Computation and Stability Analysis of Solutions of Periodic Delay Differential Algebraic Equations

2005 ◽  
Author(s):  
Koen Verheyden ◽  
Kurt Lust ◽  
Dirk Roose
Keyword(s):  
Stability Analysis ◽  
Numerical Methods ◽  
Periodic Solutions ◽  
Numerical Algorithms ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

This paper is concerned with the numerical computation, continuation and stability analysis of periodic solutions of periodic delay differential algebraic equations. We consider systems with a time-periodic right hand side function and time-periodic delays. We introduce numerical algorithms based on collocation to compute periodic solutions and their stability. The presented methods combine knowledge from numerical methods for delay equations and differential algebraic equations. Our algorithms are illustrated with numerical results for two models.

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1235-1260 ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Stability Analysis ◽  
Monodromy Matrix ◽  
Original System ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Dimensional Approximation ◽  
Delay Differential ◽  
Time Periodic ◽  
Nonlinear Delay

Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

scholarly journals On the Stability Analysis of Delay Differential-Algebraic Equations

2018 ◽  
Vol 34 (2) ◽  
Author(s):  
Phi Ha
Keyword(s):  
Stability Analysis ◽  
Exponential Stability ◽  
Linear Time ◽  
Differential Algebraic Equations ◽  
Spectral Condition ◽  
Algebraic Equations ◽  
Linear Time Invariant ◽  
Stability Of Dynamical Systems ◽  
The Stability ◽  
Delay Differential

The stability analysis of linear time invariant delay differentialalgebraic equations (DDAEs) is analyzed. Examples are delivered to demonstrate that the eigenvalue-based approach to analyze the exponential stability of dynamical systems is not valid for an arbitrarily high index system, and hence, a new concept of weakly exponential stability (w.e.s) is proposed. Then, we characterize the w.e.s in term of a spectral condition for some special classes of DDAEs.

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.

PERIODIC SOLUTIONS OF DIFFERENTIAL ALGEBRAIC EQUATIONS WITH TIME DELAYS: COMPUTATION AND STABILITY ANALYSIS

2006 ◽  
Vol 16 (01) ◽  
pp. 67-84 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
DIRK ROOSE
Keyword(s):  
Stability Analysis ◽  
Periodic Solutions ◽  
Time Delays ◽  
Differential Algebraic Equations ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Floquet Multipliers ◽  
Infinite Dimensional ◽  
Local Asymptotic Stability ◽  
Convergence Results

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations With Time Periodic Coefficients

2007 ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Similarity Transformation ◽  
Monodromy Matrix ◽  
Original System ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Dimensional Approximation ◽  
Delay Differential ◽  
Time Periodic ◽  
Nonlinear Delay

Stability theory of Nonlinear Delay Differential Algebraic Equations (DDAE) with periodic coefficients is proposed with a geometric interpretation of the evolution of the linearized system. First, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a finite dimensional approximation or a “monodromy matrix” of a compact infinite dimensional operator. The monodromy operator is essentially a map of the Chebyshev coefficients of the state form the delay interval to the next adjacent interval of time. The monodromy matrix is obtained by a similarity transformation of the momodromy matrix of the associated semi-explicit system. The computations are entirely performed in the original system form to avoid cumbersome transformations associated with the semi-explicit system. Next, two computational algorithms are detailed for obtaining solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

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