SUMMARYThis paper presents a new technique for studying the stability properties of dynamic systems modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the 'infinite-dimensional Floquet transition matrix U '. Two different formulas for the computation of the approximate U , whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second-order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is an effective way to study the stability of time-periodic DDEs.

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This article utilizes Savitzky-Golay (SG) filter to eliminate seismic random noise. This is a novel method for seismic random noise reduction in which SG filter adopts piecewise weighted polynomial via leastsquares estimation. Therefore, effective smoothing is achieved in extracting the original signal from noise environment while retaining the shape of the signal as close as possible to the original one. Although there are lots of classical methods such as Wiener filtering and wavelet denoising applied to eliminate seismic random noise, the SG filter outperforms them in approximating the true signal. SG filter will obtain a good tradeoff in waveform smoothing and valid signal preservation under suitable conditions. These are the appropriate window size and the polynomial degree. Through examples from synthetic seismic signals and field seismic data, we demonstrate the good performance of SG filter by comparing it with the Wiener filtering and wavelet denoising methods.

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