Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.     

Chaotic pitch motion of an asymmetric non-rigid spacecraft with viscous drag in circular orbit

We study the pitch motion dynamics of an asymmetric spacecraft in circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by a small aerodynamic drag torque proportional to the angular velocity of the body about its mass center. We also suppose that one of the moments of inertia of the spacecraft is a periodic function of time. Under both perturbations, we show that the system exhibits a transient chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for heteroclinic chaos in terms of the system parameters which is numerically contrasted. We also show that some periodic orbits survive for perturbation small enough.     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

万有引力场中陀螺体的混沌运动

研究万有引力场中沿圆轨道运行的非对称陀螺体的姿态运动,引入Ｄｅｐｒｉｔ正则变量建立系统的Ｈａｍｉｌｔｏｎ结构,利用Ｍｅｌｎｉｋｏｖ方法证明在万有引力短作用的昆体产生混沌运动的可能性。对Ｐｏｉｎｃａｒｅ截面的数值计算表明提高陀螺体的转子转速可对混沌起抑制作用。     

Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate

This paper presents the study on the chaotic wave and chaotic dynamics of the nonlinear wave equations for a simply supported truss core sandwich plate combined with the transverse and in-plane excitations. Based on the governing equation of motion for the simply supported sandwich plate with truss core, the reductive perturbation method is used to simplify the partial differential equation. According to the exact solution of the unperturbed equation, two different kinds of the topological structures are derived, which one structure is the resonant torus and another structure is the heteroclinic orbit. The characteristic of the singular points in the neighborhood of the resonant torus for the nonlinear wave equation is investigated. It is found that there exists the homoclinic orbit on the unperturbed slow manifold. The saddle-focus type of the singular point appears when the homoclinic orbit is broken under the perturbation. Additionally, the saddle-focus type of the singular point occurs when the resonant torus on the fast manifold is broken under the perturbation. It is known that the dynamic characteristics are well consistent on the fast and slow manifolds under the condition of the perturbation. The Melnikov method, which is called the first measure, is applied to study the persistence of the heteroclinic orbit in the perturbed equation. The geometric analysis, which is named the second measure, is used to guarantee that the heteroclinic orbit on the fast manifold comes back to the stable manifold of the saddle on the slow manifold under the perturbation. The theoretical analysis suggests that there is the chaos for the Smale horseshoe sense in the truss core sandwich plate. Numerical simulations are performed to further verify the existence of the chaotic wave and chaotic motions in the nonlinear wave equation. The damping coefficient is considered as the controlling parameter to study the effect on the propagation property of the nonlinear wave in the sandwich plate with truss core. The numerical results confirm the validity of the theoretical study.     

Homoclinic orbits in a shallow arch subjected to periodic excitation

Homoclinic orbits in a shallow arch subjected to periodic excitation are investigated in the presence of 1:1 internal resonance and external resonance. The method of multiple scales is used to obtain a set of near-integrable systems. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect equilibria in a resonance band of the slow manifold. These orbits arise from singular homoclinic orbits and are composed of alternating slow and fast pieces. The result obtained imply the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the class of shallow arch systems.     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

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.     

有界噪声激励下单摆-谐振子系统的混沌运动

研究了具有同宿轨道和周期轨道的可积单摆-谐振子系统在弱Hamilton摄动(即弱耦合摄动)和弱非Hamilton摄动(即阻尼和有界噪声微扰)下的混沌运动．用Melnikov方程预测Hamilton系统中可能存在混沌运动的参数域，并用Poincare截面验证解析结果．用数值方法计算了有阻尼与有界噪声激励下系统的最大Lyapun0V指数和Poincare截面，结果表明有界噪声在频率上的扩散减小了引发系统产生混沌运动的效应。     

Estimate for the number of perturbed ultrasubharmonics of a system with one and a half degrees of freedom close to a Hamiltonian system

We consider a two-dimensional autonomous Hamiltonian system with heteroclinic contour under the action of a time-periodic perturbation. It is shown that the number of ultrasubharmonics in the perturbed system is estimated from below by a function proportional to the square of the logarithm of the perturbation parameter when this parameter tends to zero.     

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.     

Shil’nikov chaos in the 4D Lorenz–Stenflo system modeling the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere

The Lorenz–Stenflo system serves as a model of the time evolution of nonlinear acoustic-gravity waves in a rotating atmosphere. In the present paper, we study the Shil’nikov chaos which arises in the 4D Lorenz–Stenflo system. The analytical and numerical results constitute an application of the Shil’nikov theorems to a 4D system (whereas most results present in the literature deal with applying the Shil’nikov theorems to 3D systems), which allows for the study of chaos along homoclinic and heteroclinic orbits arising as solutions to the Lorenz–Stenflo system. We verify the observed chaos via competitive modes analysis—a diagnostic for chaotic systems. We give an analytical test, completely in terms of the model parameters, for the Smale horseshoe chaos near homoclinic orbits of the origin, as well as for the case of specific heteroclinic orbits. Numerical results are shown for other cases in which the general analytical method becomes too complicated to apply. These results can be extended to more complicated higher-dimensional systems governing plasmas, and, in particular, may be used to shed light on period-doubling and Smale horseshoe chaos that arises in such models.     

Comments on ＂Non-existence of Shilnikov chaos in continuous-time systems＂

A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Dynamics of a hyperchaotic Lorenz-type system

This paper discusses the complex dynamics of a new four-dimensional continuous-time autonomous hyperchaotic Lorenz-type system. The local dynamics, such as the stability, pitchfork bifurcation, and Hopf bifurcation at equilibria of this hyperchaotic system are analyzed by using the parameter-dependent center manifold theory and the normal form theory. The existence of homoclinic and heteroclinic orbits of this hyperchaotic system is further rigorously studied. More exactly, under some special parameter conditions, the fact that this hyperchaotic system has no homoclinic orbit but has two and only two heteroclinic orbits are proved.     

Parametric resonance,stability and heteroclinic bifurcation in a nonlinear oscillator with time-delay: Application to a quarter-car model

This work examines dynamical behavior of a nonlinear oscillator with symmetric potential that models a quarter-car forced by the road profile under parametric excitation. The parametric resonance of a harmonically excited nonlinear quarter-car model with position and velocity time-delayed active control are investigated. We focus on the influence of delay and parametric excitation in the system. The influence of parametric excitation, time-delay and feedback gain parameters on the stability of the steady state response are investigated. By means of Melnikov's method, conditions for onset of chaos resulting from heteroclinic bifurcation is derived analytically and numerically.