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.     

Periodic and Homoclinic Motions in Forced,Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.     

Global Dynamics of a Rapidly Forced Cart and Pendulum

In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, we study hovering motions, which are periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincaré maps. Second, we perform a complete global analysis on this cart and pendulum system. Toward this end, the same iteration scheme we use to establish the existence of the hovering periodic orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds that form homoclinic tangles. A quantitative global analysis of these tangles is carried out. Three distinguished limiting cases are analyzed. Melnikov theory is applied in one case, and an extension of a recent result about exponentially small splitting of separatrices is developed and applied in another case. Finally, the influence of small damping is studied. This global analysis is useful in the design of open-loop control laws.     

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.     

Melnikov's method for rigid bodies subject to small perturbation torques

Summary In this paper, the global motion of rigid bodies subjected to small perturbation torques, either conservative or dissipative, is investigated by means of Melnikov's method. Deprit's variables are introduced to transform the equations of motion into a standard form which is rendered suitable for the application of Melnikov's method. The Melnikov method is used to predict the transversal intersections of stable and unstable manifolds for the pertubed rigid-body motion. The chosen examples are a self-excited rigid body subject to a small periodic torque in a viscous medium, and the heavy rigid body. It is shown in both cases that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.     

A Melnikov Method for Homoclinic Orbits with Many Pulses

We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near‐integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist. (Accepted December 2, 1996)     

Homoclinic Tubes in Discrete Nonlinear Schrödinger Equation under Hamiltonian Perturbations

In this paper, we study the discrete cubic nonlinear Schrödinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Bäcklund–Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type calculation and an implicit function argument. We also discuss symbolic dynamics of invariant tubes each of which consists of a doubly infinite sequence of curve segments when the lattice is four dimensional. Structures inside the asymptotic manifolds of the transversal homoclinic tubes are studied, special orbits, in particular homoclinic orbits and heteroclinic orbits when the lattice is four dimensional, are studied.     

Chaos control in a pendulum system with excitations and phase shift

Melnikov methods are used for suppressing homoclinic and heteroclinic chaos of a pendulum system with a phase shift and excitations. This method is based on the addition of adjustable amplitude and phase-difference of parametric excitation. Theoretically, we give the criteria of suppression of homoclinic and heteroclinic chaos, respectively. Numerical simulations are given to illustrate the effect of the chaos control in this system. Moreover, we calculate the maximum Lyapunov exponents (LEs) in parameter plane, and study how to vary the maximum LE when the parametric frequency varies.     

Chaotic behavior in the nonlinear elastic beam

The dynamic response of the non-linear elastic simply supported beam subjected to axial forces and transverse periodic load is studied. Melnikov method is used to consider the dynamic behavior of the system whose post-buckling path is steady. The effect of the higher order terms in the controlling equation is taken into account. It is found that the fifth-order terms have a great influence on the dynamic behavior of the system. The result shows that there exist either homoclinic orbits or heteroclinic orbits in the system. In this paper, the critical values of the system entering chaotic states are given. The diagram of an example is shown. The project is supported by the National Natural Sciences Foundation of China.     

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.     

Complex dynamics in the stretch-twist-fold flow

The system associated with fluid particle motions of the stretch-twist-fold (STF) flow has displayed rich and attractive dynamic properties. Detailed research on the system has been done in this work. By using a high-dimensional generalization of the Melnikov method, the explicit parametric conditions for the existence of periodic solutions in the system can be determined. Then, by using the new-KAM-like theorems for perturbations of a three-dimensional generalized Hamiltonian system, the criteria for the existence of invariant tori in the STF flow have been obtained. In addition, one new first integral is found. On the basis of it, nonexistence of chaos in the system at α=0 is rigorously proved. Nonexistence of homoclinic orbits is also proved in the system if some conditions hold. And an interesting phenomenon is found, where the unit circle in the (y,z)-plane is filled with heteroclinic orbits of the system at α=0. The system with α=0 is also successfully reduced to a generalized Hamiltonian system, and further transformed to slowly varying oscillators.     

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。     

Geometric aspects of the parametrically driven pendulum

The behaviour of the parametrically driven pendulum is very complex. Therefore, a global study is carried out to cover all possible situations. The study is mainly numeric, though primary bifurcations of subharmonic motions, as well as the homoclinic intersection of the hilltop saddle, are evaluated according to the Melnikov theory. Extended use is made of the cell-to-cell mapping algorithm to evaluate attracting basins of the various periodic motions. Heteroclinic intersections are always present, independently of the excitation intensity, so that the boundaries of attracting basins are always very complicated, even below the homoclinic tangency of the hilltop saddle. The oscillator exhibits various kinds of rotating and oscillating motions. All these motions lead to chaos after a period doubling cascade. It is shown that chaos usually occurs at a much greater excitation level than at that which produces homoclinic tangency of the hilltop saddle; the greater the damping, the greater the difference. The oscillatory chaotic motion is associated with the first change in the period two Birkhoff signature.     

Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling

A method for controlling non-linear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic space varying time-periodic transversal excitation. The homoclinic bifurcation of the hilltop saddle is identified as the undesired dynamical event, because it triggers, e.g., cross-well scattered (possibly chaotic) dynamics. Its elimination is then pursued by a control strategy which consists in choosing the best spatial and temporal shape of the excitation permitting the maximum shift of the homoclinic bifurcation threshold in the excitation amplitude-frequency parameters space.The homoclinic bifurcation is detected by the Holmes and Marsden's theorem [A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal. 76 (1981) 135-165] constituting a generalization of the classical Melnikov's theory. Two classes of boundary conditions (b.c.) are identified: for the first, the Melnikov function is exactly the same as obtained with the reduced order models, while for the second, which is more general, this is no longer true, and the non-linear normal modes theory is used. Based on this distinction, the control method is then separately applied to the two cases, and the optimal spatial and temporal shapes of the excitation are determined.A detailed comparison of the infinite vs finite dimensional models is performed with respect to the control features, and it is shown that, depending on the b.c., the control based on the reduced order model provides either exact or engineering acceptable results, although more systematic investigations are required to generalize the last conclusion.     

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.     

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.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

Melnikov方法在输流管混沌运动研究中的应用

对基础简谐运动激励下两端固定输流管道的混沌运动进行了研究，推导出了系统的运动方程，确定了系统存在的平衡点及其稳定性，计算出了未扰系统的同宿轨道，并利用Melnikov方法得到了系统发生混沌运动时参数需满足的临界条件，同时还利用相平面图和：Poincare映射等方法对管道的混沌运动进行了数值模拟，通过比较发现，由Melnikov方法确定的临界参数值要稍小于数值模拟中首次观察到混沌运动时对应的临界值。     

Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.     

A group of chaotic motion of soft spring quadratic duffing equations

In this paper,we use the Melnikov function method to study a kind of soft Duffing equations(?) Af((?),x) x-x~(2k 1)=r[M(x,(?))cosωt N(x,(?))sinωt](k=1,2,3…)and give the condition that the equations have chaotic motion and bifurcation.The method used in this paper is effective for dealing with the Melnikov function integral of the system whose explict expression of the homoclinic or heteroclinic orbit cannot be given.