Limit cycle oscillation and multiple entrainment phenomena in a duffing oscillator under time-delayed displacement feedback

Abstract

The Duffing oscillator under time-delayed displacement feedback is investigated to study the effect of intentional time-delay on the global dynamics of the oscillator. From the free vibration study performed by employing the describing function method it is observed that for the undamped oscillator, an infinite number of limit cycles is present for all possible values of gain and delay. The number of stable and unstable limit cycles in the gain versus delay plane is studied region wise with the help of limit cycle stability lines. Secondly, in a damped system, the number of limit cycles is finite and depends upon the values of gain, delay and damping coefficient from which the maximum number of limit cycles, their frequencies and amplitudes are obtained. When the system is excited by harmonic forcing, these limit cycles exhibit the phenomena of multiple entrainments and their frequency response curves become very complex and most often results in the very high amplitude oscillations. The study of the forced damped oscillator is therefore carried out by applying the method of slowly varying parameter and the frequency response curves for period-1 responses are analyzed. Further, with the a priori knowledge of possible stable and unstable limit cycles obtained by the application of semi-analytical methods, the various instability phenomena due to subharmonic and quasiperiodic responses have also been investigated by numerical simulation using Simulink in the different parametric ranges.

Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators

“…For the delayed Duffing oscillator (1.1) stability of large amplitude rapidly oscillating periodic solutions has been observed numerically, and supported by formal asymptotic expansions. See [WaCha04,HaBe12,MChB15,DaShRa17]. Similar methods have been applied by [XuChu03] towards delayed feedback control of a forced van der Pol -Duffing oscillator.…”

Unbounded sequences of stable limit cycles in the delayed Duffing equation: an exact analysis

“…In their DDE, the time delay was fixed at T = 1. Mitra&al [MiChaBa17] studied the same DDE with an added linear stiffness. By assuming an approximate solution in harmonic form x(t) = A sin(ωt), they claimed that the system exhibits an infinite number of stable limit cycles for any value of the time delay T .…”