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Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain
Preservation of homoclinic orbits under discretization of delay differential equations
1. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China |
2. | School of Mathematics, Jilin University, Changchun 130012, China |
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. |
show all references
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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