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

Article

*Physica A: Statistical Mechanics and its Applications.*387 (1), pp. 12-38. https://doi.org/10.1016/j.physa.2007.08.023

Article Title | Normal form transforms separate slow and fast modes in stochastic dynamical systems |
---|---|

ERA Journal ID | 351 |

Article Category | Article |

Authors | |

Author | Roberts, A. J. |

Journal Title | Physica A: Statistical Mechanics and its Applications |

Journal Citation | 387 (1), pp. 12-38 |

Number of Pages | 27 |

Year | 2008 |

Place of Publication | Netherlands |

ISSN | 0378-4371 |

1873-2119 | |

Digital Object Identifier (DOI) | https://doi.org/10.1016/j.physa.2007.08.023 |

Web Address (URL) | http://www.sciencedirect.com/science/article/pii/S0378437107008655 |

Abstract | Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate \emph{all} slow processes from \emph{all} fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. Some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system. The results will help us accurately model, interpret and simulate multiscale stochastic systems. |

Keywords | stochastic dynamical systems; multiscale modelling |

ANZSRC Field of Research 2020 | 490510. Stochastic analysis and modelling |

499999. Other mathematical sciences not elsewhere classified | |

490409. Ordinary differential equations, difference equations and dynamical systems | |

Public Notes | File reproduced in accordance with the copyright policy of the publisher/author. |

Byline Affiliations | Computational Engineering and Science Research Centre |

https://research.usq.edu.au/item/9y7v5/normal-form-transforms-separate-slow-and-fast-modes-in-stochastic-dynamical-systems

## Download files

##### 2144

total views##### 380

total downloads##### 1

views this month##### 2

downloads this month

## Export as

## Related outputs

##### Co-ordinate transforms underpin multiscale modelling and reduction in deterministic and stochastic systems

Roberts, A. J.. 2008. "Co-ordinate transforms underpin multiscale modelling and reduction in deterministic and stochastic systems." Abbott, Derek, Aste, Tomaso, Batchelor, Murray and Dewar, Robert (ed.)*Complex Systems II: SPIE Symposium on Microelectronics, MEMS, and Nanotechnology 2007.*Canberra, Australia 05 - 07 Dec 2007 United States. SPIE. https://doi.org/10.1117/12.767596

##### A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions

Roberts, A. J., MacKenzie, T. and Bunder, J. E.. 2014. "A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions."*Journal of Engineering Mathematics.*86 (1), pp. 175-207. https://doi.org/10.1007/s10665-013-9653-6

##### Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions

MacKenzie, Tony and Roberts, A. J.. 2014.*Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions.*Adelaide, Australia. University of Adelaide.

##### Accurate macroscale modelling of spatial dynamics in multiple dimensions

Roberts, A. J., MacKenzie, Tony and Bunder, J. E.. 2012.*Accurate macroscale modelling of spatial dynamics in multiple dimensions.*Adelaide, Australia. University of Adelaide.

##### The style files: use the most informative synonym

Roberts, Tony. 2007. "The style files: use the most informative synonym."*Gazette of the Australian Mathematical Society.*34 (4), pp. 208-209.

##### Modelling the dynamics of turbulent floods

Mei, Z., Roberts, A. J. and Li, Zhenquan. 2002. "Modelling the dynamics of turbulent floods."*SIAM Journal on Applied Mathematics.*63 (2), pp. 423-458. https://doi.org/10.1137/S0036139999358866

##### Reproductive pair correlations and the clustering of organisms

Young, W. R., Roberts, A. J. and Stuhne, Gordan. 2001. "Reproductive pair correlations and the clustering of organisms."*Nature.*412 (6844), pp. 328-331. https://doi.org/10.1038/35085561

##### The style files: write what you mean

Roberts, Tony. 2007. "The style files: write what you mean."*Gazette of the Australian Mathematical Society.*34 (3), pp. 156-157.

##### General tooth boundary conditions for equation free modeling

Roberts, A. J. and Kevrekidis, I. G.. 2007. "General tooth boundary conditions for equation free modeling."*SIAM Journal on Scientific Computing.*29 (4), pp. 1495-1510. https://doi.org/10.1137/060654554

##### The style files: appearance affects communication; but not necessarily as you like

Roberts, Tony. 2007. "The style files: appearance affects communication; but not necessarily as you like."*Gazette of the Australian Mathematical Society.*34 (2), pp. 78-80.

##### Two-zone model of shear dispersion in a channel using centre manifolds

Roberts, A. J. and Strunin, Dmitry V.. 2004. "Two-zone model of shear dispersion in a channel using centre manifolds."*The Quarterly Journal of Mechanics and Applied Mathematics.*57 (3), pp. 363-378. https://doi.org/10.1093/qjmam/57.3.363

##### The style files: omit redundant words

Roberts, Tony. 2007. "The style files: omit redundant words."*Gazette of the Australian Mathematical Society.*34 (1), pp. 20-21.

##### Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation

MacKenzie, T. and Roberts, A. J.. 2006. "Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretisation."*SIAM Journal on Applied Dynamical Systems.*5 (3), pp. 365-402. https://doi.org/10.1137/050627733

##### Low-dimensional modelling of a generalised Burgers equation

Li, Zhenquan and Roberts, A. J.. 2007. "Low-dimensional modelling of a generalised Burgers equation."*Global Journal of Pure and Applied Mathematics.*3 (3), pp. 203-218.

##### A flexible error estimate for the application of centre manifold theory

Li, Zhenquan and Roberts, A. J.. 2007. "A flexible error estimate for the application of centre manifold theory."*Global Journal of Pure and Applied Mathematics.*3 (3), pp. 241-249.

##### Fractal landscape method: an alternative approach to measuring area-restricted searching behavior

Tremblay, Yann, Roberts, Antony J. and Costa, Daniel P.. 2007. "Fractal landscape method: an alternative approach to measuring area-restricted searching behavior."*The Journal of Experimental Biology.*210 (6), pp. 935-945. https://doi.org/10.1242/jeb.02710

##### A normal form of thin fluid film equations solves the transient paradox

Roberts, A. J.. 2006. "A normal form of thin fluid film equations solves the transient paradox."*Physica D: Nonlinear Phenomena.*223 (1), pp. 69-81. https://doi.org/10.1016/j.physd.2006.08.018

##### Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics

Roberts, A. J.. 2003. "Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics." Ball, Rowena and Akhmediev, Nial (ed.)*Nonlinear dynamics from lasers to butterflies: selected lectures from the 15th Canberra International Physics Summer School*. World Scientific Publishing. pp. 257-313

##### Modelling nonlinear dynamics of shape-memory-alloys with approximate models of coupled thermoelasticity

Melnik, Roderick V. N. and Roberts, A. J.. 2003. "Modelling nonlinear dynamics of shape-memory-alloys with approximate models of coupled thermoelasticity."*Journal of Applied Mathematics and Mechanics (ZAMM).*83 (2), pp. 93-104. https://doi.org/10.1002/zamm.200310009

##### Modelling turbulent flow from dam break using slow manifolds

Georgiev, D. J., Roberts, A. J. and Strunin, D. V.. 2009. "Modelling turbulent flow from dam break using slow manifolds ."*Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal.*50 (S), pp. C1033-C1051.

##### Low Prandtl number fluid convection modelled using symbolic algebra (REDUCE) and Matlab

Passmore, Tim and Roberts, A. J.. 2003. "Low Prandtl number fluid convection modelled using symbolic algebra (REDUCE) and Matlab."*ANZIAM 2003: Computational Techniques and Applications.*Toowoomba, Australia 16 - 18 Jul 2003 Cambridge, United Kingdom .

##### Low-dimensional boundary-layer model of turbulent dispersion in a channel

Strunin, Dmitry V. and Roberts, Anthony J.. 2009. "Low-dimensional boundary-layer model of turbulent dispersion in a channel." Ao, S. I., Gelman, Len, Hukins, David W. L., Hunter, Andrew and Korsunsky, A. M. (ed.)*WCE 2009: World Congress of Engineering .*London, United Kingdom 01 - 03 Jul 2009 London, UK.

##### A mathematical model to account for quantitative iterative effect of TNF-α positive feedback mechanism on inflammatory cascade in TLR-4 mediated TNF-α signaling pathway

Roberts, Tony, Essien, Bryan, Kotiw, Michael, Butler, Harry and Strunin, Dmitry. 2008. "A mathematical model to account for quantitative iterative effect of TNF-α positive feedback mechanism on inflammatory cascade in TLR-4 mediated TNF-α signaling pathway." Ragan, Mark (ed.)*19th International Conference on Genome Informatics (GIW 2008).*Gold Coast, Australia 01 - 03 Dec 2008 London, United Kingdom.

##### Advection-dispersion in symmetric field-flow fractionation channels

Suslov, Sergey A. and Roberts, A. J.. 1998.*Advection-dispersion in symmetric field-flow fractionation channels.*Toowoomba, Australia. University of Southern Queensland.

##### The style files: teach explicit skills with feedback

Roberts, Tony. 2008. "The style files: teach explicit skills with feedback."*Gazette of the Australian Mathematical Society.*35 (3), pp. 156-157.

