ISSN:
1573-269X
Keywords:
slow and fast invariant manifolds
;
fast and slow chaos
;
energy transfer
;
separatrix
;
Lyapunov exponents
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008356204490
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