Subharmonic resonance analysis of asymmetrical stiffness nonlinear systems with time delay

doi:10.1007/s10483-025-3273-8

Abstract

Abstract:

Incorporating asymmetric quadratic and cubic stiffnesses into a time-delayed Duffing oscillator provides a more accurate representation of practical systems, where the resulting nonlinearities critically influence subharmonic resonance phenomena, yet comprehensive investigations remain limited. This study employs the generalized harmonic balance (HB) method to conduct an analytical investigation of the subharmonic resonance behavior in asymmetric stiffness nonlinear systems with time delay. To further examine the switching behavior between primary and subharmonic resonances, a numerical continuation approach combining the shooting method and the parameter continuation algorithm is developed. The analytical and numerical continuation solutions are validated through direct numerical integration. Subsequently, the switching behavior and associated bifurcation points are analyzed by means of the numerical continuation results in conjunction with the Floquet theory. Finally, the effects of delay parameters on the existence range of subharmonic responses are discussed in detail, and the influence of initial conditions on system dynamics is explored with basin of attraction plots. This work establishes a comprehensive framework for the analytical and numerical study on time-delayed nonlinear systems with asymmetric stiffness, providing valuable theoretical insights into the stability management of such dynamic systems.

Key words: time delay feedback, bifurcation, asymmetrical stiffness, subharmonic resonance, harmonic balance (HB) method

2010 MSC Number:

O322

Xinliang LIU, Bin FANG, Shaoke WAN, Xiaohu LI. Subharmonic resonance analysis of asymmetrical stiffness nonlinear systems with time delay. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1347-1364.

TrendMD

Figures/Tables 11

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 1

Response type, presence range, and bifurcation type"

Stage	I	II	III	IV	V	VI	VII	VIII
Response type	PS	PU	PS	SS	SU	SS	SU	PS
Frequency range	11.04 $*$	[11.04, 3.27]	[3.27, 3.43]	[3.43, 3.72]	[3.72, 4.55]	[4.55, 5.98]	[5.98, 3.56]	3.56 $*$
Bifurcation type	NS	TP		NS	PD	NS	NS	NS
(i) The response types are defined as follows: PS for primary resonance stability, PU for primary resonance instability, SS for subharmonic resonance stability, and SU for subharmonic resonance instability. (ii) The bifurcation type refers to the bifurcation leading to the subsequent stage. For instance, NS bifurcation transitions Stage I to Stage II, and thus Stage I corresponds to NS bifurcation. (iii) The superscript $*$ represents the boundary (start or end) of the analyzed frequency range

