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

B. Fiedler; A. López Nieto; R.H. Rand; S.M. Sah; I. Schneider; B. de Wolff

doi:10.1016/j.jde.2019.11.015

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

B. Fiedler, A. López Nieto, R.H. Rand, S.M. Sah, I. Schneider, B. de Wolff

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

© 2019 Elsevier Inc.We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillator x″(t)+ax(t)+bx(t−T)+x3(t)=0. Fix T>0. We target periodic solutions xn(t) of small minimal periods pn=2T/n, for integer n→∞, and with correspondingly large amplitudes. Note how xn(t) are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators x″+ax+(−1)nbx+x3=0. Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large n≥n0, we obtain local exponential stability for (−1)nb<0, and exponential instability for (−1)nb>0, provided that 0≠(−1)n+1bT2<3/2π2. We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.

Original language	English
Pages (from-to)	5969-5995
Journal	Journal of Differential Equations
Volume	268
Issue number	10
DOIs	https://doi.org/10.1016/j.jde.2019.11.015
Publication status	Published - 5 May 2020
Externally published	Yes

Funding

Acknowledgment. This work originated at the International Conference on Structural Nonlinear Dynamics and Diagnosis 2018, in memoriam Ali Nayfeh, at Tangier, Morocco. We are deeply indebted to Mohamed Belhaq, Abderrahim Azouani, to all organizers, and to all helpers of this outstanding conference series. They indeed keep providing a unique platform of inspiration and highest level scientific exchange, over so many years, to the benefit of all participants. Original typesetting was patiently accomplished by Patricia Hăbăşescu. This work was partially supported by the Deutsche Forschungsgemeinschaft through SFB 910 project A4. Authors RHR and SMS gratefully acknowledge support by the National Science Foundation under grant number CMMI-1634664.

Funders	Funder number
National Science Foundation	CMMI-1634664
Deutsche Forschungsgemeinschaft	SFB 910

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@article{2d57738b2cc94921a66d1b9bb2a810d9,

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

abstract = "{\textcopyright} 2019 Elsevier Inc.We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillator x″(t)+ax(t)+bx(t−T)+x3(t)=0. Fix T>0. We target periodic solutions xn(t) of small minimal periods pn=2T/n, for integer n→∞, and with correspondingly large amplitudes. Note how xn(t) are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators x″+ax+(−1)nbx+x3=0. Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large n≥n0, we obtain local exponential stability for (−1)nb<0, and exponential instability for (−1)nb>0, provided that 0≠(−1)n+1bT2<3/2π2. We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.",

author = "B. Fiedler and {L{\'o}pez Nieto}, A. and R.H. Rand and S.M. Sah and I. Schneider and {de Wolff}, B.",

year = "2020",

month = may,

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doi = "10.1016/j.jde.2019.11.015",

language = "English",

volume = "268",

pages = "5969--5995",

journal = "Journal of Differential Equations",

issn = "0022-0396",

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TY - JOUR

T1 - Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators

AU - Fiedler, B.

AU - López Nieto, A.

AU - Rand, R.H.

AU - Sah, S.M.

AU - Schneider, I.

AU - de Wolff, B.

PY - 2020/5/5

Y1 - 2020/5/5

N2 - © 2019 Elsevier Inc.We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillator x″(t)+ax(t)+bx(t−T)+x3(t)=0. Fix T>0. We target periodic solutions xn(t) of small minimal periods pn=2T/n, for integer n→∞, and with correspondingly large amplitudes. Note how xn(t) are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators x″+ax+(−1)nbx+x3=0. Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large n≥n0, we obtain local exponential stability for (−1)nb<0, and exponential instability for (−1)nb>0, provided that 0≠(−1)n+1bT2<3/2π2. We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.

AB - © 2019 Elsevier Inc.We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillator x″(t)+ax(t)+bx(t−T)+x3(t)=0. Fix T>0. We target periodic solutions xn(t) of small minimal periods pn=2T/n, for integer n→∞, and with correspondingly large amplitudes. Note how xn(t) are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators x″+ax+(−1)nbx+x3=0. Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large n≥n0, we obtain local exponential stability for (−1)nb<0, and exponential instability for (−1)nb>0, provided that 0≠(−1)n+1bT2<3/2π2. We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results.

U2 - 10.1016/j.jde.2019.11.015

DO - 10.1016/j.jde.2019.11.015

M3 - Article

SN - 0022-0396

VL - 268

SP - 5969

EP - 5995

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -

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

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