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Non-integrability of restricted double pendula. (English) Zbl 1364.37122

Summary: We consider two special types of double pendula, with the motion of masses restricted to various surfaces. In order to get quick insight into the dynamics of the considered systems the Poincaré cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggests that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system’s variational equations along a particular non-equilibrium solution.

MSC:

37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C23 Bifurcation theory for ordinary differential equations
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References:

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