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2:1 Resonance in the delayed nonlinear Mathieu equation

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Abstract

We investigate the dynamics of a delayed nonlinear Mathieu equation:

$$\ddot{x}+(\delta+\varepsilon\alpha\,\cos t)x +\varepsilon\gamma x^3=\varepsilon\beta x(t-T)$$

in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of averaging (valid for small ɛ) is used to obtain a slow flow that is analyzed for stability and bifurcations. We show that the 2:1 instability region associated with parametric excitation can be eliminated for sufficiently large delay amplitudes β, and for appropriately chosen time delays T. We also show that adding delay to an undamped parametrically excited system may introduce effective damping.

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, USA

    Tina M. Morrison & Richard H. Rand

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  1. Tina M. Morrison
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  2. Richard H. Rand
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Correspondence to Richard H. Rand.

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Morrison, T.M., Rand, R.H. 2:1 Resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn 50, 341–352 (2007). https://doi.org/10.1007/s11071-006-9162-5

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  • DOI: https://doi.org/10.1007/s11071-006-9162-5

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