the calculations. The analysis procedure is firstly applied to a rotating orthogonal double pendulum, which serves as a test for the correctness of the implementation and the viability of the approach. Then, the procedure is used for the analysis of the dynamics of a motorcycle. For running straight ahead, the nominal solution undergoes Hopf bifurcations if the forward velocity is varied, which lead to periodic wobble and weave motions. For stationary cornering, wobble instabilities are found at much lower speeds, while the maximal speed is limited by the saturation of the tyre forces.     

Three-to-One Internal Resonances in Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.     

Nonlinear nonplanar vibration of a functionally graded box beam

A functionally graded material (FGM) is a type of material designed to change continuously within the solid. It can be designed for specific applications such as thermal barrier coatings, corrosion protection, biomedical materials, space/aerospace industries, automotive applications, compliant mechanisms etc. In these applications, many primary and secondary structural elements can be idealized as beams. So, the aim of the present work is to study the nonlinear nonplanar vibration of a clamped-free slender box beam made of a FGM. More specifically, the cross section consisting of two isotropic materials, connected by a FG layer, is considered. To correctly describe the dynamic characteristics of the system, the nonlinear integro-differential equations used in this work, which consider the flexural–flexural–torsional couplings that occur in the nonplanar motions of the beam, include both geometric and inertial nonlinearities. In addition, the Galerkin method is applied to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge–Kutta method. A detailed parametric analysis using several tools of nonlinear dynamics, unveils the complex dynamics of the FG beam in the main resonance region. The FG beam displays a complex nonlinear dynamic behavior with several coexisting planar and nonplanar solutions, leading to an intricate bifurcation scenario. Special attention is given to the symmetry breaking of beam dynamics and its influence on the bifurcations and instabilities. The results show that even small variations in cross section and material gradation have profound influence on the bifurcation diagrams and the dynamic behavior of the structure.     

Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays

The congestion control algorithm, which has dynamic adaptations at both user ends and link ends, with heterogeneous delays is considered and analyzed. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as the bifurcation parameter, and when the delay exceeds a critical value, a limit cycle emerges via a Hopf bifurcation. Resonant double Hopf bifurcation is also found to occur in this model. An efficient perturbation-incremental method is presented to study the delay-induced resonant double Hopf bifurcation. For the bifurcation parameter close to a double Hopf point, the approximate expressions of the periodic solutions are updated iteratively by use of the perturbation-incremental method. Simulation results have verified and demonstrated the correctness of the theoretical results.     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.     

Chaotic Dynamics of a Harmonically Excited Spring-Pendulum System with Internal Resonance

An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

神经网络动力系统吸引子对称性的变化

用余维２的双Ｈｏｐｆ分叉的规范形方程设计了期望存储振荡型记忆模式的模拟四阶关系神经网络，所设计的网络向量场具有中心对称性。研究表明该网络发生二阶双Ｈｏｐｆ分叉后可以出现不变二环面。观察二环面上的销相运动，发现了系统出现对称破裂的规律，邓把双Ｈｏｐｆ分叉的两个频率比表示成既约分数的形式，当该分数的分子和分母均为奇数时，网络的吸引子保持对称性，而当分子主分母中任一个为偶数时，就会发生对称破裂。     

Hopf bifurcation and chaotic motions of a tubular cantilever subject to cross flow and loose support

The Hopf bifurcation and chaotic motions of a tubular cantilever impacting on loose support is studied using an analytic model that involves delay differential equations. By using the damping-controlled mechanism, a single flexible cantilever in an otherwise rigid square array of cylinders is analyzed. The analytical model, after Galerkin discretization to five d.o.f., exhibits interesting dynamical behavior. Numerical solutions show that, with increasing flow beyond the critical, the amplitude of motion grows until impacting with the loose support placed at the tip end of the cylinder occurs; more complex motions then arise, leading to chaos and quasi-periodic motions for a sufficiently high flow velocity. The effect of location of the loose support on the global dynamics of the system is also investigated.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Stability and Oscillations in a Time-Delayed Vehicle System with Driver Control

In this paper, linear stability and chaotic motion of a time-delayednonlinear vehicle system are studied. The stability is determined bycomputing the spectrum associated with a system of linear retardedfunctional differential equations, which reveals that a loss ofstability occurs following a Hopf bifurcation. Beyond the critical valuefor linear stability, the system exhibits limit cycle motions.Subharmonic, quasi-periodic and chaotic motions are observed for asystem excited by a periodic disturbance.     

Nonlinear dynamics of imperfectly-supported pipes conveying fluid

In this paper, the nonlinear dynamics of a pipe imperfectly supported at the upstream end and free at the other and conveying fluid is investigated. The imperfect support is modelled via cubic translational and rotational springs. The equation of motion is obtained via Hamilton’s principle for an open system, and the Galerkin method is used for discretizing the resulting partial differential equation. The dynamics of a system with either strong rotational or strong translational stiffness is examined in details. Numerical results show that similarly to a cantilevered pipe, the system undergoes a supercritical Hopf bifurcation leading to period-1 limit cycle oscillations. The Hopf bifurcation may, however, occur at a much lower flow velocity compared to the perfect system. At higher flow velocities, quasi-periodic and chaotic-like motions may be observed. The amplitude of transverse displacement is generally much higher than that for a cantilevered pipe, mainly due to large-amplitude rigid-body motion. In addition, effects of the mass ratio, internal dissipation, hardening- or softening-type nonlinearity, as well as concentrated- or distributed-type nonlinearity on the dynamics of the system are examined.     

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.     

Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

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.