# American Institute of Mathematical Sciences

September  2018, 23(7): 2727-2741. doi: 10.3934/dcdsb.2018092

## Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

 CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics - University of Udine, via delle scienze 206, 33100 Udine, Italy

* Corresponding author

Received  December 2016 Revised  November 2018 Published  March 2018

Fund Project: 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)

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.

Citation: Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092
##### References:
 [1] L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, no. 146 in Transl. Math. Monographs, AMS, Providence, 1995. Google Scholar [2] 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. Google Scholar [3] 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. Google Scholar [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathemathics and Scientifing Computing series, Oxford University Press, 2003. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] D. Breda, Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] 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. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] 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.Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] 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.Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. Google Scholar [26] 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. Google Scholar [27] D. Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp., 36 (1981), 107-118. doi: 10.1090/S0025-5718-1981-0595045-1. Google Scholar [28] 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. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] F. Kappel, Semigroups and Delay Equations, no. 152 (Trieste, 1984) in Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986. Google Scholar [32] 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. Google Scholar [33] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790. Google Scholar [34] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [35] 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. Google Scholar [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, no. 44 in Applied Mathematical Sciences, Springer Verlag, New York, 1983. Google Scholar [37] A. Prasad, Amplitude Death in coupled chaotic oscillators, Phys. Rev. E, 72 (2005), 056204-10pp. Google Scholar [38] 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. Google Scholar [39] 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. Google Scholar [40] J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083. Google Scholar [41] 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. Google Scholar [42] L. N. Trefethen, Spectral Methods in MATLAB, Software -Environment -Tools series, SIAM, Philadelphia, 2000. Google Scholar

show all references

##### References:
 [1] L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, no. 146 in Transl. Math. Monographs, AMS, Providence, 1995. Google Scholar [2] 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. Google Scholar [3] 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. Google Scholar [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathemathics and Scientifing Computing series, Oxford University Press, 2003. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] D. Breda, Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] 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. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] 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. Google Scholar [16] 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. Google Scholar [17] 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.Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] 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.Google Scholar [23] 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. Google Scholar [24] 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. Google Scholar [25] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. Google Scholar [26] 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. Google Scholar [27] D. Gottlieb, The stability of pseudospectral-Chebyshev methods, Math. Comp., 36 (1981), 107-118. doi: 10.1090/S0025-5718-1981-0595045-1. Google Scholar [28] 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. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] F. Kappel, Semigroups and Delay Equations, no. 152 (Trieste, 1984) in Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986. Google Scholar [32] 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. Google Scholar [33] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790. Google Scholar [34] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [35] 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. Google Scholar [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, no. 44 in Applied Mathematical Sciences, Springer Verlag, New York, 1983. Google Scholar [37] A. Prasad, Amplitude Death in coupled chaotic oscillators, Phys. Rev. E, 72 (2005), 056204-10pp. Google Scholar [38] 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. Google Scholar [39] 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. Google Scholar [40] J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083. Google Scholar [41] 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. Google Scholar [42] L. N. Trefethen, Spectral Methods in MATLAB, Software -Environment -Tools series, SIAM, Philadelphia, 2000. Google Scholar
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 $\circ$).
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 $M = 20$.
Projection of the attractor of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 14$ (left), $\tau = 17$ (right).
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).
 $5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$ $3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$ $0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$ $-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$ $-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$ $-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
 $5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$ $3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$ $0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$ $-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$ $-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$ $-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
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 $\mathbb{R}^{6}$.
 $-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$ $-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$ $-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$ $-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$ $-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$ $-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
 $-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$ $-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$ $-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$ $-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$ $-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$ $-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
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