Subharmonic resonance analysis of asymmetrical stiffness nonlinear systems with time delay

doi:10.1007/s10483-025-3273-8

摘要/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

中图分类号:

O322

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1347-1364.

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

图/表 11

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

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