Suppression of chaos in nonlinear oscillators using a linear vibration absorber

Abstract

This paper proposes a nonfeedback control to suppress the chaotic response of nonlinear oscillators. A linear vibration absorber is used as nonfeedback method, whose key idea is to find conditions under which the nonlinear oscillator response converges to an equilibrium point or stay oscillating around it. Theoretical results show that if the ratio between the natural frequencies of the primary system $(\omega _1)$ and the undamped absorber $(\omega _2)$, that is, $\omega _r=\frac{\omega _2}{\omega _1}$ is tuned to be equal to the excitation frequency $(\Omega )$, then the chaotic behavior of a nonlinear oscillator is driven to a stable hyperbolic equilibrium point. Numerical results are presented for the Duffing oscillator shown that if $\omega _r$ is tuned close to excitation frequency, then the chaotic response of the Duffing oscillator is driven to periodic orbits. In the case of damped absorber, $\omega _r$ to be tuned close to the excitation frequency does not guarantee that the response of the primary system is periodic, it is also necessary to take into account the amplitude of the excitation.