This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree.
The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.
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Figure 1.
Parameterization of a 2D local unstable manifold attached to the origin for Wright's equation with
Figure 2.
Boundary torus of a parameterized local unstable manifold attached to a periodic orbit for Wright's equation with
Figure 3.
The attractor (yellow) and the unstable manifold (red) of the cubic Ikeda equation at
Figure 8. The attractor (yellow) and two-dimensional manifolds (blue) of the periodic orbits (magenta). The attractor is computed simply by integrating arbitrary initial conditions using a standard DDE integrator. The periodic orbit is computed using 140 Fourier modes. The unstable manifold is parameterized to Taylor order 51, where again each Taylor coefficient is computed with 140 Fourier modes
Figure 12.
The two-dimensional fast manifold (red) and the two-dimensional slow manifold (green) manifold at
Figure 17.
The unstable manifold of the state-dependent perturbation of Wright's equation, for several values of
[1] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013, First paperback reprint of the 2003 original [MR1997488].
![]() ![]() |
[2] |
M. Breden, J. Lessard and J. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indagationes Mathematicae, 27 (2016), 340-367.
doi: 10.1016/j.indag.2015.11.001.![]() ![]() ![]() |
[3] |
M. Breden, J.-P. Lessard and J. D. Mireles James, Computation of maximal local (un)stable manifold patches by the parameterization method, Indag. Math. (N.S.), 27 (2016), 340-367.
doi: 10.1016/j.indag.2015.11.001.![]() ![]() ![]() |
[4] |
X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds ⅰ: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245.![]() ![]() ![]() |
[5] |
X. Cabré, E. Fontich and R. de la Llave, The parametrization method for invariant manifolds ⅱ: Regularity with respect to parameters, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245.![]() ![]() ![]() |
[6] |
X. Cabré, E. Fontich and R. de la Llave, The parametrization method for invariant manifolds ⅲ: Overview and applications, Journal of Differential Equations, 218 (2005), 444-515.
![]() |
[7] |
R. C. Calleja, A. Celletti and R. de la Llave, A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, J. Differential Equations, 255 (2013), 978-1049.
doi: 10.1016/j.jde.2013.05.001.![]() ![]() ![]() |
[8] |
R. C. Calleja, T. Humphries and B. Krauskopf, Resonance phenomena in a scalar delay differential equation with two state-dependent delays Submitted, arXiv: 1607.02683.
doi: 10.1137/16M1087655.![]() ![]() ![]() |
[9] |
S. A. Campbell, Calculating center manifolds for delay differential equations using MapleTM, in Delay Differential Equations, Springer, New York, 2009,221-244.
![]() ![]() |
[10] |
M. J. Capiński, Computer assisted existence proofs of Lyapunov orbits at L2 and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst., 11 (2012), 1723-1753.
doi: 10.1137/110847366.![]() ![]() ![]() |
[11] |
R. Castelli, J.-P. Lessard and J. Mireles-James, Parametrization of invariant manifolds for periodic orbits (ⅰ): Efficient numerics via the Floquet normal form, SIAM Journal on Applied Dynamical Systems, 14 (2015), 132-167.
doi: 10.1137/140960207.![]() ![]() ![]() |
[12] |
R. Castelli, J. -P. Lessard and J. D. Mireles James, Parameterization of invariant manifolds for periodic orbits (ⅱ): a-posteriori analysis and computer assisted error bounds, To appear in Journal of Dynamics and Differential Equations.
