Periodic Motions to Chaos in Pendulum (original) (raw)

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.

Chaos and transient chaos in an experimental nonlinear pendulum

Journal of Sound and Vibration, 2006

Pendulum is a mechanical device that instigates either technological or scientific studies, being associated with the measure of time, stabilization devices as well as ballistic applications. Nonlinear characteristic of the pendulum attracts a lot of attention being used to describe different phenomena related to oscillations, bifurcation and chaos. The main purpose of this contribution is the analysis of chaos in an experimental nonlinear pendulum. The pendulum consists of a disc with a lumped mass that is connected to a rotary motion sensor. This assembly is driven by a string-spring device that is attached to an electric motor and also provides torsional stiffness to the system. A magnetic device provides an adjustable dissipation of energy. This experimental apparatus is modeled and numerical simulations are carried out. Free and forced vibrations are analyzed showing that numerical results are in close agreement with those obtained from experimental data. This analysis shows that the experimental pendulum has a rich response, presenting periodic response, chaos and transient chaos. r

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.

Deterministic chaos in the elastic pendulum: A simple laboratory for nonlinear dynamics

American Journal of Physics, 1992

The chaotic motion of the elastic pendulum is studied by means of four indicators, the Paineare section, the maximum Lyapunov exponent, the correlation function, and the power spect ru~. It is shown that for very low and very large energies the motion is regular while it is very irregular for intennediate energies. Analytical considerations and graphical representati~ns concerning the applicability of KAM theorem are also presented. This system and the type of description used are very suitable to introduce undergraduate students to nonlinear dynamics.

Converting the chaotic motion of the double pendulum into linear motion by changing its masses

The present paper deals with the linear and nonlinear dynamic analysis of a double pendulum. This paper mainly deals with converting the non-linear motion of the pendulums, which have the same length, into linear motion by changing the values of the mass of the pendulums. The paper contains an experimental analysis, along with a simulation by Pro-E (Creo) software. A three-step procedure has been carried out for dynamic analysis. In the first step, the equation of motion has been developed by the Lagrangian method and the authors have derived new relations between the masses of the pendulums in order to convert the non-linear motion of the pendulum into a linear motion in terms of the initial displacement. In the second step, an analysis has been carried out by different analysis tools, like time displacement plot and phase portrait. The authors have found different nonlinear behaviors like periodic, period 2T, quasiperiodic, multi-periodic, and chaotic. Finally, in the third step, the results have been verified experimentally.

Real-time demonstration of the main characteristics of chaos in the motion of a real double pendulum

Real-time demonstration of the main char. of chaos in the motion of a real double pendulum 1 Abstract. Several studies came to the conclusion that chaotic phenomena are worth including in high school and undergraduate education. The double pendulum is one of the simplest systems which are chaotic, therefore, the numerical simulations and theoretical studies of it have been given large publicity, and thanks to its spectacular motion, it has become one of the most famous demonstration tools of chaos, either through simulations or in real experiments. Although several attempts have been made to use the experiment in laboratory exercises, as the friction in the real experiment changes the nature of the motion and the values of characteristic parameters during the motion, examining the measured (dissipative) motion and to compare with theoretical results raise several questions. In our review, we are presenting a measurement system which is able to analyse these questions. The system, which consists of simple yet precise data acquisition electronics, easily attainable sensors, a Bluetooth module (to the communication with the PC), and an open-source software, demonstrates on-line the main characteristics of chaos and the methods of its study and allows to analyse the dissipative motion. Further information (including downloadable software) is provided on a dedicated page, http://www.inf.u-szeged.hu/noise/Research/DoublePendulum/.

Regular and Chaotic Motions of the P arametrically Eorced Pendulum: Theory and Simulations

Lecture Notes in Computer Science, 2002

New types of regular and chaotic behaviour of the parametrically driven pendulum are discovered with the help of computer simulations. A simple qualitative physical explanation is suggested to the phenomenon of subharmonic resonances. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and their boundaries in the parameter space are determined. The conditions of the inverted pendulum stability are determined with a greater precision than they have been known earlier. A close relationship between the upper limit of stability of the dynamically stabilized inverted pendulum and parametric resonance of the hanging down pendulum is established. Most of the newly discovered modes are waiting a plausible physical explanation.

On Nonlinear Dynamics of the Pendulum with Periodically Varying Length

2009

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of child's swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Two types of transitions to chaos of the pendulum depending on problem parameters are investigated numerically.

Chaotic orbits of a pendulum with variable length

Electronic Journal of Differential Equations

The main purpose of this investigation is to show that a pendulum, whose pivot oscillates vertically in a periodic fashion, has uncountably many chaotic orbits. The attribute chaotic is given according to the criterion we now describe. First, we associate to any orbit a finite or infinite sequence as follows. We write 1 or −1 every time the pendulum crosses the position of unstable equilibrium with positive (counterclockwise) or negative (clockwise) velocity, respectively. We write 0 whenever we find a pair of consecutive zero's of the velocity separated only by a crossing of the stable equilibrium, and with the understanding that different pairs cannot share a common time of zero velocity. Finally, the symbol ω, that is used only as the ending symbol of a finite sequence, indicates that the orbit tends asymptotically to the position of unstable equilibrium. Every infinite sequence of the three symbols {1, −1, 0} represents a real number of the interval [0, 1] written in base 3 when −1 is replaced with 2. An orbit is considered chaotic whenever the associated sequence of the three symbols {1, 2, 0} is an irrational number of [0, 1]. Our main goal is to show that there are uncountably many orbits of this type.