Pendulum Equation Research Articles

Autoresonance in a Model of a Terahertz Wave Generator

We study a model of an electromagnetic wave generator based on a system of coupled Josephson junctions. The model is a chain of coupled sine-Gordon equations for the phases of the electric field in the junctions under dissipation and constant pumping. We find conditions for a resonant field excitation under various parameters of the system. It turns out that the chain of sine-Gordon equations evokes an autoresonance with a certain dependence of the frequency on the magnitude of the Josephson pumping current. We construct an asymptotic expansion for a solution of the chain under a large resonant frequency. The leading terms of the expansion for the phases of the electric field are linear in time, which is typical of an autoresonance in a system of coupled oscillators. The key role here is played by the main resonance equation, which defines the mode of the resonant excitation of the chain. This equation is the equation of a mathematical pendulum with periodically changing mass. A class of solutions of this equation is studied in detail, and classes of separatrix solutions corresponding to the zero velocity of the pendulum are described. It is proved that there exists a separatrix π-kink type solution on which the autoresonance mode is realized in the original chain of sine- Gordon equations.

Almost global asymptotic stability of a grid-connected synchronous generator

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an “infinite bus,” i.e., a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $$2\pi $$ ). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the exact swing equation, which resembles a forced pendulum equation and is equivalent to our fourth-order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500-kW synchronverter.

Precessional switching of antiferromagnets by electric field induced Dzyaloshinskii-Moriya torque

Antiferromagnetic insulators (AFIs) have attracted much interest from many researchers as promising candidates for use in ultrafast, ultralow-dissipation spintronic devices. As a fast method of reversing magnetization, precessional switching is realized when antiferromagnetic N\'eel orders $l=({\mathbf{s}}_{1}+{\mathbf{s}}_{2})/2$ surmount the magnetic anisotropy or potential barrier in a given magnetic system, which is described well by the antiferromagnetic plane pendulum (APP) model. Here, we report that, as an alternative switching scenario, the direct coupling of an electric field with Dzyaloshinskii-Moriya (DM) interaction, which stems from spin-orbit coupling, is exploited for optimal switching. We derive the pendulum equation of motion of antiferromagnets, where DM torque is induced by a pulsed electric field. The temporal DM interaction is found to not only be in the form of magnetic torques (e.g., spin-orbit torque or magnetic field) but also modifies the magnetic potential that limits $l$ 's activity; as a result, appropriate controls (e.g., direction, magnitude, and pulse shape) of the induced DM vector realize deterministic reversal in APP. The results present an approach for the control of a magnetic storage device by means of an electric field.

The almost-periodic solutions of the weakly coupled pendulum equations

In this paper, it is proved that, for the networks of weakly coupled pendulum equations \t\t\td2xndt2+λn2sinxn=ϵWn(xn−1,xn,xn−1),n∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\frac{d^{2} x_{n}}{d t^{2}}+\\lambda_{n}^{2} \\sin x_{n}= \\epsilon W_{n}(x_{n-1},x_{n},x_{n-1}),\\quad n \\in\\mathbb {Z}, $$\\end{document} there are many (positive Lebesgue measure) normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions.

Light rays and the tidal gravitational pendulum

Null geodesic deviation in classical general relativity is expressed in terms of a scalar function, defined as the invariant magnitude of the connecting vector between neighbouring light rays in a null geodesic congruence projected onto a two-dimensional screen space orthogonal to the rays, where λ is an affine parameter along the rays. We demonstrate that η satisfies a harmonic oscillator-like equation with a λ-dependent frequency, which comprises terms accounting for local matter affecting the congruence and tidal gravitational effects from distant matter or gravitational waves passing through the congruence, represented by the amplitude, of a complex Weyl driving term. Oscillating solutions for η imply the presence of conjugate or focal points along the rays. A polarisation angle, is introduced comprising the orientation of the connecting vector on the screen space and the phase, of the Weyl driving term. Interpreting β as the polarisation of a gravitational wave encountering the light rays, we consider linearly polarised waves in the first instance. A highly non-linear, second-order ordinary differential equation, (the tidal pendulum equation), is then derived, so-called due to its analogy with the equation describing a non-linear, variable-length pendulum oscillating under gravity. The variable pendulum length is represented by the connecting vector magnitude, whilst the acceleration due to gravity in the familiar pendulum formulation is effectively replaced by . A tidal torque interpretation is also developed, where the torque is expressed as a coupling between the moment of inertia of the pendulum and the tidal gravitational field. Precessional effects are briefly discussed. A solution to the tidal pendulum equation in terms of familiar gravitational lensing variables is presented. The potential emergence of chaos in general relativity is discussed in the context of circularly, elliptically or randomly polarised gravitational waves encountering the null congruence.

