Harmonic oscillator minimal models

For elastomers, a bifurcation due to the nonlinearities is observed with the emergence of a sub-harmonic mode, qH (Fig. 8.8). This bifurcation clearly corresponds to a period-doubling sequence similar to what is observed for Faraday instability of shaken fluid layers, except that this bifurcation refers to time and not space [20]. For the Faraday instability, the period-doubling is usually described with a nonlinear parametric oscillator model. Interestingly, the Euler-Lagrange equation describing the minimization of energy for the compressed sheet on elastomer bears also some resemblance with a parametric oscillator [14]. 

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