##### The style files: write to read breadth first, not depth first

Roberts, Tony. 2008. "The style files: write to read breadth first, not depth first."*Gazette of the Australian Mathematical Society.*35 (1), pp. 17-19.

##### Computer algebra derives discretisations via self-adjoint multiscale modelling

Roberts, A. J.. 2008. "Computer algebra derives discretisations via self-adjoint multiscale modelling."*Unpublihsed.*

##### Computer algebra describes flow of turbulent floods via the Smagorinski large eddy closure

Roberts, A. J.. 2008.*Computer algebra describes flow of turbulent floods via the Smagorinski large eddy closure.*Toowoomba, Australia. University of Southern Queensland.

##### The inertial dynamics of thin film flow of non-Newtonian fluids

Roberts, A. J.. 2008. "The inertial dynamics of thin film flow of non-Newtonian fluids."*Physics Letters A.*372 (10), pp. 1607-1611. https://doi.org/10.1016/j.physleta.2007.10.014

##### Computer algebra models dynamics on a multigrid across multiple length and time scales

Roberts, A. J.. 2007.*Computer algebra models dynamics on a multigrid across multiple length and time scales.*

##### Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model

Georgiev, Dian J., Roberts, A. J. and Strunin, Dmitry V.. 2007. "Nonlinear dynamics on centre manifolds describing turbulent floods: k-\omega model."*Discrete and Continuous Dynamical Systems Series A.*

##### The dynamics of the vertical structure of turbulence in flood flows

Georgiev, D. J., Roberts, A. J. and Strunin, D. V.. 2007. "The dynamics of the vertical structure of turbulence in flood flows."*Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal.*48, pp. C573-C590. https://doi.org/10.0000/anziamj.v48i0.124

##### Mathematical and numerical analysis of Falk-Konopka-type models for shape-memory alloys

Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2000. "Mathematical and numerical analysis of Falk-Konopka-type models for shape-memory alloys."*International Journal of Differential Equations and Applications.*1A (3), pp. 291-300.

##### Computing dynamics of copper-based SMA via center manifold reduction of 3D models

Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2000. "Computing dynamics of copper-based SMA via center manifold reduction of 3D models."*Computational Materials Science.*18 (3/4), pp. 255-268.

##### Branching behavior of standing waves - the signatures of resonance

Smith, D. H. and Roberts, A. J.. 1999. "Branching behavior of standing waves - the signatures of resonance."*Physics of Fluids.*11 (5), pp. 1051-1064. https://doi.org/10.1063/1.869976

##### Models encompassing hydraulic jumps in radial flows over a horizontal plate

Strunin, Dmitry and Roberts, Tony. 2001. "Models encompassing hydraulic jumps in radial flows over a horizontal plate." Kluev, Vitaly and Mastorakis, Nikos (ed.)*2nd WSEAS Multiconference on Applied and Theoretical Mathematics (WSEAS 2001).*Cairns, Australia 17 - 23 Dec 2001 Greece.

##### Dip coating process for hot metal castings

McGuinness, M. and Roberts, Anthony. 2000. "Dip coating process for hot metal castings." Hewitt, John (ed.)*16th Australian and New Zealand Mathematics In Industry Study Group (MISG 1999).*Brisbane, Australia 01 - 05 Feb 1999 Adelaide, Australia.

##### Coupled thermomechanical waves in hyperbolic thermoelasticity

Strunin, D. V., Melnik, R. V. N. and Roberts, A. J.. 2001. "Coupled thermomechanical waves in hyperbolic thermoelasticity."*Journal of Thermal Stresses.*24 (2), pp. 121-140. https://doi.org/10.1080/01495730150500433

##### A corrected quadrature formula and applications

Ujevic, Nenad and Roberts, A. J.. 2004. "A corrected quadrature formula and applications."*Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal.*45 (E), pp. E41-E56.

##### Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models

Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2002. "Phase transitions in shape memory alloys with hyperbolic heat conduction and differential-algebraic models."*Computational Mechanics.*29 (1), pp. 16-26. https://doi.org/10.1007/s00466-002-0311-5

##### Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena

Melnik, R. V. N., Roberts, A. J. and Thomas, K. A.. 2001. "Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena."*Mechanics Research Communications.*28 (6), pp. 637-651. https://doi.org/10.1016/S0093-6413(02)00216-1

##### A holistic finite difference approach models linear dynamics consistently

Roberts, A. J.. 2003. "A holistic finite difference approach models linear dynamics consistently."*Mathematics of Computation.*72 (241), pp. 247-262. https://doi.org/10.1090/S0025-5718-02-01448-5

