Preservation of homoclinic orbits under discretization of delay differential equations

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March 2011, 31(1): 275-299. doi: 10.3934/dcds.2011.31.275

Preservation of homoclinic orbits under discretization of delay differential equations

Yingxiang Xu ^1, and Yongkui Zou ^2,

1.	School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.	School of Mathematics, Jilin University, Changchun 130012, China

Received February 2010 Revised November 2010 Published June 2011

Abstract
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In this paper, we propose a nondegenerate condition for a homoclinic orbit with respect to a parameter in delay differential equations. Based on this nondegeneracy we describe and investigate the regularity of the homoclinic orbit together with parameter. Then we show that a forward Euler method, when applied to a one-parameteric system of delay differential equations with a homoclinic orbit, also exhibits a closed loop of discrete homoclinic orbits. These discrete homoclinic orbits tend to the continuous one by the rate of $O(\varepsilon)$ as the step-size $\varepsilon$ goes to $0$. And the corresponding parameter varies periodically with respect to a phase parameter with period $\varepsilon$ while the orbit shifts its index after one revolution. We also show that at least two homoclinic tangencies occur on this loop. By numerical simulations, the theoretical results are illustrated, and the possibility of extending theoretical results to the implicit and higher order numerical schemes is discussed.

Keywords: homoclinic orbits, homoclinic tangencies., Delay differential equations, forward Euler method.

Mathematics Subject Classification: Primary: 37C29, 37N30; Secondary: 65P3.

Citation: Yingxiang Xu, Yongkui Zou. Preservation of homoclinic orbits under discretization of delay differential equations. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 275-299. doi: 10.3934/dcds.2011.31.275

References:

[1]	W.-J. Beyn, The effect of discretization on homoclinic orbits, in "Bifurcation: Analysis, Algorithms, Applications," Internat. Ser. Numer. Math., 79, Birkhäuser, Basel, (1987), 1-8.
[2]	W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: 10.1093/imanum/10.3.379.
[3]	W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal., 34 (1997), 1207-1236. doi: 10.1137/S0036142995281693.
[4]	W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin-New York, 1978.
[5]	K. Engelborghs and E. J. Doedel, Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations, Numer. Math., 91 (2002), 627-648. doi: 10.1007/s002110100313.
[6]	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.
[7]	G. Farkas, Unstable manifolds for RFDEs under discretization: The Euler method, Comput. Math. Appl., 42 (2001), 1069-1081. doi: 10.1016/S0898-1221(01)00222-X.
[8]	G. Farkas, A numerical $C^1$-shadowing result for retarded functional differential equations, J. Compt. Appl. Math., 145 (2002), 269-289. doi: 10.1016/S0377-0427(01)00581-7.
[9]	G. Farkas, Nonexistence of uniform exponential dichotomies for delay equations, J. Differential Equations, 182 (2002), 266-268.
[10]	B. Fiedler and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and 'invisible' chaos, Mem. Amer. Math. Soc., 119 (1996).
[11]	J. K. Hale and S. M. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[12]	J. K. Hale and W. Zhang, On uniformity of exponential dichotomies for delay equations, J. Differential Equations, 204 (2004), 1-4.
[13]	K. In't Hout and C. Lubich, Periodic orbits of delay differential equations under discretization, BIT, 38 (1998), 72-91. doi: 10.1007/BF02510918.
[14]	U. Kirehgraber, F. Lasagni, K. Nipp and D. Stoffer, On the application of invariant manifold theory, in particular to numerical analysis, in "Bifurcation and Chaos: Analysis, Algorithms, Applications," Internat. Ser. Numer. Math. (eds. R. Seydel et al.), 97, Birkhäuser, Basel, (1991), 189-197.
[15]	X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations, 63 (1986), 227-254.
[16]	K. J. Palmer, "Shadowing in Dynamical Systems. Theory and Applications," Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000.
[17]	M. L. Peña, Exponential dichotomy for singularly perturbed linear functional-differential equations with small delays, Appl. Anal., 47 (1992), 213-225. doi: 10.1080/00036819208840141.
[18]	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.
[19]	Y.-K. Zou and W.-J. Beyn, On manifolds of connecting orbits in discretizations of dynamical systems, Nonlinear Anal., 52 (2003), 1499-1520. doi: 10.1016/S0362-546X(02)00269-9.
[20]	Y.-K. Zou and W.-J. Beyn, On the existence of transversal heteroclinic orbits in discretized dynamical systems, Nonlinearity, 17 (2004), 2275-2292. doi: 10.1088/0951-7715/17/6/014.

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References:

[1]	W.-J. Beyn, The effect of discretization on homoclinic orbits, in "Bifurcation: Analysis, Algorithms, Applications," Internat. Ser. Numer. Math., 79, Birkhäuser, Basel, (1987), 1-8.
[2]	W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: 10.1093/imanum/10.3.379.
[3]	W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal., 34 (1997), 1207-1236. doi: 10.1137/S0036142995281693.
[4]	W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, 629, Springer-Verlag, Berlin-New York, 1978.
[5]	K. Engelborghs and E. J. Doedel, Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations, Numer. Math., 91 (2002), 627-648. doi: 10.1007/s002110100313.
[6]	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.
[7]	G. Farkas, Unstable manifolds for RFDEs under discretization: The Euler method, Comput. Math. Appl., 42 (2001), 1069-1081. doi: 10.1016/S0898-1221(01)00222-X.
[8]	G. Farkas, A numerical $C^1$-shadowing result for retarded functional differential equations, J. Compt. Appl. Math., 145 (2002), 269-289. doi: 10.1016/S0377-0427(01)00581-7.
[9]	G. Farkas, Nonexistence of uniform exponential dichotomies for delay equations, J. Differential Equations, 182 (2002), 266-268.
[10]	B. Fiedler and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and 'invisible' chaos, Mem. Amer. Math. Soc., 119 (1996).
[11]	J. K. Hale and S. M. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[12]	J. K. Hale and W. Zhang, On uniformity of exponential dichotomies for delay equations, J. Differential Equations, 204 (2004), 1-4.
[13]	K. In't Hout and C. Lubich, Periodic orbits of delay differential equations under discretization, BIT, 38 (1998), 72-91. doi: 10.1007/BF02510918.
[14]	U. Kirehgraber, F. Lasagni, K. Nipp and D. Stoffer, On the application of invariant manifold theory, in particular to numerical analysis, in "Bifurcation and Chaos: Analysis, Algorithms, Applications," Internat. Ser. Numer. Math. (eds. R. Seydel et al.), 97, Birkhäuser, Basel, (1991), 189-197.
[15]	X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations, 63 (1986), 227-254.
[16]	K. J. Palmer, "Shadowing in Dynamical Systems. Theory and Applications," Mathematics and its Applications, 501, Kluwer Academic Publishers, Dordrecht, 2000.
[17]	M. L. Peña, Exponential dichotomy for singularly perturbed linear functional-differential equations with small delays, Appl. Anal., 47 (1992), 213-225. doi: 10.1080/00036819208840141.
[18]	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.
[19]	Y.-K. Zou and W.-J. Beyn, On manifolds of connecting orbits in discretizations of dynamical systems, Nonlinear Anal., 52 (2003), 1499-1520. doi: 10.1016/S0362-546X(02)00269-9.
[20]	Y.-K. Zou and W.-J. Beyn, On the existence of transversal heteroclinic orbits in discretized dynamical systems, Nonlinearity, 17 (2004), 2275-2292. doi: 10.1088/0951-7715/17/6/014.

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