Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-29T11:05:39.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive noise effects in active nonlinear spatially extended systems

Published online by Cambridge University Press:  17 May 2012

M. PRADAS ,
G.A. PAVLIOTIS ,
S. KALLIADASIS ,
D.T. PAPAGEORGIOU  and
D. TSELUIKO
M. PRADAS
Affiliation:
Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, UK email: m.pradas-gene@imperial.ac.uk, s.kalliadasis@imperial.ac.uk
G.A. PAVLIOTIS
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK email: g.pavliotis@imperial.ac.uk, d.papageorgiou@imperial.ac.uk
S. KALLIADASIS
Affiliation:
Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, UK email: m.pradas-gene@imperial.ac.uk, s.kalliadasis@imperial.ac.uk
D.T. PAPAGEORGIOU
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK email: g.pavliotis@imperial.ac.uk, d.papageorgiou@imperial.ac.uk
D. TSELUIKO
Affiliation:
School of Mathematics, Loughborough University, Leicestershire, LE11 3TU, UK email: D.Tseluiko@lboro.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

Keywords

Stochastic partial differential equationsCritical phenomenaMultiple scale methodsSingular perturbations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 5 , October 2012 , pp. 563 - 591
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barabási, A.-L. & Stanley, H. E. (1995) Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[2]Bezrukov, S. M. & Vodyanoy, I. (1995) Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature (London) 378, 362–364.CrossRefGoogle ScholarPubMed
[3]Blömker, D. (2007) Amplitude Equations for Stochastic Partial Differential Equations, Interdisciplinary Mathematical Sciences, vol. 3, World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
[4]Blömker, D., Hairer, M. & Pavliotis, G. A. (2007) Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity 20, 1721–1744.CrossRefGoogle Scholar
[5]Blömker, D., Hairer, M. & Pavliotis, G. A. (2009) Some remarks on stabilization by additive noise, in Stochastic Partial Differential Equations and Applications. In: Da Prato, G. and Tubaro, L. (editors), Vol 25, Quaderni di Matematica, pp. 37–50.Google Scholar
[6]Blömker, D. & Mohammed, W. W. (2009) Amplitude equation for SPDEs with quadratic nonlinearities. Electron. J. Probab. 14 (88), 2527–2550.CrossRefGoogle Scholar
[7]Cross, M. C. & Hohenberg, P. C. (1993) Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
[8]Cuerno, R. & Barabási, A.-L. (1995) Dynamic scaling of ion-sputtered surfaces. Phys. Rev. Lett. 74, 4746.CrossRefGoogle ScholarPubMed
[9]Cuerno, R., Makse, H. A., Tomassone, S., Harrington, S. T. & Stanley, H. E. (1995) Stochastic erosion for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology. Phys. Rev. Lett. 75, 4464–4467.CrossRefGoogle ScholarPubMed
[10]Duan, J. & Ervin, V. J. (2001) On the stochastic Kuramoto–Sivashinsky equation. Nonlinear Anal. 44, 205–216.CrossRefGoogle Scholar
[11]Duprat, C., Giorgiutti-Dauphiné, F., Tseluiko, D., Saprykin, S. & Kalliadasis, S. (2009) Liquid film coating a fiber as a model system for the formation of bound states in active dispersive-dissipative nonlinear media. Phys. Rev. Lett. 103, 234501.CrossRefGoogle Scholar
[12]Frost, F. & Rauschenbach, B. (2003) Nano-structuring of solid surfaces by ion beam erosion. Appl. Phys. A 77, 1–9.CrossRefGoogle Scholar
[13]García-Ojalvo, J., Hernández-Machado, A. & Sancho, J. M. (1993) Effects of external noise on the Swift–Hohenberg equation. Phys. Rev. Lett. 71, 1542–1545.CrossRefGoogle ScholarPubMed
