Quasi-periodic bursters and chaotic dynamics in a shallow arch subject to a fast–slow parametric excitation

In this paper, various nonlinear dynamics of a one-degree-of-freedom shallow arch model are investigated. The arch is subject to an imposed displacement of its support that is composed of slow and fast harmonic motions. The corresponding mathematical model consists of a nonlinear quasi-periodic Mathieu–Duffing equation. The dynamics are analyzed using the singular perturbation theory and the Melnikov method. It is shown that invariant slow manifolds of the averaged system, over the fast dynamics, are slaving the dynamics of the system under the condition of non-hyperbolicity of the undeformed state of the arch. These manifolds correspond to the buckled, the unbuckled and the undeformed solutions of the arch. Various kinds of quasi-periodic and chaotic bursters relating these slow manifolds are obtained, and quasi-periodic bursters doubling and tripling sequences leading to hysteretic chaos are observed. Using the Melnikov method and the Lyapunov exponents computations, it was demonstrated that chaos induced by the slow excitation can be suppressed by the fast harmonic excitation in large domains of the control parameters space, especially in the regions where the undeformed configuration of the arch is not hyperbolic.

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