### Abstract

This 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.

Original language | English (US) |
---|---|

Pages (from-to) | 895-922 |

Number of pages | 28 |

Journal | International Journal for Numerical Methods in Engineering |

Volume | 59 |

Issue number | 7 |

State | Published - Feb 21 2004 |

Externally published | Yes |

### Fingerprint

### Keywords

- Chebyshev polynomials
- Delay-differential equations (DDEs)
- Stability properties

### ASJC Scopus subject areas

- Engineering (miscellaneous)
- Applied Mathematics
- Computational Mechanics

### Cite this

*International Journal for Numerical Methods in Engineering*,

*59*(7), 895-922.

**Stability of linear time-periodic delay-differential equations via Chebyshev polynomials.** / Butcher, Eric; Ma, Haitao; Bueler, Ed; Averina, Victoria; Szabo, Zsolt.

Research output: Contribution to journal › Article

*International Journal for Numerical Methods in Engineering*, vol. 59, no. 7, pp. 895-922.

}

TY - JOUR

T1 - Stability of linear time-periodic delay-differential equations via Chebyshev polynomials

AU - Butcher, Eric

AU - Ma, Haitao

AU - Bueler, Ed

AU - Averina, Victoria

AU - Szabo, Zsolt

PY - 2004/2/21

Y1 - 2004/2/21

N2 - This 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.

AB - This 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.

KW - Chebyshev polynomials

KW - Delay-differential equations (DDEs)

KW - Stability properties

UR - http://www.scopus.com/inward/record.url?scp=1442313921&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1442313921&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:1442313921

VL - 59

SP - 895

EP - 922

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 7

ER -