Homoclinic, subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length

Dynamics of the pendulum with periodically varying length

Oscillations of a pendulum with a periodically varying length and a model of swing

Qualitative analysis of a pendulum with a periodically varying length is conducted. It is proved that there are two periodic solutions having a prescribed amplitude A(n and a period 1 which is an even multiple k of the excitation period. Stability analysis is carried out for the principal parametric oscillations (k"2). In this connection it is shown that such a pendulum cannot serve as a mathem...

Integrability of Superharmonic Functions and Subharmonic Functions

We apply the coarea formula to obtain integrability of superharmonic functions and nonintegrability of subharmonic functions. The results involve the Green function. For a certain domain, say Lipschitz domain, we estimate the Green function and restate the results in terms of the distance from the boundary.

Homoclinic Bifurcations

1. Introduction. We say that a one-parameter family of diffeomorphisms ip^: M — • M, p G R, has a homoclinic bifurcation, or a homoclinic tangency, for p = 0 if ipo has an orbit of nontransverse intersection of a stable and an unstable manifold, both of the same hyperbolic fixed point (or periodic point), which splits, for p > 0, into two orbits of transverse intersection of these stable and un...

Homoclinic Bifurcations for the H

Chaotic dynamics can be eeectively studied by continuation from an anti-integrable limit. We use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations. We nd a bound on the parameter range for which the H enon map exhibits a complete binary horseshoe as well as a subshift of nite type, and study these numerically. We classify homoclinic bi...