Control of Bistability in a Delayed Duffing Oscillator

Abstract

The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding.

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.…”

Delay-induced resonance suppresses damping-induced unpredictability

“…In order to provide analytical support for the numerical results presented above, we perform a linear stability analysis [17,18] for the fixed points x∗=x±∗=±3.606. Even though the analysis is performed for x * = +3.606, for symmetry reasons it is equivalent for x * = −3.606.…”

Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

“…When ̸ = 0 and linear time delay feedback Φ( , ( − )) = ( − ), the following topics have been studied for various types of Duffing oscillators with time delayed feedback: in [38] authors constructed a low-order approximate solution under weak feedback gain parameter; about the low-and high-order approximations see also [39]; in [40] with = 0, the Hopf bifurcation diagrams have been explored for the approximate periodic solutions (amplitude versus time delay and feedback gain versus time delay ); moreover, in [41] authors made an analysis on the effect of the control gain and time delay parameters on the amplitude of approximate period solution from the theoretical and numerical points of view; see also [42]; in [43] authors studied the chaotic behaviour with respect to gains and time delay parameters; see also [44]. Equations under time delay control such as (34) (especially with damped term) are used as a model for various controlled physical, mechanical, and engineering systems with time delays; see, for instance, [39,[45][46][47][48] and the references therein.…”

“…In Section 2 we state a fundamental lemma proposing a new oscillation criterion that plays a crucial role in the formulation of the main results illustrated on some suitable chosen examples. In Section 3 we consider an application of the main results to the Duffing type quasilinear equations with time delayed feedback, taking into account the known results in applied sciences concerning such kind of nonlinear oscillators without time delay, see , and with time delay, see [38][39][40][41][42][43][44][45][46][47][48] and the references therein. In Section 5 we present some open questions and comments for further study that can follow our main results.…”