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Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

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    * Corresponding author 

The first author is a member of INdAM Research group GNCS and is supported by INdAM GNCS projects "Analisi numerica di sistemi dinamici infinito-dimensionali e non regolari" (2015) and "Analisi numerica di certi tipi non classici di equazioni di evoluzione" (2016) and by the project PSD 2015 2017 DIMA PRID 2017 ZANOLIN "SIDIA – SIstemi DInamici e Applicazioni" (UNIUD)

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  • A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included.

    Mathematics Subject Classification: Primary: 37M25, 65L03, 65L07.


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  • Figure 1.  First five rightmost characteristic roots ($\times$) of (14) and first five dominant Lyapunov exponents computed with the current method ($\bullet$) and with the method in [14] ($\circ$), both for $M = 20$ and $T = 10^{5}$ (left); relevant absolute errors for increasing $M$ (right): current method (solid $\bullet$) and method in [14] (dashed $\circ$).

    Figure 2.  Absolute error with respect to $1$ of the largest exponent of (14) plotted against the final truncation time $T$, computed with the current method for varying $M = 10,15,20$ (solid $\bullet$, top-to-bottom) and with the method in [14] (dashed $\circ$) for $M = 20$.

    Figure 3.  Projection of the attractor of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 14$ (left), $\tau = 17$ (right).

    Table 1.  First six exponents of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 50$ computed with $M = 20$ (first column), from [14] (second column) and from [39] (third column); the reference solution corresponds to the initial function of constant value $\varphi\equiv2$ in (2).

    $5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$
    $3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$
    $0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$
    $-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$
    $-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$
    $-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
     | Show Table
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    Table 2.  First three exponents of (16) for $a = b = 0.1$, $c = 14$ and $\epsilon = 0.5$ and varying coupling delay $\tau = 1$ (first column), $1.5$ (second column) and $2$ (third column), computed with the current method for $M = 5$ and $T = 10^3$ (first three rows) and with the method in [14] for $M = 20$ and $T = 10^{4}$ (second three rows); the reference solution of (2) corresponds to the initial function of constant value a (pseudo)random vector in $\mathbb{R}^{6}$.

