Normal form transforms separate slow and fast modes in stochastic dynamical systems

Physica A: Statistical Mechanics and its Applications

Volumn 387, Issue 1, 2008, Pages 12-38

  Roberts, A J ^a  

^a UNIVERSITY OF SOUTHERN QUEENSLAND   (Australia)

Author keywords

Multiscale modelling; Stochastic dynamical systems

Indexed keywords

MATHEMATICAL MODELS; NONLINEAR PROGRAMMING; RANDOM PROCESSES;

FAST TIME PROCESSES; MACROSCOPIC MODELING; MULTISCALE MODELING; NORMAL FORM TRANSFORMS; STOCHASTIC DYNAMICAL SYSTEMS;

DYNAMICAL SYSTEMS;

EID: 35748935082     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2007.08.023     Document Type: Article

References (39)

1. C. Elphick, E. Tirapeugi, M.E. Brachet, P. Coullet, G. Iooss, A simple global characterisation for normal forms of singular vector fields, Physica D 29 (1987) 95-127, doi:10.1016/0167-2789(87)90049-2.
2. S.M. Cox, A.J. Roberts, Initial conditions for models of dynamical systems, Physica D 85 (1995) 126-141, doi:10.1016/0167-2789(94)00201-Z.
3. F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescales, Texts in Applied Mathematics, vol. 50, Springer, Berlin, 2005.
4. G.A. Pavliotis, A.M. Stuart, Multiscale methods: averaging and homogenization. Technical Report 〈http://www.ma.ic.ac.uk/∼pavl〉, 2007.
5. D. Givon, I.G. Kevrekidis, R. Kupferman, Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Technical Report, Department of Mathematics, LBNL, Berkeley, CA 94720, 2006.
6. Murdock J. Normal Forms and Unfoldings for Local Dynamical Systems, Springer Monographs in Mathematics (2003), Springer, Berlin
7. Arnold L. Random Dynamical Systems, Springer Monographs in Mathematics (2003), Springer, Berlin
8. L. Arnold, P. Imkeller, Normal forms for stochastic differential equations, Probab. Theory Relat. Fields 110 (1998) 559-588, doi:10.1007/s004400050159.
9. X. Chao, A.J. Roberts, On the low-dimensional modelling of Stratonovich stochastic differential equations, Physica A 225 (1996) 62-80, doi:10.1016/0378-4371(95)00387-8.
10. Roberts A.J. Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. LMS J. Comput. Math. 9 (2006) 193-221. http://www.lms.ac.uk/jcm/9/lms2005-032 〈http://www.lms.ac.uk/jcm/9/lms2005-032〉.
11. A. Du, J. Duan, Invariant manifold reduction for stochastic dynamical systems, Technical Report 〈http://arXiv.org/abs/math.DS/0607366〉, 2006.
12. Temam R. Inertial manifolds. Math. Intelligencer 12 (1990) 68-74
13. Bensoussan A., and Flandoli F. Stochastic inertial manifold. Stochastics Stochastics Rep. 53 (1995) 13-39
14. Coullet P.H., Elphick C., and Tirapegui E. Normal form of a Hopf bifurcation with noise. Phys. Lett. 111A 6 (1985) 277-282
15. A.J. Roberts, Normal form of stochastic or deterministic multiscale differential equations, Technical Report 〈http://www.sci.usq.edu.au/staff/robertsa/sdenf.html〉, 2007.
16. Sri Namachchivaya N., and Leng G. Equivalence of stochastic averaging and stochastic normal forms. J. Appl. Mech. 57 (1990) 1011-1017
17. Sri Namachchivaya N., and Lin Y.K. Method of stochastic normal forms. Int. J. Nonlinear Mech. 26 (1991) 931-943
18. D.W. Pierce, Distinguishing coupled ocean-atmosphere interactions from background noise in the North Pacific, Technical Report 〈http://meteora.ucsd.edu/∼pierce/docs/Pierce_2001_Prog_Oceanog.pdf〉, 2001.
19. W. Just, H. Kantz, C. Rodenbeck, M. Helm, Stochastic modelling: replacing fast degrees of freedom by noise, J. Phys. A 34 (2001) 3199-3213, doi:10.1088/0305-4470/34/15/302.
20. Roberts A.J. Low-dimensional modelling of dynamics via computer algebra. Comput. Phys. Commun. 100 (1997) 215-230
21. Boxler P. A stochastic version of the centre manifold theorem. Probab. Theory Relat. Fields 83 (1989) 509-545
22. N. Berglund, B. Gentz, Geometric singular perturbation theory for stochastic differential equations, J. Differential Equations 191 (2003) 1-54, doi:10.1016/S0022-0396(03)00020-2.
23. J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Science, vol. 35, Springer, Berlin, 1981.
24. Robinson J.C. The asymptotic completeness of inertial manifolds. Nonlinearity 9 (1996) 1325-1340. http://www.iop.org/EJ/abstract/0951-7715/9/5/013 〈http://www.iop.org/EJ/abstract/0951-7715/9/5/013〉
25. Arnold L., and Kedai X. Simultaneous normal form and center manifold reduction for random differential equations. In: Perello C., Simo C., and Sola-Morales J. (Eds). Equadiff-91 (1993) 68-80
26. A. Papavasiliou, I.G. Kevrekidis, Variance reduction for the equation-free simulation of multiscale stochastic systems, Technical Report, Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK, June 2006.
27. Kevrekidis I.G., Gear C.W., Hyman J.M., Kevrekidis P.G., Runborg O., and Theodoropoulos K. Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system level tasks. Commun. Math. Sci. 1 (2003) 715-762
28. C.W. Gear, J. Li, I.G. Kevrekidis, The gap-tooth method in particle simulations, Phys. Lett. A 316 (2003) 190-195, doi:10.1016/j.physleta.2003.07.004.
29. Roberts A.J., and Kevrekidis I.G. General tooth boundary conditions for equation free modelling. SIAM J. Sci. Comput. 29 4 (2007) 1495-1510
30. Arnold L., Sri Namachchivaya N., and Schenk-Hoppé K.R. Toward an understanding of stochastic Hopf bifurcation: a case study. Int. J. Bifurcation Chaos 6 (1996) 1947-1975
31. Keller H., and Ochs G. Numerical approximation of random attractors. In: Crauel H., and Gundlach M. (Eds). Stochastic Dynamics (1999), Springer, Berlin. http://www.math.uni-bremen.de/Math-Net/preprints/ohne-metadaten/Report431.ps.gz 〈http://www.math.uni-bremen.de/Math-Net/preprints/ohne-metadaten/Report431.ps.gz〉
32. M. Dellnitz, A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math. 75 (1997) 293-317, doi:10.1007/s002110050240.
33. A.J. Roberts, Create useful low-dimensional models of complex dynamical systems, Technical Report 〈http://www.sci.usq.edu.au/staff/robertsa/Modelling/〉, 2006.
34. Olarrea J., and Javier de la Rubia F. Stochastic Hopf bifurcation: the effect of colored noise on the bifurcation interval. Phys. Rev. E 53 (1996) 268-271
35. A.J. Roberts, Computer algebra derives normal forms of stochastic differential equations, Technical Report 〈http://eprints.usq.edu.au/archive/00001873〉, 2007.
36. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.
37. D.J. Higham, P.E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math. 101 (2005) 101-119, doi:10.1007/s00211-005-0611-8.
38. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, vol. 23, Springer, Berlin, 1992.
39. Khasminskii R.Z. A limit theorem for the solutions of differential equations with random right hand sides. Theor. Probab. Appl. 11 3 (1966) 390-405