![]() |
[13] |
G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520.
doi: 10.1090/S0002-9947-96-01501-2.![]() ![]() ![]() |
[14] |
R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895.
doi: 10.1088/0951-7715/18/2/020.![]() ![]() ![]() |
[15] |
R. de la Llave and J. D. Mireles James, Connecting orbits for compact infinite dimensional maps: Computer assisted proofs of existence, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1268-1323.
doi: 10.1137/15M1053608.![]() ![]() ![]() |
[16] |
J. De Luca, N. Guglielmi, T. Humphries and A. Politi, Electromagnetic two-body problem: Recurrent dynamics in the presence of state-dependent delay, J. Phys. A, 43 (2010), 205103, 20.
doi: 10.1088/1751-8113/43/20/205103.![]() ![]() ![]() |
[17] |
M. Dellnitz, M. Hessel-Von Molo and A. Ziessler, On the computation of attractors for delay differential equations Submitted.
doi: 10.3934/jcd.2016005.![]() ![]() ![]() |
[18] |
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21.
doi: 10.1145/513001.513002.![]() ![]() ![]() |
[19] |
G. Farkas, Unstable manifolds for RFDEs under discretization: the Euler method, Comput. Math. Appl., 42 (2001), 1069-1081, Numerical Methods and Computational Mechanics (Miskolc, 1998).
doi: 10.1016/S0898-1221(01)00222-X.![]() ![]() ![]() |
[20] |
G. Farkas, Discretizing hyperbolic periodic orbits of delay differential equations, ZAMM Z. Angew. Math. Mech., 83 (2003), 38-49.
doi: 10.1002/zamm.200310003.![]() ![]() ![]() |
[21] |
J. -L. Figueras, M. Gameiro, J. -P. Lessard and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations Submitted.
doi: 10.1137/16M1073777.![]() ![]() ![]() |
[22] |
H. Froehling, J. P. Crutchfield, D. Farmer, N. H. Packard and R. Shaw, On determining the dimension of chaotic flows, Phys. D, 3 (1981), 605-617.
doi: 10.1016/0167-2789(81)90043-9.![]() ![]() ![]() |
[23] |
R. H. Goodman and J. K. Wróbel, High-order bisection method for computing invariant manifolds of two-dimensional maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 2017-2042.
doi: 10.1142/S0218127411029604.![]() ![]() ![]() |
[24] |
K. Green, B. Krauskopf and K. Engelborghs, One-dimensional unstable eigenfunction and manifold computations in delay differential equations, J. Comput. Phys., 197 (2004), 86-98.
doi: 10.1016/j.jcp.2003.11.018.![]() ![]() ![]() |
[25] |
A. Guillamon and G. Huguet, A computational and geometric approach to phase resetting curves and surfaces, SIAM J. Appl. Dyn. Syst., 8 (2009), 1005-1042.
doi: 10.1137/080737666.![]() ![]() ![]() |
[26] |
J. Hale,
Theory of Functional Differential Equations, 2nd edition, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.
![]() ![]() |
[27] |
J. K. Hale and S. M. Verduyn Lunel,
Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7.![]() ![]() ![]() |
[28] |
A. Haro, Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points, IMA Note.
![]() |
[29] |
A. Haro, M. Canadell, J. -L. Figueras, A. Luque and J. -M. Mondelo,
The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Applied Mathematical Sciences, 195, Springer, 2016.
doi: 10.1007/978-3-319-29662-3.![]() ![]() ![]() |
[30] |
À. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300 (electronic),
doi: 10.3934/dcdsb.2006.6.1261.![]() ![]() ![]() |
[31] |
A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results, J. Differential Equations, 228(2006), 530-579,
doi: 10.1016/j.jde.2005.10.005.![]() ![]() ![]() |
[32] |
A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Explorations and mechanisms for the breakdown of hyperbolicity SIAM J. Appl. Dyn. Syst., 6(2007), 142-207 (electronic),
doi: 10.1137/050637327.![]() ![]() ![]() |
[33] |
F. Hartung, T. Krisztin, H. -O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations: Ordinary Differential Equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006,435-545.
doi: 10.1016/S1874-5725(06)80009-X.![]() ![]() ![]() |
[34] |
X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅰ: Finitely differentiable, hyperbolic case, hyperbolic case, Journal of Dynamics and Differential Equations (2016). Available from: https://www.ma.utexas.edu/mp_arc/c/15/15-105.pdf.
