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Dynamics of a parametrically excited double pendulum. (English) Zbl 0820.34020
Summary: The 2:2 mode interaction of a parametrically excited double pendulum is explored in the excitation frequency/excitation amplitude plane. To determine the bifurcation structure at small amplitudes of oscillation, the method of averaging combined with centre manifold reduction is used. The full equations are solved numerically to extend the bifurcation set to larger amplitudes of response.
Numerical centre manifold reduction is employed to derive two maps, valid near two multiple bifurcation points which organize the dynamical phenomena of the mode interaction region. Iteration of these maps shows the existence of global bifurcations. These results are discussed in the light of numerical integrations which show that a whole range of interesting behaviour occurs including torus doubling, torus ‘gluing’ and chaos.

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C29 Averaging method for ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
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