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Attraction-based computation of hyperbolic Lagrangian coherent structures
Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof
1. | Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Poland |
References:
[1] |
G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation,, Arch. Rational Mech. An., 197 (2010), 1033.
doi: 10.1007/s00205-010-0309-7. |
[2] |
, CAPD - Computer assisted proofs in dynamics, a package for rigorous numerics,, Available from: , (). Google Scholar |
[3] |
L. Cesari, Functional analysis and Galerkin's method,, Mich. Math. Jour., 11 (1964), 385.
doi: 10.1307/mmj/1028999194. |
[4] |
S.-N. Chow and J. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982).
|
[5] |
F. Christiansen, P. Cvitanovic and V. Putkaradze, Spatiotemporal chaos in terms of unstable recurrent patterns,, Nonlinearity, 10 (1997), 55.
doi: 10.1088/0951-7715/10/1/004. |
[6] |
P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation,, Physica D, 67 (1993), 321.
doi: 10.1016/0167-2789(93)90168-Z. |
[7] |
E. J. Doedel, AUTO: a program for the bifurcation analysis of autonomous system,, Congr. Numer., 30 (1981), 265.
|
[8] |
E. J. Doedel and R. C. Paffenroth, The AUTO2000: command line user interface,, in Proceedings of the 9-th Python Conference, (2001), 233. Google Scholar |
[9] |
C. Foias, B. Nicolaenko, G. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension,, J. Math. Pures Appl., 67 (1988), 197.
|
[10] |
J. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation; A bridge between PDEs and dynamical systems,, Physica D, 18 (1986), 113.
doi: 10.1016/0167-2789(86)90166-1. |
[11] |
J. S. Il'yashenko, Global Analysis of the Phase Portrait for the Kuramoto-Sivashinsky equation,, J. Dyn. Diff. Eq., 4 (1992), 585.
doi: 10.1007/BF01048261. |
[12] |
M. Jolly, I. Kevrekidis and E. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations,, Physica D, 44 (1990), 38.
doi: 10.1016/0167-2789(90)90046-R. |
[13] |
M. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky Equation,, Adv. Differential Equations, 5 (2000), 31.
|
[14] |
M. Jolly, R. Rosa and R. Temam, Acurate computations on inertial manifolds,, SIAM J. Sci. Compt., 22 (2000), 2216.
doi: 10.1137/S1064827599351738. |
[15] |
I. Kevrekidis, B. Nicolaenko and C. Scovel, Back in saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation,, SIAM J. Appl. Math., 50 (1990), 760.
doi: 10.1137/0150045. |
[16] |
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theor. Phys., 55 (1976), 356.
doi: 10.1143/PTP.55.356. |
[17] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square,, Revista Matematica Complutense, 21 (2008), 351.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[18] |
R. E. Moore, Interval Analysis,, Prentice Hall, (1966).
|
[19] |
A. Neumeier, Interval Methods for Systems of Equations,, Cambrigde University Press, (1990).
|
[20] |
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics,, Texts in Applied Mathematics, (2000).
|
[21] |
G. I. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames - 1. Derivation of basic equations,, Acta Astron, 4 (1977), 1177.
doi: 10.1016/0094-5765(77)90096-0. |
[22] |
C. Scovel, I. Kevrekidis and B. Nicolaenko, Scaling laws and the prediction of bifurcations in systems modeling pattern formation,, Physics Letters A, 130 (1988), 73.
doi: 10.1016/0375-9601(88)90242-3. |
[23] |
P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation,, Foundations of Computational Mathematics, 1 (2001), 255.
doi: 10.1007/s002080010010. |
[24] |
P. Zgliczyński, Trapping regions and an ODE-type proof of existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane,, Univ. Iag. Acta Math., 41 (2003), 89.
|
[25] |
P. Zgliczyński, On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach,, J. Diff. Eq., 195 (2003), 271.
doi: 10.1016/j.jde.2003.07.009. |
[26] |
P. Zgliczyński, Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof,, SIAM Journal on Applied Dynamical Systems, 1 (2002), 215.
