Linear time complexity analysis of time-periodic delayed systems with multiplication free semi-discretization method

This study introduces the multiplication free semi-discretization method (MFSD), a novel approach for the efficient stability and fixed-point analysis of time-periodic delayed differential equations (DDEs). Efficient stability prediction is critical for mitigating instabilities in mechanical systems, such as machine tool chatter in milling and vibration in delayed control systems, where high-accuracy modelling often leads to prohibitive computational costs. Unlike traditional semi-discretization methods that are burdened by high computational demands due to repeated matrix multiplication, the MFSD method leverages a composite mapping structure inspired by collocation techniques to achieve linear time complexity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> .The performance of the method is validated through extensive spectral analysis of the delayed Mathieu equation and further demonstrated on three practical engineering cases: a seasonal biological model, a multi-cutter turning system with spindle speed variation, and a finite-element-method-based longitudinal vibration of an elastic beam under delayed boundary feedback. Results demonstrate that MFSD maintains linear scaling even for system resolutions up to 10 6 , enabling high-fidelity stability mapping even for high-dimensional structural dynamics that were previously computationally inaccessible. The algorithm is provided as an open-source tool via the SemiDiscretizationMethod.jl package to support the vibration and control research community.