![]() |
[35] |
X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method Ⅱ: Analytic case, Journal of Differential Equations, 261 (2016), 2068-2108.
doi: 10.1016/j.jde.2016.04.024.![]() ![]() ![]() |
[36] |
T. Heil, I. Fischer, W. Elsäßer, B. Krauskopf, A. Gavrielides and K. Green, Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms, Phys. Rev. E (3), 67(2003), 066214, 11.
doi: 10.1103/PhysRevE.67.066214.![]() ![]() ![]() |
[37] |
G. Huguet and R. de la Llave, Computation of limit cycles and their isochrons: Fast algorithms and their convergence, SIAM J. Appl. Dyn. Syst., 12 (2013), 1763-1802.
doi: 10.1137/120901210.![]() ![]() ![]() |
[38] |
A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete Contin. Dyn. Syst., 32 (2012), 2701-2727.
doi: 10.3934/dcds.2012.32.2701.![]() ![]() ![]() |
[39] |
T. Johnson and W. Tucker, A note on the convergence of parametrised non-resonant invariant manifolds, Qual. Theory Dyn. Syst., 10 (2011), 107-121.
doi: 10.1007/s12346-011-0040-2.![]() ![]() ![]() |
[40] |
G. S. Jones, The existence of periodic solutions of $f^{\prime} (x)=-α f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl., 5 (1962), 435-450.
doi: 10.1016/0022-247X(62)90017-3.![]() ![]() ![]() |
[41] |
G. S. Jones, On the nonlinear differential-difference equation $f^{\prime} (x)=-α f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl., 4 (1962), 440-469.
doi: 10.1016/0022-247X(62)90041-0.![]() ![]() ![]() |
[42] |
À. Jorba and M. Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904.![]() ![]() ![]() |
[43] |
G. Kiss and J.-P. Lessard, Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115.
doi: 10.1016/j.jde.2011.11.020.![]() ![]() ![]() |
[44] |
G. Kiss and J. -P. Lessard, Rapidly and slowly oscillating periodic oscillations of a delayed van der pol oscillator, Submitted.
![]() |
[45] |
D. E. Knuth,
The Art of Computer Programming. Vol. 2 Seminumerical Algorithms, Third edition [of MR0286318], Addison-Wesley, Reading, MA, 1998.
![]() ![]() |
[46] |
B. Krauskopf, H. Osinga, E. Doedel, M. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bifurcation & Chaos, 15 (2005), 763-791.
doi: 10.1142/S0218127405012533.![]() ![]() ![]() |
[47] |
B. Krauskopf and K. Green, Computing unstable manifolds of periodic orbits in delay differential equations, J. Comput. Phys., 186 (2003), 230-249.
doi: 10.1016/S0021-9991(03)00050-0.![]() ![]() ![]() |
[48] |
J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations, 248 (2010), 992-1016.
doi: 10.1016/j.jde.2009.11.008.![]() ![]() ![]() |
[49] |
J. -P. Lessard, Delay Differential Equations and Continuation, Lecture notes for the AMS short course on rigorous numerics in dynamics.
![]() |
[50] |
J.-P. Lessard, J. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.
doi: 10.1007/s10884-014-9367-0.![]() ![]() ![]() |
[51] |
J. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for fourier series and the radii polynomial approach To appear in Physica D.
doi: 10.1016/j.physd.2016.02.007.![]() ![]() ![]() |
[52] |
X. Li and R. de la Llave, Construction of quasi-periodic solutions of delay differential equations via kam techniques, Journal of Differential Equations, 247 (2009), 822-865.
doi: 10.1016/j.jde.2009.03.009.![]() ![]() ![]() |
[53] |
J. Mireles-James and K. Mischaikow, Rigorous a-posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM Journal on Applied Dynamical Systems, 12 (2013), 957-1006.
doi: 10.1137/12088224X.![]() ![]() ![]() |
[54] |
J. D. Mireles James, Computer assisted error bounds for linear approximation of (un)stable manifolds and rigorous validation of higher dimensional transverse connecting orbits, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1102-1133.