Subharmonic Solutions with Prescribed Minimal Period of a Forced Pendulum Equation with Impulses

This paper is mainly concerned with a forced pendulum equation with impulses and the length of the pendulum is variable. The main tool utilized to establish our results on the subharmonic solutions with prescribed minimal period is the theorem of the least action principle due to Mawhin and Willem. As an application, three examples are given, and the corresponding numerical subharmonic solutions for the examples are obtained by applying Matlab software. Some results in the literature can be generalized and improved.

Equivalent representation form in the sense of Lyapunov, of nonlinear forced, damped second-order differential equations

This paper focuses on finding in the sense of Lyapunov, the equivalent forced, damped cubic–quintic Duffing equation of nonlinear forced damped, second-order ordinary differential equations that could contain rational or irrational restoring elastic terms. The accuracy obtained from the equivalent expressions of dynamical systems such as the generalized pendulum equation, the power-form elastic term oscillator, oscillatory systems with irrational elastic restoring forces that modeled the motion of a mass attached to two stretched elastic springs, and the motion of the mechanism of dipteran flight motor, is numerically evaluated by computing their corresponding Lyapunov characteristic exponents, the amplitude–frequency, the amplitude–time, phase portraits, Poincare’s maps, and their Kaplan–Yorke dimension plots.

Nonlinear dynamics of resonant electrons interacting with coherent Langmuir waves

We study the nonlinear dynamics of resonant particles interacting with coherent waves in space plasmas. Magnetospheric plasma waves such as whistler-mode chorus, electromagnetic ion cyclotron waves, and hiss emissions contain coherent wave structures with various discrete frequencies. Although these waves are electromagnetic, their interaction with resonant particles can be approximated by equations of motion for a charged particle in a one-dimensional electrostatic wave. The equations are expressed in the form of nonlinear pendulum equations. We perform test particle simulations of electrons in an electrostatic model with Langmuir waves and a non-oscillatory electric field. We solve equations of motion and study the dynamics of particles with different values of inhomogeneity factor S defined as a ratio of the non-oscillatory electric field intensity to the wave amplitude. The simulation results demonstrate deceleration/acceleration, thermalization, and trapping of particles through resonance with a single wave, two waves, and multiple waves. For two-wave and multiple-wave cases, we describe the wave-particle interaction as either coherent or incoherent based on the probability of nonlinear trapping.

Nonlinear pendulum equations and space plasma reveal potential radiation belt trick

By pairing space plasma wave phenomena with nonlinear pendulum equations, researchers discovered a potential way to artificially generate coherent waves to control conditions within the Van Allen radiation belts.

Prevalence of stable periodic solutions in the forced relativistic pendulum equation

We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.

Unstable Mechanical Objects: Motion Control, Stabilization

We consider here mechanical objects that have desired operating regime unstable without additional control signal. The resources of control are assumed limited. First of all the motion equations of a multi-link pendulum mounted on a moving base – wheel are designed. Equations that describe only the pendulum motion are separated. The problem of controlling a single inverted pendulum on a wheel is studied. Control law that ensures large basin of attraction of the top unstable equilibrium is developed. The 'inverse' system – the inertia wheel pendulum with fixed suspension point is investigated. For this system the control algorithm is developed to stabilize globally the top equilibrium of the pendulum. This control law is tested successfully in simulation and experiments. The double pendulum with fixed suspension point is considered. The limited in absolute value control torque is applied in the inter-link joint. The control algorithm to ensure global stabilization of the inverted pendulum is designed. For maximizing basin of attraction of unstable system, we use all control resources to suppress the unstable modes. At the end of the paper, the problem of gyroscopic stabilizing of the upright unstable position of a robot-bicycle is studied.

