Experiments on periodic and chaotic motions of a parametrically forced pendulum

Results on the dynamics of the planar pendulum with parametric vertical timeperiodic forcing are reviewed and extended. Numerical methods are employed to study the various dynamical features of the system about its equilibrium positions. Furthermore, the dynamics of the system far from its equilibrium points is systematically investigated by using phase portraits and Poincaré sections. The attractors and the associated basins of attraction are computed. We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions.

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“…1983;Leven et al . 1985;Michaelis 1985) obtained by removing one of the phase space dimensions (in our case the time 'τ ').…”

Section: Poincaré Sections and Strange Attractorsmentioning

confidence: 99%

“…Analogous pictures have also been found numerically for similar models (see Doerner et al . 1994;Leven et al . 1985;Smith & Blackburn 1989;Blackburn et al .…”

Section: Poincaré Sections and Strange Attractorsmentioning

confidence: 99%

On the dynamics of a vertically driven damped planar pendulum

Bartuccelli

Gentile

Georgiou

2001

Proc. R. Soc. Lond. A

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“…Mechanical systems in general [Awrejcewicz, 2014;Slotine & Li, 1991] and the pendulum-like oscillators in particular [Capecchi, 1995;Alevras et al, 2015;Avanço et al, 2015;Leven et al, 1985], have been of interest in different studies in the field of nonlinear dynamic systems. In addition to its intrinsic interest, the pendulum is one of the most common models for many oscillators with many applications in engineering such as, marine structures, auto suspension systems, damping devices in civil engineering, etc.…”

Section: Introductionmentioning

confidence: 99%

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

Luque

Barbancho

Cañete

et al. 2017

Int. J. Bifurcation Chaos

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Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360• . In these cases the use of the phase portrait does not properly depict the system's evolution. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait's disadvantages. In this research a new graphic representation is introduced: the phase shadow. Its use clearly reveals the system behavior while overcoming the drawback of the existing plots. Through the paper the method to obtain the graphic is stated. Additionally, to show the phase shadow's expressiveness, a rotating pendulum is considered. The work exposes that the new graph is an enhanced representational tool for systems having equilibrium points, limit cycles, chaotic attractors and/or bifurcations.

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“…This system has attracted great attention [12][13][14][15][16][17]. It presents a wide range of dynamical behaviour, such as: the stabilization of the hilltop saddle [18][19][20][21][22][23]; the occurrence of chaotic behaviour [24][25][26][27][28][29]; the observation of period-doubling cascades [30,31] and the existence of resonance regions [11,32]. Moreover, it can be used as qualitative analogue for more complex systems [31,33,34].…”

Section: Introductionmentioning

confidence: 99%

Tilted excitation implies odd periodic resonances

et al. 2016

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Our aim is to unveil how resonances of parametric systems are affected when symmetry is broken. We showed numerically and experimentally that odd resonances indeed come about when the pendulum is excited along a tilted direction. Applying the Melnikov subharmonic function, we not only determined analytically the loci of saddle-node bifurcations delimiting resonance regions in parameter space, but also explained these observations by demonstrating that, under the Melnikov method point of view, odd resonances arise due to an extra torque that appears in the asymmetric case.

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Experiments on periodic and chaotic motions of a parametrically forced pendulum

Cited by 89 publications

References 13 publications

On the dynamics of a vertically driven damped planar pendulum

On the dynamics of a vertically driven damped planar pendulum

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

Tilted excitation implies odd periodic resonances

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