Abstract
The chaos controls of a Duffing system with impacts are investigated using appropriate random phase. Based on the discontinuous map and Wedig’s algorithm, a procedure of calculating top Lyapunov exponent is presented for stochastic Duffing system with impacts. As the random phase changes, two kinds of chaos control are achieved for Duffing system with impacts by the criterion of the top Lyapunov exponent: one is chaos suppression, and the other is chaos generation. In addition, the obtained results are verified by numerical simulations including the phase portrait, Poincare surface of section and time history.
Highlights
Many non-smooth systems can be described as kinds of discontinuity such as impacts, jumps, switch, dry frictions, and sliding transition.[1,2,3,4] Their motion is characterized by smooth evolutions with discontinuous interrupted
The core of researches is focused on the novel dynamics induced by sliding, grazing, chattering and corner collision
In order to calculate the top Lyapunov exponent, we show the following equations of the first-order perturbation δxi(i = 1, 2) with xi(i = 1, 2)
Summary
Many non-smooth systems can be described as kinds of discontinuity such as impacts, jumps, switch, dry frictions, and sliding transition.[1,2,3,4] Their motion is characterized by smooth evolutions with discontinuous interrupted. The Lyapunov exponent is a numerical tool to identify the chaotic transitions.[20,21] it falls outside the usual methodology due to the discontinuity in non-smooth systems. For deterministic system with impacts, using the DM, several methods of calculating Lyapunov exponent for specific non-smooth systems have been proposed.[22,23,24] For non-smooth systems with noise, the results are scanty. Feng[25] presented a numerical method of Lyapunov exponent for a general vibro-impact system with white noise. Wei[29] explored the Lyapunov exponent and chaos for a Duffing system with noise. The chaos control is carried out for a Duffing system with impacts by adjusting the random phase.
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