Singular Perturbations of Volterra Equations with Periodic Nonlinearities. Stability and Oscillatory Properties

Singularly perturbed integro-differential Volterra equations with MIMO periodic nonlinearities are considered, which describe synchronization circuits (such as phase- and frequency-locked loops) and many other “pendulum-like” systems. Similar to the usual pendulum equation, such systems are typically featured by infinite sequences of equilibria points, and none of which can be globally asymptotically stable. A natural extension of the global asymptotic stability is the gradient-like behavior, that is, convergence of any solution to one of the equilibria. In this paper, we offer an efficient frequency-domain criterion for gradientlike behavior. This criterion is not only applicable to a broad class of infinite-dimensional systems with periodic nonlinearities, but in fact ensures the equilibria set stability under singular perturbation. In particular, the proposed criterion guarantees the absence of periodic solutions that are considered to be undesirable in synchronization systems. In this paper we also discuss a relaxed version of this criterion, which guarantees the absence of “high-frequency” periodic solutions, whose frequencies lie beyond a certain bounded interval.

English

Based on the energy balance method (EBM), a more accurate analytical solution of the pendulum equation with rotating support was presented. The results were compared with those obtained by the differential transformation method (DTM) and He’s improved energy balance method. It was shown that the results are more accurate than the said methods. Key words: Energy balance method, approximate solutions, nonlinear oscillators, pendulum with rotating support.

Calculation of radiant energy in free electron lasers without inversion

The equations of motion of particles in free electron lasers without inversion (FELWI) were obtained using the Hamilton formalism. An one-dimensional decoupled phase equation of the type of mathematical pendulum equation was derived in the case of weak signal. For practical estimates, this equation was solved together with the equation of electron energy change within the frameworks of perturbation theory, and expressions for gain were obtained in FEL regime and for the output angle from the first undulator. Expressions for the gain coefficient, the phase and the energy change were found as functions of the parameters of beam and of the device.

Fast converging exact power series for the time and period of the simple pendulum

A time explicit fast converging exact power series solution to the pendulum equation is derived in this paper. A novel series for the period results from it. The approximate formula that comprises the first three terms gives an accuracy of 99.99% up to the amplitude of 90°. The accuracy was compared with that of 11 other approximate period formulas.

The nonlinear pendulum always oscillates

It is shown that the nonlinear pendulum equation can be transformed into a linear harmonic oscillator in the phase space thanks to Kerner’s method [12]. Moreover, as a mathematical divertissement, the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum is quantized.

Systematic Computer-Assisted Proof of Branches of Stable Elliptic Periodic Orbits and Surrounding Invariant Tori

We present a concurrent algorithm for rigorous validation of the existence of continuous branches of stable elliptic fixed points for area-preserving planar maps. The method utilizes a classical theorem of Siegel and Moser combined with computed-assisted estimation of higher order derivatives of maps, continuation along the parameter range, and concurrent scheduling of tasks. We apply the algorithm to certain exemplary Poincare maps coming from reversible or Hamiltonian systems: the periodically forced pendulum equations, the Michelson system, and the Henon--Heiles Hamiltonian. Moreover, our algorithm provides at once a computer-assisted proof of the existence of wide branches of stable elliptic periodic solutions and the existence of invariant tori surrounding them.

Formation of plasmon pulses in the cooperative decay of excitons of quantum dots near a metal surface

The formation of pulses of surface electromagnetic waves in a metal/dielectric interface is considered in the process of cooperative decay of excitons of quantum dots distributed near a metal surface in a dielectric layer. It is shown that the efficiency of exciton energy transfer to excited plasmons can be increased by selecting the dielectric material with specified values of the complex permittivity. It is found that in the mean field approximation the semiclassical model of formation of plasmon pulses in the system under study is reduced to the pendulum equation with the additional term of nonlinear losses.

Study of the Bifurcation of a Multiple Limit Cycle of the Second Kind by Means of a Dulac–Cherkas Function: A Case Study

We consider a generalized pendulum equation depending on the scalar parameter [Formula: see text] having for [Formula: see text] a limit cycle [Formula: see text] of the second kind and of multiplicity three. We study the bifurcation behavior of [Formula: see text] for [Formula: see text] by means of a Dulac–Cherkas function.

Bistability of patterns of synchrony in Kuramoto oscillators with inertia.

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.