Inverted oscillations of a driven pendulum

Abstract

Inverted oscillations of a parametrically driven planar pendulum are considered, together with their relationship to the inverted solution. In particular, a horseshoe structure of the associated manifolds is identified which explains the similarity between the bifurcations of the inverted position and the hanging position. This allows us to apply a large body of existing knowledge to the dynamics enabling a lower bound on the forcing required to achieve inverted oscillations to be established.

Comparisons between the pendulum with varying length and the pendulum with oscillating support

“…Here, a first difference between the two systems appears. Whilst it is well known [61,40,1,2,6,13,30] that for α sufficiently small and δ chosen accordingly, it is possible to stabilise the upward fixed point (θ,θ) = (π, 0) for the system (2), on the contrary we show that, as observed numerically in [18], stability of the upward fixed point cannot be achieved for any values of the parameters in the system (3). In Section 3 we calculate sufficient conditions on the damping coefficient in each system for the origin to achieve global attraction (up to a zero measure set).…”

“…The first and second period doubling bifurcations are marked in Figure 9(a) by the curves B and C, respectively. The curve above curve C marks the birth of a rotating chaotic attractor, which persists only in a very small interval of parameter values and disappears by a catastrophic bifurcation [19,30,32]. The appearance of chaotic attractors for small sets of parameter values and with small basins of attraction has been observed in similar contexts of multistable dissipative systems close to the conservative limit [34].…”

“…For this reason perturbed pendula are often used as toy models to study new phenomena in dynamics -for instance in [3,21,49,60,59,64] -or act as simplified models for more complex real world systems -see [46,57,62]. One of the most commonly studied perturbations is the pendulum with vertically oscillating support [6,12,13,30,64], often also referred to as the parametrically forced pendulum, see for example [21,29,32,44,66]. Another much studied perturbation of the simple pendulum is the pendulum with periodically varying length [8,16,58,18,56].…”

Tilted excitation implies odd periodic resonances

“…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].…”

Dynamics of a Parametrically Excited System with Two Forcing Terms

“…Authors have particularly concentrated on identifying the oscillatory orbits and analytically obtaining the approximate boundaries of the so-called "escape zone". For instance, numerical and analytical investigations into the chaotic behaviour of a parametrically excited pendulum system were carried out by Bishop and Clifford [1][2][3][4][5], a study into symmetry-breaking in the response of this model by Bishop, Sofroniou and Shi [6], breaking the symmetry of the parametrically excited pendulum system via a bias term by Sofroniou and Bishop [7] and studies into the dynamics of the harmonically excited pendulum with rotational orbits by Xu, Wiercigroch and Cartmell [8] and by Garira and Bishop [9].…”