Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

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.     

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.     

Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1∶2 internal resonance

The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos. Project supported by the National Natural Science Foundation of China (19472046)     

The Response of a Rayleigh–Liénard Oscillator to a Fundamental Resonance

A Rayleigh–Liénard oscillator excited by a fundamentalresonance is investigated by using an asymptotic perturbation method based on Fourier expansion and time rescaling. Two first-order nonlinear ordinarydifferential equations governing the modulation of the amplitude andthe phase of solutions are derived. These equations are used todetermine steady-state responses and their stability. Excitationamplitude-response and frequency-response curves are shown and checkedby numerical integration. Dulac's criterion, the Poincaré–Bendixsontheorem, and energy considerations are used in order to study the existenceand characteristics of limit cycles of the two modulation equations. Alimit cycle corresponds to a modulated motion for the Rayleigh–Liénardoscillator. For small excitation amplitude, the analytical results arein excellent agreement with the numerical solutions. In certain caseswhen the excitation amplitude is very low, an approximate analyticsolution corresponding to a modulated motion can be obtained andnumerically checked. Moreover, if the excitation amplitude is increased,an infinite-period bifurcation occurs because the modulation periodlengthens and becomes infinite, while the modulation amplitude remainsfinite and suddenly the attractor settles down into a periodic motion.     

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.     

窄带噪声作用下二自由度非线性系统的响应

研究二自由度非线性系统在窄带随机噪声激励下的主共振响应,用多尺度法分离了系统的快变项,讨论了系统的阻尼项、随机项等对系统响应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象,饱和现象也存在。数值模拟表明文中所提出的方法是有效的。     

窄带随机噪声激励下多自由度非线性系统的响应

在随机振动的研究中，研究较多的是系统在宽带噪声作用下的响应问题，对于非线性系统特别是多自由度非线性系统在窄带随机噪声作用下的响应问题则研究较少。本文研究了三自由度非线性系统在窄带随机噪声激励下的主共振响应和稳定性问题。用多尺度法分离了系统的快变项，给出了系统响应的振幅和相位角满足的方程。用摄动法讨论了系统随机项对系统响应的影响。当随机扰动较小时，在一定的参数范围内，对应于不同的初值，系统具有两个均方响应值，随机饱和现象也存在。当随机扰动增大时，系统可从一个大的响应突跳为一个小的响应，或从一个小的响应突跳为一个大的响应，即存在随机跳跃现象。数值模拟表明本文提出的方法是有效的。     

Articulated Pipes Conveying Fluid Pulsating with High Frequency

Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.     

An Alternative Approach to Study Bifurcation from a Limit Cycle in Periodically Perturbed Autonomous Systems

The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov–Schmidt method.     

一类混沌系统中的簇发振荡及其延迟叉形分岔行为

由于多时间尺度问题在实际工程系统中广泛存在,关于其复杂动力学行为及其产生机制的研究已成为当前国内外的热点课题之一.簇发振荡是多时间尺度系统复杂动力学行为的典型代表,而分岔延迟又是簇发振荡中的常见现象.本文为探讨非线性系统中分岔延迟所引发的簇发振荡的分岔机制,在一个三维混沌系统中引入参数激励,当激励频率远小于系统的固有频率时,系统产生了两时间尺度簇发振荡.将整个激励项看做慢变参数,激励系统转化为广义自治系统也即快子系统,分析快子系统平衡点的稳定性以及分岔条件,并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理.文中考察了4组参数条件下系统的动力学行为,研究发现当慢变激励项周期性地通过分岔点时,系统产生了明显的超临界叉形分岔延迟行为,随着参数激励振幅的增大,分岔延迟的时间也逐渐延长,当这种延迟的动态行为终止于不同的参数区域时,导致系统轨线围绕不同稳定吸引子(平衡点,极限环)运动,从而得到了不同的簇发振荡行为.      

Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam

The sub- and super-critical dynamics of an axially moving beam subjected to a transverse harmonic excitation force is examined for the cases where the system is tuned to a three-to-one internal resonance as well as for the case where it is not. The governing equation of motion of this gyroscopic system is discretized by employing Galerkin’s technique which yields a set of coupled nonlinear differential equations. For the system in the sub-critical speed regime, the periodic solutions are studied using the pseudo-arclength continuation method, while the global dynamics is investigated numerically. In the latter case, bifurcation diagrams of Poincaré maps are obtained via direct time integration. Moreover, for a selected set of system parameters, the dynamics of the system is presented in the form of time histories, phase-plane portraits, and Poincaré maps. Finally, the effects of different system parameters on the amplitude-frequency responses as well as bifurcation diagrams are presented.     

Effects of thermo hydrodynamic (THD) floating ring bearing model on rotordynamic bifurcation

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.     

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.     

A Second-Order Approximation of Multi-Modal Interactions in Externally Excited Circular Cylindrical Shells

We investigate the nonlinear response of an infinitely long, circularcylindrical shell to a primary-resonance excitation of one of itsflexural modes, which is involved in a one-to-two internal resonancewith the breathing mode. The excited flexural mode is involved in aone-to-one internal resonance with its orthogonal flexural mode. Thereare two simultaneous internal (autoparametric) resonances: two-to-oneand one-to-one. The method of multiple scales is directly applied to thepartial-differential equations to obtain a system of six first-ordernonlinear ordinary-differential equations governing modulation of theamplitudes and phases of the three interacting modes. In the absence ofdamping, the modulation equations are derivable from a Lagrangian,reflecting the conservative nature of the system. The modulationequations are used to study the equilibrium and dynamic solutions andtheir stability and hence their bifurcations. The response may be eithera two-mode or a three-mode solution. For certain excitation parameters,the equilibrium three-mode solutions undergo Hopf bifurcations. Acombination of a shooting technique and Floquet theory is used tocalculate limit cycles and their stability, and hence theirbifurcations.     

Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.     

Nonlinear Dynamics and Stability of a Machine Tool Traveling Joint

In this work, we investigate the nonlinear dynamics and stability of a machine tool traveling joint. The dynamical system considered includes contacting elements of a lathe joint and the cutting process where the onset of instability is governed by mode coupling. The equilibrium equations of the dynamical system yield a unique fixed point that can change its stability via a Hopf bifurcation. The unstable domain is primarily governed by the cutting tool location, the contact stiffness of the joint and the depth of material to be removed. Self excited vibrations due to a mode coupling instability evolve around the unstable fixed point and one or more limit cycles may coexist in the statically unstable domain. Stability and accuracy of the approximate analytical solutions are analyzed by applying Floquet analysis. Perturbation of the dynamical system with weak periodic excitation results with periodic and aperiodic solutions.     

黏弹性圆柱形壳动力学高余维分岔、普适开折问题

讨论两端受到谐波激励的黏弹性圆柱形壳的非线性动力学行为,利用奇异性理论,研究了分岔方程的普适开折问题,严格证明了它是一个高余维分岔问题。余维数为5（含有一个模参数）,给出了它的所有可能的普适开折形式。在分岔参数满足某些条件时得到该分岔问题的转迁集及分岔图,展示了一些新的动力学行为,改进和完善了奇异性分析方法。     

Nonlinear analysis of chatter vibration in a cylindrical transverse grinding process with two time delays using?a?nonlinear time transformation method

A nonlinear dynamic system of cylindrical transverse grinding process is studied in this paper. The system consists of a grinding wheel and a workpiece, which are connected to the base by spring-damper elements, interacting with nonlinear normal forces. This two DOF model includes two time delays originated from the regenerative effects of the workpiece and the grinding wheel. Bifurcation points are located using a numerical algorithm by which we can find all the eigenvalues in a given rectangular region on the complex plane for the delayed differential equations. Supercritical bifurcation has been found for some sets of system parameter values. The amplitudes of the limit cycles are predicted using a nonlinear time transformation method, which is similar to the harmonic balance approach in that a periodic solution is approximated by a Fourier series. However, the main difference is that a nonlinear time ? is introduced in the Fourier series rather than the physical time t. The analytical solutions of stable limit cycles up to the third harmonics are compared with numerical simulations for the retarded system. It is shown that the proposed method gives accurate approximate solutions.     

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.