The parametrically forced pendulum: A case study in \(1\frac12\) degree of freedom
The global nonchaotic dynamics of the parametrically forced pendulum in a Hamiltonian \(1\frac{1}{2}\) degree of freedom setting with two parameters \(\ddot{x}=(\alpha+\beta \cos t)\sin x\) is investigated. Here, \(x\in \mathcal{S}^1\) is the deviation from the upper equilibrium with range over the whole circle, \(x=0\) corresponds to the upper equilibrium, \(x=\pi\) denotes the lower equilibrium, \(t\in \mathcal{S}^1\) is the independent variable. The parameters \(\alpha=\sqrt{\frac{g}{l}}\;\) (\(g\) is the gravitational acceleration, \(l\) is the length of the pendulum) and \(\beta\) correspond to the square of the eigenfrequency at the lower equilibrium and the amplitude of the forcing and they generally are nonnegative. At the absence of forcing the system is integrable. The authors are interested in all periodic and quasi-periodic dynamics and their bifurcations with special attention to orbits of low period as usually generating largest stability islands. Two main results are obtained. Firstly, at any parameters \((\alpha,\beta)\) there exists a bounded region outside of which the dynamics consists of a set of large measure invariant tori with thin layers of resonant dynamics between them. Secondly, the parameter plane can be divided into four different types of fundamental domains, based on the stability types of the upper and lower equilibria. The stability boundaries are curves of pitchfork or period doubling bifurcations, where the pitchfork bifurcation has codimension one due to spatial symmetry of the system. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrality. The authors give a broad review of the predecessor's works.
scientific article; zbMATH DE number 2134826
scientific article