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

Abstract

We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillatorFix T > 0. We target periodic solutions x n (t) of small minimal periods p n = 2T /n, for integer n → ∞, and with correspondingly large amplitudes. Simultaneously, for sufficiently large n ≥ n 0 , we obtain local exponential stability for (−1) n b < 0, and exponential instability for (−1) n b > 0, provided thatWe interpret our results in terms of noninvasive delayed feedback stabilization and destabilization of large amplitude rapidly periodic solutions in the standard Duffing oscillator. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker-Sell torus bifurcation at the validity boundary of our theoretical results.

Non-Spiking Laser Controlled by a Delayed Feedback

“…However, there are other problems where the damping rate could be moderate to large. Consider, for example, the delayed Duffing equation, which has benefited from recent mathematical interests [24][25][26]. Duffing equation models an elastic pendulum whose spring's stiffness is nonlinear.…”

Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control

“…As a consequence, Pyragas control has many different successful applications, e.g., in atomic force microscopes 28 , un-manned helicopters 19 , complex robots 26 , semiconductor lasers 22,23 , and the enzymatic peroxidaseoxidase reaction 17 , among others. The success of Pyragas control has been verified for a large number of specific theoretical models as well, including spiral break-up in cardiac tissues 21 , flow alignment in sheared liquid crystals 27 , near Hopf bifurcation 3 and unstable foci 12,29 , synchrony in networks of coupled Stuart-Landau oscillators 24,25 , delay equations 6,7 , in quantum systems 2,10 , the Duffing oscillator 5 and Turing patterns 16 .…”

Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control

“…As a consequence, Pyragas control has many different successful applications, e.g., in atomic force microscopes, 32 un-manned helicopters, 21 complex robots, 29 semiconductor lasers, 24,25 and the enzymatic peroxidase-oxidase reaction, 19 among others. The success of Pyragas control has been verified for a large number of specific theoretical models as well, including spiral breakup in cardiac tissues, 23 flow alignment in sheared liquid crystals, 30 near Hopf bifurcation 3 and unstable foci, 12,33 synchrony in networks of coupled Stuart-Landau oscillators, 26,27 delay equations, 6,7 in quantum systems, 2,10 the Duffing oscillator, 5 and Turing patterns. 18 General conditions on the success or failure of Pyragas control are hard to obtain because the time delay adds infinitely many dimensions to the complexity of the dynamical system.…”