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Rigorous enclosures of rotation numbers by interval methods
On the computation of attractors for delay differential equations
1. | Institute for Mathematics, University of Paderborn, D-33095 Paderborn |
2. | Department of Mathematics, Paderborn University, 33095 Paderborn, Germany, Germany |
References:
[1] |
A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.
doi: 10.1016/0167-2789(85)90093-4. |
[2] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).
|
[3] |
C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.
doi: 10.1016/S0022-0396(02)00148-1. |
[4] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).
doi: 10.1007/978-1-4612-3506-4. |
[5] |
J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.
doi: 10.1016/0167-2789(84)90275-6. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[7] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.
doi: 10.1007/s002110050240. |
[8] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[9] |
M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).
doi: 10.1103/PhysRevLett.94.231102. |
[10] |
R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.
doi: 10.1137/1010058. |
[11] |
J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.
doi: 10.2140/pjm.1951.1.353. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957). Google Scholar |
[13] |
J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.
doi: 10.1016/0167-2789(82)90042-2. |
[14] |
C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[15] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.
doi: 10.1137/S106482750238911X. |
[16] |
G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69. Google Scholar |
[17] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.
doi: 10.1016/j.physd.2010.03.009. |
[18] |
C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.
doi: 10.1137/040608295. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[20] |
B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.
doi: 10.1088/0951-7715/12/5/303. |
[21] |
B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.
doi: 10.1063/1.166450. |
[22] |
I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.
doi: 10.1016/j.physd.2004.04.004. |
[23] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.
doi: 10.1126/science.267326. |
[24] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[25] |
J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.
doi: 10.1088/0951-7715/18/5/013. |
[26] |
T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.
doi: 10.1137/080718772. |
[27] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.
doi: 10.1007/BF01053745. |
[28] |
C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.
|
[29] |
J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.
doi: 10.1007/s003329900072. |
[30] |
F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.
|
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.
doi: 10.1007/978-1-4612-0645-3. |
[32] |
C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.
doi: 10.1007/s10915-008-9256-y. |
[33] |
S. Willard, General Topology,, Addison-Wesley, (1970).
|
show all references
References:
[1] |
A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.
doi: 10.1016/0167-2789(85)90093-4. |
[2] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).
|
[3] |
C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.
doi: 10.1016/S0022-0396(02)00148-1. |
[4] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).
doi: 10.1007/978-1-4612-3506-4. |
[5] |
J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.
doi: 10.1016/0167-2789(84)90275-6. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[7] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.
doi: 10.1007/s002110050240. |
[8] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[9] |
M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).
doi: 10.1103/PhysRevLett.94.231102. |
[10] |
R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.
doi: 10.1137/1010058. |
[11] |
J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.
doi: 10.2140/pjm.1951.1.353. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957). Google Scholar |
[13] |
J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.
doi: 10.1016/0167-2789(82)90042-2. |
[14] |
C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[15] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.
doi: 10.1137/S106482750238911X. |
[16] |
G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69. Google Scholar |
[17] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.
doi: 10.1016/j.physd.2010.03.009. |
[18] |
C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.
doi: 10.1137/040608295. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[20] |
B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.
doi: 10.1088/0951-7715/12/5/303. |
[21] |
B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.
doi: 10.1063/1.166450. |
[22] |
I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.
doi: 10.1016/j.physd.2004.04.004. |
[23] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.
doi: 10.1126/science.267326. |
[24] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[25] |
J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.
doi: 10.1088/0951-7715/18/5/013. |
[26] |
T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.
doi: 10.1137/080718772. |
[27] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.
doi: 10.1007/BF01053745. |
[28] |
C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.
|
[29] |
J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.
doi: 10.1007/s003329900072. |
[30] |
F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.
|
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.
doi: 10.1007/978-1-4612-0645-3. |
[32] |
C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.
doi: 10.1007/s10915-008-9256-y. |
[33] |
S. Willard, General Topology,, Addison-Wesley, (1970).
|
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