Table 1

Fig. 8

Fig. 9

Fig. 10

References 51

[1]	MEI, Z. and WANG, Z. Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback. Applied Mathematics and Mechanics (English Edition), 43, 1747–1762 (2022) https://doi.org/10.1007/s10483-022-2921-8
[2]	LANI-WAYDA, B. and WALTHER, H. O. A Shilnikov phenomenon due to state-dependent delay, by means of the fixed point index. Journal of Dynamics and Differential Equations, 28, 627–688 (2016)
[3]	LI, C., CHEN, G., LIAO, X., and YU, J. Hopf bifurcation in an Internet congestion control model. Chaos, Solitons & Fractals, 19(4), 853–862 (2004)
[4]	ZHANG, Q., WEI, X., and XU, J. An analysis on the global asymptotic stability for neural networks with variable delays. Physics Letters A, 328(2-3) 163–169 (2004)
[5]	ZHANG, X. M., HAN, Q. L., SEURET, A., and GOUAISBAUT, F. An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay. Automatica, 84, 221–226 (2017)
[6]	LUO, A. C. J. and XING, S. Y. Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator. Chaos, Solitons & Fractals, 89, 405–434 (2016)
[7]	JIN, Y. F., WANG, H. T., and XU, P. F. Noise-induced enhancement of stability and resonance in a tri-stable system with time-delayed feedback. Chaos, Solitons & Fractals, 168, 113099 (2023)
[8]	GUO, S. J. Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity, 18(5), 2391–2407 (2005)
[9]	ZHANG, L. J. and OROSZ, G. Consensus and disturbance attenuation in multi-agent chains with nonlinear control and time delays. International Journal of Robust and Nonlinear Control, 27(5), 781–803 (2017)
[10]	MAZENC, F. and MALISOFF, M. Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay. Automatica, 78, 1–13 (2017)
[11]	DAI, H. L., ABDELKEFI, A., WANG, L., and LIU, W. B. Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks. Nonlinear Dynamics, 78(2), 907–919 (2014)
[12]	NAYFEH, A. H. and NAYFEH, N. A. Time-delay feedback control of lathe cutting tools. Journal of Vibration and Control, 18(8), 1106–1115 (2012)
[13]	GUO, Y. X., JIANG, W. H., and NIU, B. Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh-Rose neurons. Nonlinear Dynamics, 71(3), 515–529 (2013)
[14]	MOU, S. S., GAO, H. J., LAM, J., and QIANG, W. Y. A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay. IEEE Transactions on Neural Networks, 19(3), 532–535 (2008)
[15]	XU, W. Y., CAO, J. D., and XIAO, M. Bifurcation analysis and control in exponential RED algorithm. Neurocomputing, 129, 232–245 (2014)
[16]	ZHANG, S. and XU, J. Quasiperiodic motion induced by heterogeneous delays in a simplified internet congestion control model. Nonlinear Analysis-Real World Applications, 14(1), 661–670 (2013)
[17]	ZHAO, X. M. and OROSZ, G. Nonlinear day-to-day traffic dynamics with driver experience delay: modeling, stability and bifurcation analysis. Physica D-Nonlinear Phenomena, 275, 54–66 (2014)
[18]	VALIDO, A. A., COCCOLO, M., and SANJUÁN, M. A. F. Time-delayed Duffing oscillator in an active bath. Physical Review E, 108(6), 064205 (2023)
[19]	PYRAGAS, V. and PYRAGAS, K. State-dependent act-and-wait time-delayed feedback control algorithm. Communications in Nonlinear Science and Numerical Simulation, 73, 338–350 (2019)
[20]	KUMAR, R. and MITRA, R. K. Controlling period-doubling route to chaos phenomena of roll oscillations of a biased ship in regular sea waves. Nonlinear Dynamics, 111(15), 13889–13918 (2023)
[21]	JI, J. C. Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator. Journal of Sound and Vibration, 297(1-2), 183–199 (2006)
[22]	YU, Y., ZHANG, Z. D., and BI, Q. S. Multistability and fast-slow analysis for van der Pol-Duffing oscillator with varying exponential delay feedback factor. Applied Mathematical Modelling, 57, 448–458 (2018)
[23]	RUSINEK, R., WEREMCZUK, A., KECIK, K., and WARMINSKI, J. Dynamics of a time delayed Duffing oscillator. International Journal of Non-Linear Mechanics, 65, 98–106 (2014)
[24]	HU, H. Y. and WANG, Z. H. Singular perturbation methods for nonlinear dynamic systems with time delays. Chaos, Solitons & Fractals, 40(1), 13–27 (2009)
[25]	FAN, Y. H., JIAO, Y. J., and LI, X. L. Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms. Mathematical Methods in the Applied Sciences, 46(13), 14576–14595 (2023)
[26]	WU, H., ZENG, X. H., LIU, Y. B. A., and LAI, J. Analysis of harmonically forced Duffing oscillator with time delay state feedback by incremental harmonic balance method. Journal of Vibration Engineering & Technologies, 9(6), 1239–1251 (2021)
[27]	LIAO, H. T. Nonlinear dynamics of Duffing oscillator with time delayed term. Computer Modeling in Engineering & Sciences, 103(3), 155–187 (2014)
