A spectral element approach for the stability of delay systems

We describe a spectral element approach to study the stability and equilibria solutions of Delay differential equations (DDEs). In contrast to the prototypical temporal finite element analysis (TFEA), the described spectral element approach admits spectral rates of convergence and allows exploiting hp-convergence schemes. The described approach also avoids the limitations of analytical integrations in TFEA by using highly accurate numerical quadratures—enabling the study of more complicated DDEs. The effectiveness of this new approach is compared with well-established methods in the literature using various case studies. Specifically, the stability results are compared with the conventional TFEA and Legendre collocation methods whereas the equilibria solutions are compared with the numerical simulations and the homotopy perturbation method (HPM) solutions. Our results reveal that the presented approach can have higher rates of convergence than both collocation methods and the HPM. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  Seymour V. Parter,et al.  On the Legendre–Gauss–Lobatto Points and Weights , 1999, J. Sci. Comput..

[2]  Rifat Sipahi,et al.  Complete Stability Analysis of Neutral-Type First Order Two-Time-Delay Systems with Cross-Talking Delays , 2006, SIAM J. Control. Optim..

[3]  Z. C. Feng,et al.  Synchronization phenomena for coupled delay-line oscillators , 2004 .

[4]  Dimitri Breda,et al.  Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations , 2005, SIAM J. Sci. Comput..

[5]  J. Reddy.  An introduction to the finite element method , 1989 .

[6]  Lloyd N. Trefethen,et al.  Barycentric Lagrange Interpolation , 2004, SIAM Rev..

[7]  Gábor Stépán,et al.  Stability of up-milling and down-milling, part 1: alternative analytical methods , 2003 .

[8]  Gábor Stépán,et al.  Semi‐discretization method for delayed systems , 2002 .

[9]  Haitao Ma,et al.  Chebyshev Expansion of Linear and Piecewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations , 2003 .

[10]  Mehdi Dehghan,et al.  On the convergence of He's variational iteration method , 2007 .

[11]  Andrew Deeks,et al.  Use of higher‐order shape functions in the scaled boundary finite element method , 2006 .

[12]  Esmail Hesameddini,et al.  An Optimal Choice of Initial Solutions in the Homotopy Perturbation Method , 2009 .

[13]  Neil D. Sims,et al.  Analytical prediction of chatter stability for variable pitch and variable helix milling tools , 2008 .

[14]  John G Milton,et al.  On-off intermittency in a human balancing task. , 2002, Physical review letters.

[15]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[16]  Firas A. Khasawneh,et al.  Analysis of milling dynamics for simultaneously engaged cutting teeth , 2010 .

[17]  Wim Michiels,et al.  Topics in Time Delay Systems , 2009 .

[18]  Ji-Huan He,et al.  Variational iteration method: New development and applications , 2007, Comput. Math. Appl..

[19]  Ji-Huan He,et al.  Variational iteration method for delay differential equations , 1997 .

[20]  Ji-Huan He.  HOMOTOPY PERTURBATION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2006 .

[21]  Jian-Qiao Sun,et al.  A method of continuous time approximation of delayed dynamical systems , 2009 .

[22]  Gábor Stépán,et al.  Updated semi‐discretization method for periodic delay‐differential equations with discrete delay , 2004 .

[23]  Zhanhua Yu.  Variational iteration method for solving the multi-pantograph delay equation , 2008 .

[24]  Tatyana Luzyanina,et al.  Computing Floquet Multipliers for Functional Differential Equations , 2002, Int. J. Bifurc. Chaos.

[25]  Haitao Ma,et al.  Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials , 2004 .

[26]  Brian P Mann,et al.  An empirical approach for delayed oscillator stability and parametric identification , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[28]  Esmail Babolian,et al.  On numerical improvement of Gauss-Lobatto quadrature rules , 2005, Appl. Math. Comput..

[29]  B. Mann,et al.  Stability of Interrupted Cutting by Temporal Finite Element Analysis , 2003 .

[30]  F. Austin,et al.  THE VARIATIONAL ITERATION METHOD WHICH SHOULD BE FOLLOWED , 2010 .

[31]  K. A. Shore,et al.  Cascaded and adaptive chaos synchronization in multiple time-delay laser systems , 2009 .

[32]  L. González-Vega,et al.  Barycentric Hermite Interpolants for Event Location in Initial-Value Problems , 2008 .

[33]  A. Ranjbar,et al.  Maintaining the stability of nonlinear differential equations by the enhancement of HPM , 2008 .

[34]  R. Eddie Wilson,et al.  Collocation schemes for periodic solutions of neutral delay differential equations , 2006 .

[35]  Firas A. Khasawneh,et al.  Comparison Between Collocation Methods and Spectral Element Approach for the Stability of Periodic Delay Systems , 2010 .

[36]  N. Higham.  The numerical stability of barycentric Lagrange interpolation , 2004 .

[37]  Koen Engelborghs,et al.  Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations , 2002, Numerische Mathematik.

[38]  Gábor Stépán,et al.  Increased stability of low-speed turning through a distributed force and continuous delay model , 2009 .

[39]  Gábor Stépán,et al.  Balancing with Reflex Delay , 2000 .

[40]  Yilong Lu,et al.  Sparse matrix eigenvalue solver for finite element solution of dielectric waveguides , 1991 .

[41]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[42]  Silviu-Iulian Niculescu,et al.  Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers , 2007, SIAM J. Appl. Math..

[43]  Brian P. Mann,et al.  Stability of Delay Equations Written as State Space Models , 2010 .

[44]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[45]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[46]  Ji-Huan He.  Periodic solutions and bifurcations of delay-differential equations , 2005 .

[47]  Rifat Sipahi,et al.  A Unique Methodology for Chatter Stability Mapping in Simultaneous Machining , 2005 .

[48]  Mehdi Dehghan,et al.  Solution of delay differential equations via a homotopy perturbation method , 2008, Math. Comput. Model..

[49]  Xiaoyan Ma,et al.  He's homotopy perturbation method to periodic solutions of nonlinear Jerk equations , 2008 .

[50]  Eric A. Butcher,et al.  Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels , 2009 .

[51]  Haitao Ma,et al.  Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials , 2006 .

[52]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[53]  Claus Schneider,et al.  Hermite interpolation: The barycentric approach , 1991, Computing.

[54]  Dirk Roose,et al.  Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations , 2000, SIAM J. Sci. Comput..

[55]  Wim Michiels,et al.  Consensus Problems with Distributed Delays, with Application to Traffic Flow Models , 2009, SIAM J. Control. Optim..

[56]  J. Hale,et al.  Global geometry of the stable regions for two delay differential equations , 1993 .

[57]  Doron S. Lubinsky,et al.  Interpolatory integration rules and orthogonal polynomials with varying weights , 1992, Numerical Algorithms.

[58]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[59]  J. R. Claeyssen.  Effect of Delays on Functional Differential Equations , 1976 .

[60]  V. A. Chirikalov.  Computation of the differentiation matrix for the Hermite interpolating polynomials , 1994 .

[61]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[62]  Jianqiao Sun,et al.  Optimal Control of Time-Delayed Linear Systems With the Method of Continuous Time Approximation , 2008 .