Period doublings in coupled parametrically forced damped pendulums (original) (raw)
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Critical behavior of period doublings in coupled inverted pendulums
Physical Review E, 1998
We study the critical behaviors of period doublings in N (Nϭ2,3,4,. . .) coupled inverted pendulums by varying the driving amplitude A and the coupling strength c. It is found that the critical behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each inverted pendulum is coupled to all the other ones with equal strength, the zero-coupling critical point and an infinity of critical line segments constitute the same critical set in the A-c plane, independently of N. However, for any other nonglobal-coupling cases of shorter-range couplings, the structure of the critical set becomes different from that for the global-coupling case, because of a significant change in the stability diagram of periodic orbits born via period doublings. The critical scaling behaviors on the critical set are also found to be the same as those for the abstract system of the coupled one-dimensional maps. ͓S1063-651X͑98͒05912-1͔
Multiple transitions to chaos in a damped parametrically forced pendulum
Physical Review E, 1996
We study bifurcations associated with stability of the lowest stationary point (SP) of a damped parametrically forced pendulum by varying ω 0 (the natural frequency of the pendulum) and A (the amplitude of the external driving force). As A is increased, the SP will restabilize after its instability, destabilize again, and so ad infinitum for any given ω 0. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB's) and pitchfork bifurcations, except the first destabilization at which a supercritical or subcritical bifurcation takes place depending on the value of ω 0. For each case of the supercritical destabilizations, an infinite sequence of PDB's follows and leads to chaos. Consequently, an infinite series of period-doubling transitions to chaos appears with increasing A. The critical behaviors at the transition points are also discussed.
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.
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.
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.
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.
CONNECTING PERIOD-DOUBLING CASCADES TO CHAOS
International Journal of Bifurcation and Chaos, 2012
The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos.
Periodic Motions to Chaos in Pendulum
It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions' complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.
Symmetry and chaos in the motion of the damped driven pendulum
Zeitschrift f�r Physik B Condensed Matter, 1985
The damped, driven pendulum equation is studied numerically. A relation is pointed out between the symmetry of the initial period-m dynamical state of a m x 2" perioddoubling sequence and the form of the chaotic attractor for the final chaotic dynamical state reached after completion of the inverse-doubling sequence. Effects of extrinsic noise are also mentioned.
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