Threshold Dynamics of Scalar Linear Periodic Delay-Differential Equations

2012 ◽  
pp. 269-278 ◽  
Author(s):  
Yuming Chen ◽  
Jianhong Wu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Threshold Dynamics ◽  
Periodic Delay ◽  
Delay Differential

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

A State-Space Temporal Finite Element Approach for Stability Investigations of Delay Equations

2009 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Brian P. Mann ◽  
Bhavin Patel
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
State Space ◽  
Delay Differential Equations ◽  
State Space Model ◽  
Element Analysis ◽  
Generalized Framework ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

This paper describes a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems — systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

Stability of Delay Equations Written as State Space Models

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1067-1085 ◽  
Author(s):  
B.P. Mann ◽  
B.R. Patel
Keyword(s):  
Differential Equations ◽  
State Space ◽  
Delay Differential Equations ◽  
State Space Model ◽  
Element Analysis ◽  
New Approach ◽  
Generalized Framework ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

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