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Frequency Islands in The Primary Resonance of Nonlinear Delay Systems

The purpose of this effort is to investigate primary resonances of nonlinear delay systems. Along these lines, the response of a Duffing oscillator with delayed-state feedback to primary resonance excitations is considered and analyzed using the method of multiple scales. Unlike previous research efforts that let the coefficients of the delay states (gains) be small to allow direct implementation of the method of multiple scales, we demonstrate that the method can be adapted to analyze nonlinear delay systems with large gains. Further, we unveil very interesting dynamic responses characterized by the presence of \textit{islands} in the frequency response of the delayed Duffing oscillator. It is demonstrated that these islands grow in size and collide with the main branch of solutions (\textit{mainland}) as the magnitude of the external excitation is increased or as the gain-delay combination is chosen closer to the stability boundaries of the free response.
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