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

Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

Dimitri Breda , and  Sara Della Schiava

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  September 2018 Early access  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.

Keywords: Lyapunov exponents, delay differential equations, pseudospectral methods.
Mathematics Subject Classification: Primary: 37M25, 65L03, 65L07.
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

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 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 $\circ$).
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14] (dashed $\circ$) for $M = 20$.">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 $M = 20$.
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Figure 3.  Projection of the attractor of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 14$ (left), $\tau = 17$ (right).
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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).
$5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$
$3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$
$0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$
$-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$
$-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$
$-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
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$5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$
$3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$
$0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$
$-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$
$-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$
$-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
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 $\mathbb{R}^{6}$.
$-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$
$-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$
$-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$
$-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$
$-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$
$-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
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$-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$
$-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$
$-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$
$-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$
$-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$
$-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
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