[14]García-Ojalvo, J. & Sancho, J. M. (1999) Noise in Spatially Extended Systems, Springer-Verlag, New York.CrossRefGoogle Scholar
[15]Heagy, J. F., Platt, N. & Hammel, S. M. (1994) Characterization of on-off intermittency. Phys. Rev. E 49, 1140–1150.Google ScholarPubMed
[16]Horsthemke, W. & Lefever, R. (1984) Noise-Induced Transitions, Springer, Berlin, Germany.Google Scholar
[17]Hutt, A. (2008) Additive noise may change the stability of nonlinear systems. Europhys. Lett. 84, 34003.CrossRefGoogle Scholar
[18]Hutt, A., Longtin, A. & Schimansky-Geier, L. (2007) Additive global noise delays turing bifurcations. Phys. Rev. Lett. 98, 230601.CrossRefGoogle ScholarPubMed
[19]Hyman, J. M. & Nicolaenko, B. (1986) The Kuramoto–Sivashinsky equation: A bridge between PDEs and dynamical systems. Physica D 18, 113–126.Google Scholar
[20]Hyman, J. M., Nicolaenko, B. & Zaleski, S. (1986) Order and complexity in the Kuramoto–Sivashinsky model of weakly turbulent interfaces. Physica D 23, 265–292.Google Scholar
[21]John, T., Stannarius, R. & Behn, U. (1999) On-off intermittency in stochastically driven electrohydrodynamic convection in nematics. Phys. Rev. Lett. 83, 749–752.CrossRefGoogle Scholar
[22]Jolly, M. S., Kevrekidis, I. G. & Titi, E. S. (1990) Approximate inertial manifolds for the Kuramoto–Sivashinsky equation: Analysis and computations. Physica D 44, 38–60.Google Scholar
[23]Kalliadasis, S. & Thiele, U. (editors) (2007) Thin Films of Soft Matter, Springer-Wien, New York.CrossRefGoogle Scholar
[24]Karma, A. & Misbah, C. (1993) Competition between noise and determinism in step flow growth. Phys. Rev. Lett. 71, 3810.CrossRefGoogle ScholarPubMed
[25]Kuramoto, Y. & Tsuzuki, T. (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356–369.CrossRefGoogle Scholar
[26]Landa, P. S., Zaikin, A. A. & Schimansky-Geier, L. (1998) Influence of additive noise on noise-induced phase transitions in nonlinear chains. Chaos Solitons Fractals 9, 1367–1372.Google Scholar
[27]Lauritsen, K. B., Cuerno, R. & Makse, H. A. (1996) Noisy Kuramoto–Sivashinsky equation for an erosion model. Phys. Rev. E 54, 3577–3580.Google ScholarPubMed
[28]López, J. M., Pradas, M. & Hernández-Machado, A. (2010) Activity statistics, avalanche kinetics, and velocity correlations in surface growth. Phys. Rev. E 82, 031127.Google ScholarPubMed
[29]Lou, Y. & Christofides, P. D. (2005) Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto–Sivashinsky equation. Comput. Chem. Eng. 29, 741–759.CrossRefGoogle Scholar
[30]Lou, Y., Hu, G. & Christofides, P. D. (2008) Model predictive control of nonlinear stochastic partial differential equations with application to a sputtering process. AIChE J. 54, 2065–2081.Google Scholar
[31]Mackey, M. C., Longtin, A. & Lasota, A. (1990) Noise-induced global asymptotic stability. J. Stat. Phys. 60, 735–751.CrossRefGoogle Scholar
[32]Majda, A. J., Timofeyev, I. & Eijnden, E. Vanden (2001) A mathematical framework for stochastic climate models. Comm. Pure Appl. Math. 54 (8), 891–974.CrossRefGoogle Scholar
[33]Mecke, K. & Rauscher, M. (2005) On thermal fluctuations in thin film flow. J. Phys.: Condens. Matter 17, S3515–S3522.Google Scholar
[34]Obeid, D., Kosterlitz, J. M. & Sandstede, B. (2010) State selection in the noisy stabilized Kuramoto-Sivashinsky equation. Phys. Rev. E 81, 066205.Google ScholarPubMed
[35]Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. (1990) Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2, 340–352.CrossRefGoogle Scholar
[36]Papageorgiou, D. T. & Smyrlis, Y.-S. (1991) The route to chaos for the Kuramoto–Sivashinsky equation. Theoret. Comput. Fluid Dyn. 3, 15–42.CrossRefGoogle Scholar
