Multiple transitions to chaos in a damped parametrically forced pendulum (original) (raw)
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Bifurcations and transitions to chaos in an inverted pendulum
Physical Review E, 1998
We consider a parametrically forced pendulum with a vertically oscillating suspension point. It is well known that, as the amplitude of the vertical oscillation is increased, its inverted state ͑corresponding to the vertically-up configuration͒ undergoes a cascade of ''resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum. We make a detailed numerical investigation of the bifurcations associated with such resurrections of the inverted pendulum by varying the amplitude and frequency of the vertical oscillation. It is found that the inverted state stabilizes via alternating ''reverse'' subcritical pitchfork and period-doubling bifurcations, while it destabilizes via alternating ''normal'' supercritical perioddoubling and pitchfork bifrucations. An infinite sequence of period-doubling bifurcations, leading to chaos, follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed. ͓S1063-651X͑98͒03809-4͔
The Transition to Chaos of Pendulum Systems
Applied Sciences
We examine the nonlinear response of two planar pendula under external and kinematic excitations, which are very relevant as paradigmatic models in nonlinear dynamics. These pendula act under the action of an additional constant torque, and are subjected to one of the following excitations: a further external periodic torque, and a vertically periodic forcing of the point of suspension. Here, we show the influence of the constant torque strength on the transition to chaotic motions of the pendulum using both Melnikov analysis and the computation of the basins of attraction. The global bifurcations are illustrated by the erosion of the corresponding basins of attraction.
Period doublings in coupled parametrically forced damped pendulums
Physical Review E, 1996
We study period doublings in N (N = 2, 3, 4,. . .) coupled parametrically forced damped pendulums by varying A (the amplitude of the external driving force) and c (the strength of coupling). With increasing A, the stationary point undergoes multiple period-doubling transitions to chaos. We first investigate the two-coupled case with N = 2. For each period-doubling transition to chaos, the critical set consists of an infinity of critical line segments and the zero-coupling critical point lying on the line A = A * i in the A − c plane, where A * i is the ith transition point for the uncoupled case. We find three kinds of critical behaviors, depending on the position on the critical set. They are the same as those for the coupled one-dimensional maps. Finally, the results of the N = 2 case are extended to many-coupled cases with N ≥ 3, in which the critical behaviors depend on the range of coupling.
Routes of periodic motions to chaos in a periodically forced pendulum
In this paper, with varying excitation amplitude, bifurcation trees of periodic motions to chaos in a periodicallydrivenpendulumareobtainedthroughasemi-analytical method. This method is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures are developed for specific periodic motions, and the corresponding nonlinear algebraic equations of such mappingstructuresaresolved.Further,semi-analyticalbifurcation trees of periodic motions to chaos are also obtained, and the corresponding eigenvalue analysis is carried out for the stability and bifurcation of the periodic motions. Finally, numerical illustrations of periodic motions on the bifurcation trees are presented in verification of the analytical prediction. Harmonic amplitude spectra are also presented for demonstrating harmonic effects on the periodic motions. The bifurcation trees of period-1 motions to chaos possess a doublespiralstructure.Thetwosetsofsolutionsofperiod-2l motions (l = 0,1,2,...)to chaos are based on the center around2mπ and(2m−1)π(m =1,2,3,...)inphasespace. Other independent bifurcation trees of period-m motions to chaos are presented. Through this investigation, the motion complexity and nonlinearity of the periodically forced pendulum can be further understood.
Onset of chaos in an extensible pendulum
Physics Letters A, 1990
A numerical study of the onset of chaos in an extensible pendulum at resonance is undertaken. We found that the system goes from regular to chaotic and back to regular behaviour as the total energy is increased. The existence of a localized region of negative curvature on the potential energy surface has been proposed to be related to this behaviour. We compare our results with the predictions of this proposal.
Experiments on periodic and chaotic motions of a parametrically forced pendulum
Physica D: Nonlinear Phenomena, 1985
An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic rou~e to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.
Bifurcation Dynamics of a Damped Parametric Pendulum
Synthesis Lectures on Mechanical Engineering, 2019
Synthesis Lectures on Mechanical Engineering series publishes 60-150 page publications pertaining to this diverse discipline of mechanical engineering. The series presents Lectures written for an audience of researchers, industry engineers, undergraduate and graduate students. Additional Synthesis series will be developed covering key areas within mechanical engineering.
American Journal of Physics, 1992
A novel demonstration of chaos in the double pendulum is discussed. Experiments to evaluate the sensitive dependence on initial conditions of the motion of the double pendulum are described. For typical initial conditions, the proposed experiment exhibits a growth of uncertainties which is exponential with exponent λ=7.5±1.5 s−1. Numerical simulations performed on an idealized model give good agreement, with the value λ=7.9±0.4 s−1. The exponents are positive, as expected for a chaotic system.
Chaos in Mechanical Pendulum-like System Near Main Parametric Resonance
Procedia IUTAM, 2012
Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal parametric resonance region, are investigated in this paper. The aim of the work is to show the chaotic motion in instability region. Two kinds of chaotic motion are detected: chaotic swings and chaotic motion composed of swings and rotation of pendulum. The results are confirmed experimentally on especially designed laboratory model. Additionally, the latest methods of chaos identification are applied to confirm chaotic dynamics experimentally.
Chaos, Solitons & Fractals, 1996
We study a pendulum parametrically excited by nonharmonic perturbations. Instead of using a circular harmonic function to perturb the pendulum, we use a Jacobi elliptic function as a perturbation, which encompasses it as a limit. Melnikov analysis provides the general condition for the onset of homoclinic bifurcations, adding now the elliptic modulus as a new parameter. Using the elliptic modulus, which is responsible for the shape of the perturbation, as a control parameter, new transitions order-chaos may occur.