[28]	LUO, A. C. J. and XING, S. Y. Symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator. Nonlinear Dynamics, 85(2), 1141–1166 (2016)
[29]	SUN, J. Q. Random vibration analysis of time-delayed dynamical systems. Probabilistic Engineering Mechanics, 29, 1–6 (2012)
[30]	WEN, S. F., SHEN, Y. J., YANG, S. P., and WANG, J. Dynamical response of Mathieu-Duffing oscillator with fractional-order delayed feedback. Chaos, Solitons & Fractals, 94, 54–62 (2017)
[31]	WEN, S. F., SHEN, Y. J., and YANG, S. P. Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay. Acta Physica Sinica, 65(9), 094502 (2016)
[32]	FENG, C. S. and CHEN, S. L. Stochastic stability of Duffing-Mathieu system with delayed feedback control under white noise excitation. Communications in Nonlinear Science and Numerical Simulation, 17(10), 3763–3771 (2012)
[33]	XU, B., ZHANG, W., and MA, J. M. Stability and Hopf bifurcation of a two-dimensional supersonic airfoil with a time-delayed feedback control surface. Nonlinear Dynamics, 77(3), 819–837 (2014)
[34]	HUANG, D. M., ZHOU, S. X., LI, R. H., and YURCHENKO, D. On the analysis of the tristable vibration isolation system with delayed feedback control under parametric excitation. Mechanical Systems and Signal Processing, 164, 108207 (2022)
[35]	RUTTANATRI, P., COLE, M. O. T., and PONGVUTHITHUM, R. Structural vibration control using delayed state feedback via LMI approach: with application to chatter stability problems. International Journal of Dynamics and Control, 9(1), 85–96 (2021)
[36]	ZHANG, C., LI, C. Y., XU, M. T., YAO, G., LIU, Z. D., and DAI, W. B. Cutting force and nonlinear chatter stability of ball-end milling cutter. The International Journal of Advanced Manufacturing Technology, 120(9-10), 5885–5908 (2022)
[37]	SHENG, L. C., SUN, Q., YE, G., and LI, W. Dynamic analysis of grinding electric spindle bearing-rotor system under eccentric action. Journal of Advanced Mechanical Design Systems and Manufacturing, 17(2), JAMDSM0028 (2023)
[38]	BALTAZAR-TADEO, L. A., COLÍN-OCAMPO, J., ABÚNDEZ-PLIEGO, A., MENDOZA-LARIOS, J. G., MARTÍNEZ-RAYÓN, E., and GARCÍA-VILLALOBOS, A. Balancing of asymmetric rotor-bearing systems using modal masses array calculated by algebraic identification of modal unbalance. Journal of Vibration Engineering & Technologies, 12(3), 4765–4788 (2024)
[39]	JI, J. and LEUNG, A. Resonances of a non-linear SDOF system with two time-delays in linear feedback control. Journal of Sound and Vibration, 253(5), 985–1000 (2002)
[40]	LIU, X. L., WAN, S. K., FANG, B., and LI, X. H. Dynamics analysis of time-delayed nonlinear system with asymmetric stiffness. Chaos, Solitons & Fractals, 189, 115624 (2024)
[41]	HU, H. Y., DOWELL, E. H., and VIRGIN, L. N. Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dynamics, 15(4), 311–327 (1998)
[42]	XU, Y. X., HU, H. Y., and WEN, B. C. 1/3 Pure sub-harmonic solution and fractal characteristic of transient process for Duffing's equation. Applied Mathematics and Mechanics (English Edition), 27, 1171–1176 (2006) https://doi.org/10.1007/s10483-006-0903-1
[43]	DOMBOVARI, Z., BARTON, D. A. W., WILSON, R. E., and STEPAN, G. On the global dynamics of chatter in the orthogonal cutting model. International Journal of Non-Linear Mechanics, 46(1), 330–338 (2011)
[44]	MIAO, H. H., LI, C. Y., YU, C. P., HUA, C. L., WANG, C. Y., ZHANG, X. L., and XU, M. T. A fully analytical nonlinear dynamic model of spindle-holder-tool system considering contact characteristics of joint interfaces. Mechanical Systems and Signal Processing, 202, 110693 (2023)
[45]	LUO, A. C. Continuous Dynamical Systems, L & H Scientific Pub. and Higher Education Press Limited, Beijing (2012)
[46]	LUO, A. C. J. and JIN, H. X. Period-1 motions in a time-delayed Duffing oscillator with periodic excitation. ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, American Society of Mechanical Engineers, Buffalo, NY (2014)
[47]	LUO, A. C. and JIN, H. Period-m motions to chaos in a periodically forced, Duffing oscillator with a time-delayed displacement. International Journal of Bifurcation and Chaos, 24(10), 1450126 (2014)
[48]	HAO, Z. and CAO, Q. The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness. Journal of Sound and Vibration, 340, 61–79 (2015)
[49]	AKIMA, H. A new method of interpolation and smooth curve fitting based on local procedures. Journal of the ACM, 17(4), 589–602 (1970)
[50]	KRACK, M. and GROSS, J. Harmonic Balance for Nonlinear Vibration Problems, Springer Nature, Switzerland (2019)
[51]	RENSON, L., SHAW, A. D., BARTON, D. A., and NEILD, S. Application of control-based continuation to a nonlinear structure with harmonically coupled modes. Mechanical Systems and Signal Processing, 120, 449–464 (2019)