[37]Papanicolaou, G. C. (1976) Some probabilistic problems and methods in singular perturbations. Rocky Mt. J. Math. 6 (4), 653–674.CrossRefGoogle Scholar
[38]Pavliotis, G. A. & Stuart, A. M. (2008) Multiscale Methods: Averaging and Homogenization, Springer, New York.Google Scholar
[39]Platt, N., Spiegel, E. A. & Tresser, C. (1993) On-off intermittency: A mechanism for bursting. Phys. Rev. Lett. 70, 279–282.CrossRefGoogle ScholarPubMed
[40]Pradas, M., López, J. M. & Hernández-Machado, A. (2009) Avalanche dynamics in fluid imbibition near the depinning transition. Phys. Rev. E 80, 050101(R).Google ScholarPubMed
[41]Pradas, M., Tseluiko, D. & Kalliadasis, S. (2011) Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23, 044104.CrossRefGoogle Scholar
[42]Pradas, M., Tseluiko, D., Kalliadasis, S., Papageorgiou, D. T. & Pavliotis, G. A. (2011) Noise induced state transitions, intermittency, and universality in the noisy Kuramoto–Sivashinsky equation. Phys. Rev. Lett. 106, 060602.CrossRefGoogle Scholar
[43]Redner, S. (2001) A Guide to First-Passage Processes, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[44]Roberts, A. J. (2003) A step towards holistic discretisation of stochastic partial differential equations. ANZIAM J. 45 (C), C1–C15.CrossRefGoogle Scholar
[45]Sagués, F., Sancho, J. M. & García-Ojalvo, J. (2007) Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829–882.CrossRefGoogle Scholar
[46]Saprykin, S., Demekhin, E. A. & Kalliadasis, S. (2005) Two-dimensional wave dynamics in thin films. l. Stationary solitary pulses. Phys. Fluids 17, 117105.CrossRefGoogle Scholar
[47]Shew, W. L., Yang, H., Petermann, T., Roy, R. & Plenz, D. (2009) Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. J. Neurosci. 29, 15595.CrossRefGoogle ScholarPubMed
[48]Sivashinsky, G. I. (1977) Nonlinear analysis of hydrodynamic instability in laminar flames: I. derivation of basic equations. Acta Astraunaut. 4, 1176–1206.Google Scholar
[49]Smyrlis, Y.-S. & Papageorgiou, D. T. (1991) Predicting chaos for infinite-dimensional dynamical systems: The Kuramoto–Sivashinsky equation, a case study. Proc. Nat. Acad. Sci. (PNAS) 88, 11129–11132.CrossRefGoogle ScholarPubMed
[50]Tseluiko, D., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F. & Kalliadasis, S. (2010) Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory. Physica D 239, 2000–2010.Google Scholar
[51]Wan, X., Zhou, X. & E, W. (2010) Study of the noise-induced transition and the exploration of the phase space for the Kuramoto–Sivashinsky equation using the minimum action method. Nonlinearity 23, 475–493.CrossRefGoogle Scholar
[52]Wiesenfeld, K. & Moss, F. (1995) Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs. Nature (London) 373, 33–36.CrossRefGoogle ScholarPubMed
[53]Wittenberg, R. W. (2002) Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto–Sivashinsky equation. Phys. Lett. A 300, 407–416.CrossRefGoogle Scholar
[54]Wittenberg, R. W. & Holmes, P. (1999) Scale and space localization in the Kuramoto–Sivashinsky equation. Chaos 9, 452–464.CrossRefGoogle ScholarPubMed
[55]Wylock, C., Pradas, M., Haut, B., Colinet, P. & Kalliadasis, S. (2012) Disorder-induced hysteresis and nonlocality of contact line motion in chemically heterogeneous microchannels. Phys. Fluids 24, 032108.CrossRefGoogle Scholar
[56]Zaikin, A. A., García-Ojalvo, J. & Schimansky-Geier, L. (1999) Nonequilibrium first-order phase transition induced by additive noise. Phys. Rev. E 9, R6275–R6278.Google Scholar