doi: 10.1016/j.cnsns.2014.08.010.![]() ![]() |
[55] |
J. D. Mireles James, Fourier-taylor approximation of unstable manifolds for compact maps: Numerical implementation and computer assisted error bounds, Foundations of Computational Mathematics, (2016).
doi: 10.1007/s10208-016-9325-9.![]() ![]() |
[56] |
J. D. Mireles James and H. Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family, SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953.
doi: 10.1137/090776329.![]() ![]() ![]() |
[57] |
A. Neumaier and T. Rage, Rigorous chaos verification in discrete dynamical systems, Phys. D, 67 (1993), 327-346.
doi: 10.1016/0167-2789(93)90169-2.![]() ![]() ![]() |
[58] |
R. D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.
doi: 10.1307/mmj/1029001104.![]() ![]() ![]() |
[59] |
N. H. Packard, J. P. Crutchfield, D. Farmer and R. Shaw, Geometry from a time series, Physical Review Letters, 45 (1980), 712-716.
doi: 10.1007/BF02650178.![]() ![]() ![]() |
[60] |
C. Reinhardt and J. M. James, Fourier-taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation, and rigorous validation, Submitted, arXiv: 1601.00307.
![]() |
[61] |
C. Reinhardt, J. B. Van den Berg and J. D. Mireles James, Computing (un)stable manifolds with validated error bounds: Non-resonant and resonant spectra, Journal of Nonlinear Science, 26 (2016), 1055-1095.
doi: 10.1007/s00332-016-9298-5.![]() ![]() ![]() |
[62] |
D. Roose, T. Luzyanina, K. Engelborghs and W. Michiels, Software for stability and bifurcation analysis of delay differential equations and applications to stabilization, in Advances in Time-Delay Systems, Lect. Notes Comput. Sci. Eng., 38, Springer, Berlin, 2004,167-181.
doi: 10.1007/978-3-642-18482-6_12.![]() ![]() ![]() |
[63] |
T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 1116-1135.
doi: 10.1137/080718772.![]() ![]() ![]() |
[64] |
G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delay differential equations, Numer. Algorithms, 30 (2002), 335-352.
doi: 10.1023/A:1020102317544.![]() ![]() ![]() |
[65] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Statist. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745.![]() ![]() ![]() |
[66] |
J. Sieber and B. Krauskopf, Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity, Nonlinearity, 17 (2004), 85-103.
doi: 10.1088/0951-7715/17/1/006.![]() ![]() |
[67] |
J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007), 397-402.
doi: 10.1016/j.physleta.2007.01.083.![]() ![]() ![]() |
[68] |
R. Szczelina and P. Zgliczyński, Algorithm for rigorous integration of delay differential equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation, Submitted, http://ww2.ii.uj.edu.pl/~zgliczyn/papers/delay/main-dde.pdf.
![]() |
[69] |
F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., 898, Springer, Berlin-New York, 1981,366-381.
![]() ![]() |
[70] |
J. B. van den Berg and J. D. Mireles James, Parametrization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation, Discrete Contin. Dyn. Syst., 36 (2016), 4637-4664.
doi: 10.3934/dcds.2016002.![]() ![]() ![]() |
[71] |
H.-O. Walther, Topics in delay differential equations, Jahresber. Dtsch. Math.-Ver., 116 (2014), 87-114.
doi: 10.1365/s13291-014-0086-6.![]() ![]() ![]() |
[72] |
D. Wilczak and P. Zgliczyński, Heteroclinic connections between periodic orbits in planar restricted circular three body problem. Ⅱ, Comm. Math. Phys., 259 (2005), 561-576.
doi: 10.1007/s00220-005-1374-x.![]() ![]() ![]() |
[73] |
E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87.
doi: 10.1515/crll.1955.194.66.![]() ![]() ![]() |
[74] |
J. K. Wróbel and R. H. Goodman, High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1734-1745.
doi: 10.1016/j.cnsns.2012.10.017.![]() ![]() ![]() |