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Two-step collocation methods for fractional differential equations
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 |
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.
References:
[1] |
L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, no. 146 in Transl. Math. Monographs, AMS, Providence, 1995. |
[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. |
[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. |
[4] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathemathics and Scientifing Computing series, Oxford University Press, 2003. |
[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. |
[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. |
[7] |
D. Breda,
Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. |
[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. |
[27] |
D. Gottlieb,
The stability of pseudospectral-Chebyshev methods, Math. Comp., 36 (1981), 107-118.
doi: 10.1090/S0025-5718-1981-0595045-1. |
[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. |
[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. |
[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. |
[31] |
F. Kappel, Semigroups and Delay Equations, no. 152 (Trieste, 1984) in Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986. |
[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. |
[33] |
A. M. Lyapunov,
The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.
|
[34] |
M. C. Mackey and L. Glass,
Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[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. |
[36] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, no. 44 in Applied Mathematical Sciences, Springer Verlag, New York, 1983. |
[37] |
A. Prasad,
Amplitude Death in coupled chaotic oscillators, Phys. Rev. E, 72 (2005), 056204-10pp.
|
[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. |
[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. |
[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. |
[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. |
[42] |
L. N. Trefethen, Spectral Methods in MATLAB, Software -Environment -Tools series, SIAM, Philadelphia, 2000. |
show all references
References:
[1] |
L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, no. 146 in Transl. Math. Monographs, AMS, Providence, 1995. |
[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. |
[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. |
[4] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathemathics and Scientifing Computing series, Oxford University Press, 2003. |
[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. |
[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. |
[7] |
D. Breda,
Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. |
[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. |
[27] |
D. Gottlieb,
The stability of pseudospectral-Chebyshev methods, Math. Comp., 36 (1981), 107-118.
doi: 10.1090/S0025-5718-1981-0595045-1. |
[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. |
[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. |
[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. |
[31] |
F. Kappel, Semigroups and Delay Equations, no. 152 (Trieste, 1984) in Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986. |
[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. |
[33] |
A. M. Lyapunov,
The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.
|
[34] |
M. C. Mackey and L. Glass,
Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[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. |
[36] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, no. 44 in Applied Mathematical Sciences, Springer Verlag, New York, 1983. |
[37] |
A. Prasad,
Amplitude Death in coupled chaotic oscillators, Phys. Rev. E, 72 (2005), 056204-10pp.
|
[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. |
[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. |
[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. |
[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. |
[42] |
L. N. Trefethen, Spectral Methods in MATLAB, Software -Environment -Tools series, SIAM, Philadelphia, 2000. |


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