    $-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$
    $-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$
    $-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$
    $-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$
    $-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$
    $-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
     | Show Table
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  •   L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, no. 146 in Transl. Math. Monographs, AMS, Providence, 1995.
      H. T. Banks  and  F. Kappel , Spline approximations for functional differential equations, J. Diff. Equations, 34 (1979) , 496-522.  doi: 10.1016/0022-0396(79)90033-0.
      A. Bellen  and  S. Maset , Numerical solution of constant coefficient linear delay differential equations as abstract cauchy problems, Numer. Math., 84 (2000) , 351-374.  doi: 10.1007/s002110050001.
      A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathemathics and Scientifing Computing series, Oxford University Press, 2003.
      G. Benettin , L. Galgani , A. Giorgilli  and  J. M. Strelcyn , Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15 (1980) , 9-20.  doi: 10.1007/BF02128236.
      G. Benettin , L. Galgani , A. Giorgilli  and  J. M. Strelcyn , Lyapunov exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2: Numerical applications, Meccanica, 15 (1980) , 21-30.  doi: 10.1007/BF02128237.
      D. Breda , Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010) , 935-956. 
      D. Breda , O. Diekmann , M. Gyllenberg , F. Scarabel  and  R. Vermiglio , Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Sys., 15 (2016) , 1-23.  doi: 10.1137/15M1040931.
      D. Breda , O. Diekmann , D. Liessi  and  F. Scarabel , Numerical bifurcation analysis of a class of nonlinear renewal equations, Electron. J. Qual. Theory Differ. Equ., 65 (2016) , 1-24. 
      D. Breda , S. Maset  and  R. Vermiglio , Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005) , 482-495.  doi: 10.1137/030601600.
      D. Breda , S. Maset  and  R. Vermiglio , Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Appl. Numer. Math., 56 (2006) , 318-331.  doi: 10.1016/j.apnum.2005.04.011.
      D. Breda , S. Maset  and  R. Vermiglio , Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012) , 1456-1483.  doi: 10.1137/100815505.
      D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations -A Numerical Approach with MATLAB, Springer Briefs in Control, Automation and Robotics, Springer, New York, 2015.
      D. Breda  and  E. S. Van Vleck , Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math., 126 (2014) , 225-257.  doi: 10.1007/s00211-013-0565-1.
      M. D. Chekroun , M. Ghil , H. Liu  and  S. Wang , Low-dimensional Galerkin approximations of nonlinear delay differential euqations, Discrete Contin. Dyn. S., 36 (2016) , 4133-4177.  doi: 10.3934/dcds.2016.36.4133.
      F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995.
      L. Dieci, M. S. Jolly and E. S. Van Vleck, Numerical techniques for approximating Lyapunov exponents and their implementation, J. Comput. Nonlinear Dynam. 6 (2010), 011003, 7pp.
      L. Dieci , R. D. Russell  and  E. S. Van Vleck , On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal., 34 (1997) , 402-423.  doi: 10.1137/S0036142993247311.
      L. Dieci  and  E. S. Van Vleck , Computation of few Lyapunov exponents for continuous and discrete dynamical systems, Appl. Numer. Math., 17 (1995) , 275-291.  doi: 10.1016/0168-9274(95)00033-Q.
      L. Dieci  and  E. S. Van Vleck , Lyapunov spectral intervals: Theory and computation, SIAM J. Numer. Anal., 40 (2002) , 516-542.  doi: 10.1137/S0036142901392304.
      L. Dieci  and  E. S. Van Vleck , Orthonormal integrators based on Householder and Givens transformations, Future Gener. Comp. Sy., 19 (2003) , 363-373.  doi: 10.1016/S0167-739X(02)00163-2.
      L. Dieci and E. S. Van Vleck, LESLIS and LESLIL: Codes for approximating Lyapunov exponents of linear systems, 2004, http://www.math.gatech.edu/ dieci/software-les.html.
      O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations -Functional, Complex and Nonlinear Analysis, no. 110 in Applied Mathematical Sciences, Springer Verlag, New York, 1995.
      J. R. Dormand  and  P. J. Prince , A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980) , 19-26.  doi: 10.1016/0771-050X(80)90013-3.
      K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.
      D. Farmer , Chaotic attractors of an infinite-dimensional dynamical system, Physica D, 4 (1981/82) , 366-393.  doi: 10.1016/0167-2789(82)90042-2.
      D. Gottlieb , The stability of pseudospectral-Chebyshev methods, Math. Comp., 36 (1981) , 107-118.  doi: 10.1090/S0025-5718-1981-0595045-1.
      D. Gottlieb, M. Y. Hussaini and S. A. Orszag, Theory and applications of spectral methods, in Spectral methods for partial differential equations, SIAM, Philadelphia, Hampton, Va., 1984, 1-54.
      J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, 2nd edition, no. 99 in Applied Mathematical Sciences, Springer Verlag, New York, 1993.
      K. Ito  and  F. Kappel , A uniformly differentiable approximation scheme for delay systems using splines, Appl. Math. Opt., 23 (1991) , 217-262.  doi: 10.1007/BF01442400.
      F. Kappel, Semigroups and Delay Equations, no. 152 (Trieste, 1984) in Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986.
      T. H. Koornvinder , Orthogonal polynomials with weight functions $(1-x)^α(1+x)^β+Mδ(x+1)+Nδ(x-1)$, Canad. Math. Bull., 27 (1984) , 205-214.  doi: 10.4153/CMB-1984-030-7.
      A. M. Lyapunov , The general problem of the stability of motion, Internat. J. Control, 55 (1992) , 521-790. 
      M. C. Mackey  and  L. Glass , Oscillations and chaos in physiological control systems, Science, 197 (1977) , 287-289.  doi: 10.1126/science.267326.
      S. Maset , Numerical solution of retarded functional differential equations as abstract Cauchy problems, J. Comput. Appl. Math., 161 (2003) , 259-282.  doi: 10.1016/j.cam.2003.03.001.
      A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, no. 44 in Applied Mathematical Sciences, Springer Verlag, New York, 1983.
      A. Prasad , Amplitude Death in coupled chaotic oscillators, Phys. Rev. E, 72 (2005) , 056204-10pp. 
      L. F. Shampine  and  S. Thompson , Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001) , 441-458.  doi: 10.1016/S0168-9274(00)00055-6.
      D. E. Sigeti , Exponential decay of power spectra at high frequency and positive Lyapunov exponents, Physica D, 52 (1995) , 136-153.  doi: 10.1016/0167-2789(94)00225-F.
      J. C. Sprott , A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007) , 397-402.  doi: 10.1016/j.physleta.2007.01.083.
      A. Stefanski , A. Dabrowski  and  T. Kapitaniak , Evaluation of the largest Lyapunov exponent in dynamical systems with time delay, Chaos, Solitons and Fract., 23 (2005) , 1651-1659.  doi: 10.1016/S0960-0779(04)00428-X.
      L. N. Trefethen, Spectral Methods in MATLAB, Software -Environment -Tools series, SIAM, Philadelphia, 2000.
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