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Analysis of non-linear dynamics and bifurcations of a double pendulum. (English) Zbl 1236.70031
Summary: The dynamic behaviour of a double pendulum system in the vicinity of several compound critical points is explored through both analytical and numerical approaches. Four types of critical points are considered, which are characterized by a double zero eigenvalue, a simple zero and a pair of pure imaginary eigenvalues, and two pairs of pure imaginary eigenvalues including resonant and non-resonant cases. With the aid of normal form theory, the explicit expressions for the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. Possible bifurcations leading to 2-D and 3-D tori are also investigated. Closed form stability conditions of the bifurcation solutions are presented. A time integration scheme is used to find the numerical solutions for these bifurcation cases, which agree with the analytical results. Finally, numerical simulation is also applied to obtain double-period cascading bifurcations leading to chaos.

70K50 Bifurcations and instability for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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