The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension

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).…”

“…However, if we consider the original system (3) with ζ = 0 and transform to the coordinates ϕ = θ − π we obtain the system ϕ(t) − 2ε sin t 1 + ε cos tφ (t) − α 1 + ε cos t sin ϕ(t) = 0, which is the same as equation (3), except for a change in sign of α. Transforming to coordinates Φ(t) = (t)ϕ(t) and linearising around Φ = 0, we obtain a system of the form (6), where only the sign of α is changed. Therefore, by changing the sign of α in equation (6) we are still able to obtain the stability of the inverted position θ = π.…”

“…Transforming to coordinates Φ(t) = (t)ϕ(t) and linearising around Φ = 0, we obtain a system of the form (6), where only the sign of α is changed. Therefore, by changing the sign of α in equation (6) we are still able to obtain the stability of the inverted position θ = π. As remarked in the previous comment, there is no stability region in the half-plane α < 0, so that we conclude that the upward position is always linearly unstable for the pendulum with varying length.…”

“…In particular the pendulum with varying length cannot be stabilised around its upward fixed point for any values of the parameters. To deduce stability from linear stability, that is for values of the parameters inside the stability regions of the linearised systems, requires a more delicate analysis, involving KAM-type arguments [6,14].…”

“…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].…”