doi: 10.1137/S111111110240176X. |
[27] |
, the file containing numerical data from the bifurcation proofs,, Available from: , (). Google Scholar |
show all references
References:
[1] |
G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation,, Arch. Rational Mech. An., 197 (2010), 1033.
doi: 10.1007/s00205-010-0309-7. |
[2] |
, CAPD - Computer assisted proofs in dynamics, a package for rigorous numerics,, Available from: , (). Google Scholar |
[3] |
L. Cesari, Functional analysis and Galerkin's method,, Mich. Math. Jour., 11 (1964), 385.
doi: 10.1307/mmj/1028999194. |
[4] |
S.-N. Chow and J. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982).
|
[5] |
F. Christiansen, P. Cvitanovic and V. Putkaradze, Spatiotemporal chaos in terms of unstable recurrent patterns,, Nonlinearity, 10 (1997), 55.
doi: 10.1088/0951-7715/10/1/004. |
[6] |
P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation,, Physica D, 67 (1993), 321.
doi: 10.1016/0167-2789(93)90168-Z. |
[7] |
E. J. Doedel, AUTO: a program for the bifurcation analysis of autonomous system,, Congr. Numer., 30 (1981), 265.
|
[8] |
E. J. Doedel and R. C. Paffenroth, The AUTO2000: command line user interface,, in Proceedings of the 9-th Python Conference, (2001), 233. Google Scholar |
[9] |
C. Foias, B. Nicolaenko, G. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension,, J. Math. Pures Appl., 67 (1988), 197.
|
[10] |
J. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation; A bridge between PDEs and dynamical systems,, Physica D, 18 (1986), 113.
doi: 10.1016/0167-2789(86)90166-1. |
[11] |
J. S. Il'yashenko, Global Analysis of the Phase Portrait for the Kuramoto-Sivashinsky equation,, J. Dyn. Diff. Eq., 4 (1992), 585.
doi: 10.1007/BF01048261. |
[12] |
M. Jolly, I. Kevrekidis and E. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations,, Physica D, 44 (1990), 38.
doi: 10.1016/0167-2789(90)90046-R. |
[13] |
M. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky Equation,, Adv. Differential Equations, 5 (2000), 31.
|
[14] |
M. Jolly, R. Rosa and R. Temam, Acurate computations on inertial manifolds,, SIAM J. Sci. Compt., 22 (2000), 2216.
doi: 10.1137/S1064827599351738. |
[15] |
I. Kevrekidis, B. Nicolaenko and C. Scovel, Back in saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation,, SIAM J. Appl. Math., 50 (1990), 760.
doi: 10.1137/0150045. |
[16] |
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theor. Phys., 55 (1976), 356.
doi: 10.1143/PTP.55.356. |
[17] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square,, Revista Matematica Complutense, 21 (2008), 351.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[18] |
R. E. Moore, Interval Analysis,, Prentice Hall, (1966).
|
[19] |
A. Neumeier, Interval Methods for Systems of Equations,, Cambrigde University Press, (1990).
|
[20] |
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics,, Texts in Applied Mathematics, (2000).
|
[21] |
G. I. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames - 1. Derivation of basic equations,, Acta Astron, 4 (1977), 1177.
doi: 10.1016/0094-5765(77)90096-0. |
[22] |
C. Scovel, I. Kevrekidis and B. Nicolaenko, Scaling laws and the prediction of bifurcations in systems modeling pattern formation,, Physics Letters A, 130 (1988), 73.
doi: 10.1016/0375-9601(88)90242-3. |
[23] |
P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation,, Foundations of Computational Mathematics, 1 (2001), 255.
doi: 10.1007/s002080010010. |
[24] |
P. Zgliczyński, Trapping regions and an ODE-type proof of existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane,, Univ. Iag. Acta Math., 41 (2003), 89.
|
[25] |
P. Zgliczyński, On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach,, J. Diff. Eq., 195 (2003), 271.
doi: 10.1016/j.jde.2003.07.009. |
[26] |
P. Zgliczyński, Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof,, SIAM Journal on Applied Dynamical Systems, 1 (2002), 215.
doi: 10.1137/S111111110240176X. |
[27] |
, the file containing numerical data from the bifurcation proofs,, Available from: , (). Google Scholar |
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