##### Holistic discretisation ensures fidelity to Burger's equation

Roberts, A. J.. 2001. "Holistic discretisation ensures fidelity to Burger's equation."*Applied Numerical Mathematics.*37 (3), pp. 371-396. https://doi.org/10.1016/S0168-9274(00)00053-2

##### A lubrication model of coating flows over a curved substrate in space

Roy, R. Valery, Roberts, A. J. and Simpson, M. E.. 2002. "A lubrication model of coating flows over a curved substrate in space."*Journal of Fluid Mechanics.*454 (1), pp. 235-261. https://doi.org/10.1017/S0022112001007133

##### Computer algebra models the inertial dynamics of a thin film flow of power law fluids and other non-Newtonian fluids

Roberts, A. J.. 2007.*Computer algebra models the inertial dynamics of a thin film flow of power law fluids and other non-Newtonian fluids .*University of Southern Queensland.

##### Computer algebra derives normal forms of stochastic differential equations

Roberts, A. J.. 2007.*Computer algebra derives normal forms of stochastic differential equations .*Toowoomba, Australia. University of Southern Queensland.

##### Computer algebra derives discretisations of the stochastically forced Burgers' partial differential equation

Roberts, A. J.. 2006.*Computer algebra derives discretisations of the stochastically forced Burgers' partial differential equation.*Toowoomba, Australia. University of Southern Queensland.

##### The style files: favour the present tense

Roberts, Tony. 2006. "The style files: favour the present tense."*Gazette of the Australian Mathematical Society.*33 (5), pp. 313-314.

##### The style files: explicitly avoid false conditionals

Roberts, Tony. 2006. "The style files: explicitly avoid false conditionals."*Gazette of the Australian Mathematical Society.*33 (4), pp. 241-242.

##### The style files: inform with titles, abstracts and introductions

Roberts, Tony. 2006. "The style files: inform with titles, abstracts and introductions."*Gazette of the Australian Mathematical Society.*33 (3), pp. 169-170.

##### The style files: clarify this

Roberts, Tony. 2006. "The style files: clarify this."*Gazette of the Australian Mathematical Society.*33 (2), pp. 104-105.

##### The style files: prefer active writing to passive

Roberts, Tony. 2006. "The style files: prefer active writing to passive."*Gazette of the Australian Mathematical Society.*33 (1), pp. 22-23.

##### Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations

Roberts, A. J.. 2005.*Computer algebra resolves a multitude of microscale interactions to model stochastic partial differential equations.*Toowoomba, Australia. University of Southern Queensland.

##### Resolving the multitude of microscale interactions accurately models stochastic partial differential equations

Roberts, A. J.. 2006. "Resolving the multitude of microscale interactions accurately models stochastic partial differential equations."*LMS Journal of Computation and Mathematics.*9, pp. 193-221. https://doi.org/10.1112/S146115700000125X

##### An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Roberts, A. J. and Li, Zhenquan. 2006. "An accurate and comprehensive model of thin fluid flows with inertia on curved substrates."*Journal of Fluid Mechanics.*553 (1), pp. 33-73. https://doi.org/10.1017/S0022112006008640

##### Resolve the multitude of microscale interactions to holistically discretise the stochastically forced Burgers' partial differential equation

Roberts, A. J.. 2006.*Resolve the multitude of microscale interactions to holistically discretise the stochastically forced Burgers' partial differential equation.*Toowoomba, Australia. University of Southern Queensland.

##### General tooth boundary conditions for equation free modelling

Roberts, A. J. and Kevrekidis, I. G.. 2006.*General tooth boundary conditions for equation free modelling.*Toowoomba, Australia. University of Southern Queensland.

##### Nonlinear analysis of rubber-based polymeric materials with thermal relaxation models

Melnik, R. V. N., Strunin, D. V. and Roberts, A. J.. 2005. "Nonlinear analysis of rubber-based polymeric materials with thermal relaxation models."*Numerical Heat Transfer Part A: Applications.*47 (6), pp. 549-569. https://doi.org/10.1080/10407780590891236

##### Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators

Roberts, A. J. and Kevrekidis, I. G.. 2005. "Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators." May, Rob and Roberts, A, J. (ed.)*CTAC 2004: 12th Biennial Computational Techniques and Applications Conference.*Melbourne, Australia Sep 2004 Cambridge, United Kingdom .

##### Predicting the off-site deposition of spray drift from horticultural spraying through porous barriers on soil and plant surfaces

Mercer, Geoff and Roberts, Tony. 2005. "Predicting the off-site deposition of spray drift from horticultural spraying through porous barriers on soil and plant surfaces." Wake, Graeme (ed.)*MISG 2005: 22nd Mathematics-In-Industry Study Group.*Auckland, New Zealand 24 - 28 Jan 2005 Auckland, New Zealand.