Stroboscopic averaging methods to study autoresonance and other problems with slowly varying forcing frequencies

Abstract

Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.

Figures (4)

• Figure 1: Left panel: autoresonance in the Duffing equation with $\alpha = 0.0001$, $\epsilon = 0.05$. At $\tau\approx 0$ the amplitude of the solution starts growing due to linear effects. After that, the oscillator automatically adjusts its instantaneous amplitude so that the corresponding frequency matches the (varying) frequency of the forcing; this allows the amplitude to keep growing with $\tau$. In the right panel, the value of $\alpha$ is again $0.0001$, but $\epsilon = 0.01$. Autoresonance does not take place; a growth in amplitude at $\tau\approx 0$ occurs but the system fails to adjust thereafter the amplitude to the frequency of the forcing.
• Figure 2: Minimum value of $\epsilon$ for which autoresonance takes place for eight values of $\alpha$. The results provided by six numerical algorithms are indistinguishable. The straight line corresponds to the approximation \ref{['eq:approximation']}.
• Figure 3: CPU time required by the different techniques.
• Figure 4: Efficiency: error when finding the minimum value of $\epsilon$ for techniques 3.--6. (using technique 1. as a reference) as a function of CPU time.