Global dynamics of a rapidly forced cart and pendulum
A free swinging plane pendulum's fixing hinge moves along an inclined line in accordance with a prescribed law \((q_0= \beta\cos \omega t)\). The pendulum is subjected to the gravitational force, and the classical case of an inverted pendulum excited by high frequency vertical force is included in this setting. The authors study periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. Existence of such motions and existence of periodic orbits near saddle equilibrium of the averaged system are shown by the method of successive approximations; an analytical approximation is computed in the same way. The latter periodic orbits are related to saddle periodic orbits having stable and unstable manifolds forming homoclinic tangles. The authors apply the Melnikov's theory, the Poincaré maps and the enlarged theory of exponentially small splitting of separatrices. Three special limiting cases are analysed: a) \(q_0\to 0\), \(\omega\) is fixed; b) \(\omega\to \infty\), \(q_0\) is fixed; c) \(\omega\to \infty\), \(q_0\to 0\). Location and size of the chaotic regions are calculated, and effect of a small damping is investigated. Several diagrams visualize the phenomena. Results can be applied in the theory of nonlinear control systems.
theory of exponentially small splitting of separatrices
scientific article; zbMATH DE number 1043740