Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-15T19:55:45.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Scaling laws and warning signs for bifurcations of SPDEs

Part of: Miscellaneous topics - Partial differential equations Stochastic analysis

Published online by Cambridge University Press:  18 September 2018

CHRISTIAN KUEHN  and
FRANCESCO ROMANO
CHRISTIAN KUEHN
Affiliation:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email: ckuehn@ma.tum.de
FRANCESCO ROMANO
Affiliation:
Faculty of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85747 Garching b. Munich, Germany email: ckuehn@ma.tum.de Ludwig-Maximilians-Universität, Elite Graduate Course Theoretical and Mathematical Physics, Theresienstraße 37, 80333 Munich, Germany email: francesco1093@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

Keywords

Critical transitiontipping pointwarning signscaling lawstochastic partial differential equation

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 853 - 868
Copyright
© Cambridge University Press 2018 

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

Berglund, N. & Gentz, B. (2002) Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Relat. Fields 3, 341–388.CrossRefGoogle Scholar
Berglund, N. & Gentz, B. (2006) Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer London Ltd.Google Scholar
Blömker, D. (2007) Amplitude Equations for Stochastic Partial Differential Equations, World Scientific Publishing Co. Pte. Ltd.CrossRefGoogle Scholar
Boettinger, C. & Hastings, A. (2012) Quantifying limits to detection of early warning for critical transitions. J. R. Soc. Interface 9(75), 2527–2539.CrossRefGoogle Scholar
Carpenter, S. R. & Brock, W. A. (2006) Rising variance: a leading indicator of ecological transition. Ecol. Lett. 9, 311–318.CrossRefGoogle ScholarPubMed
Carpenter, S. R., Cole, J. J., Pace, M. L., Batt, R., Brock, W. A., Cline, T., Coloso, J., Hodgson, J. R., Kitchell, J. F., Seekell, D. A., Smith, L. & Weidel, B. (2011) Early warning signs of regime shifts: a whole-ecosystem experiment. Science 332, 1079–1082.CrossRefGoogle Scholar
Chen, Y., Kolokolnikov, T., Tzou, J. & Gai, C. (2015) Patterned vegetation, tipping points, and the rate of climate change. Eur. J. Appl. Math. 26(6), 945–958.CrossRefGoogle Scholar
Cotilla-Sanchez, E., Hines, P. & Danforth, C. M. (2012) Predicting critical transitions from time series synchrophasor data. IEEE Trans. Smart Grid 3(4), 1832–1840.CrossRefGoogle Scholar
Da Prato, G. (2004) Kolmogorov Equations for Stochastic PDEs, Birkhäuser Basel.CrossRefGoogle Scholar
Da Prato, G. & Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Dakos, V., van Nes, E. H., Donangelo, R., Fort, H. & Scheffer, M. (2009) Spatial correlation as leading indicator of catastrophic shifts. Theor. Ecol. 3(3), 163–174.CrossRefGoogle Scholar
Ditlevsen, P. D. & Johnsen, S. J. (2010) Tipping points: early warning and wishful thinking. Geophys. Res. Lett. 37, 19703.CrossRefGoogle Scholar
Donangelo, R., Fort, H., Dakos, V., Scheffer, M. & Van Nes, E. H. (2010) Early warnings for catastrophic shifts in ecosystems: comparison between spatial and temporal indicators. Int. J. Bif. Chaos 20(2), 315–321.CrossRefGoogle Scholar
Engel, K.-J. & Nagel, R. (2000) Semigroups for Linear Evolution Equations, Springer.Google Scholar
Gowda, K. & Kuehn, C. (2015) Warning signs for pattern-formation in SPDEs. Comm. Nonl. Sci. Numer. Simul. 22(1), 55–69.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY.CrossRefGoogle Scholar
Hohenberg, P. C. & Halperin, B. I. (1977) Theory of dynamic critical phenomena. Rev. Mod. Phys. 49(3), 435.CrossRefGoogle Scholar
Hoyle, R. (2006) Pattern Formation: An Introduction to Methods, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Huang, Y., Kou, G. & Peng, Y. (2017) Nonlinear manifold learning for early warnings in financial markets. Eur. J. Oper. Res. 258(2), 692–702.CrossRefGoogle Scholar
Kielhoefer, H. (2004) Bifurcation Theory: An Introduction with Applications to PDEs, Springer-Verlag New York.CrossRefGoogle Scholar
Kirrmann, P., Schneider, G. & Mielke, A. (1992) The validity of modulation equations for extended systems with cubic nonlinearities. Proc. R. Soc. Edinburgh A 122(1), 85–91.CrossRefGoogle Scholar
Kramer, J. & Ross, J. (1985) Stabilization of unstable states, relaxation, and critical slowing down in a bistable system. J. Chem. Phys. 83(12), 6234–6241.CrossRefGoogle Scholar
Kuehn, C. (2013) A mathematical framework for critical transitions: normal forms, variance and applications. J. Nonlinear Sci. 23(3), 457–510.CrossRefGoogle Scholar
Kuehn, C. (2013) Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts. Theor. Ecol. 6(3), 295–308.CrossRefGoogle Scholar
Kuehn, C. (2015) The curse of instability. Complexity 20(6), 9–14.CrossRefGoogle Scholar
Kuehn, C. (2015) Multiple Time Scale Dynamics, Springer International Publishing Switzerland, 814 p.CrossRefGoogle Scholar
Kuehn, C. (2015) Numerical continuation and SPDE stability for the 2d cubic-quintic Allen-Cahn equation. SIAM/ASA J. Uncertain. Quantif. 3(1), 762–789.CrossRefGoogle Scholar
Kuehn, C., Zschaler, G. & Gross, T. (2015) Early warning signs for saddle-escape transitions in complex networks. Sci. Rep. 5, 13190.CrossRefGoogle ScholarPubMed
Kwasniok, F. (2018) Detecting, anticipating, and predicting critical transitions in spatially extended systems. Chaos 28(3), 033614.CrossRefGoogle ScholarPubMed
Lenton, T. M., Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahmstorf, S. & Schellnhuber, H. J. (2008) Tipping elements in the Earth’s climate system. Proc. Natl. Acad. Sci. USA 105(6), 1786–1793.CrossRefGoogle ScholarPubMed
May, R., Levin, S. A. & Sugihara, G. (2008) Ecology for bankers. Nature 451, 893–895.CrossRefGoogle ScholarPubMed
McSharry, P. E., Smith, L. A. & Tarassenko, L. (2003) Prediction of epileptic seizures. Nat. Med. 9, 241–242.CrossRefGoogle ScholarPubMed
Mormann, F., Andrzejak, R. G., Elger, C. E. & Lehnertz, K. (2007) Seizure prediction: the long and winding road. Brain 130, 314–333.CrossRefGoogle Scholar
O’Regan, S. M. & Drake, J. M. (2013) Theory of early warning signals of disease emergence and leading indicators of elimination. Theor. Ecol. 6(3), 333–357.CrossRefGoogle Scholar
Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York.CrossRefGoogle Scholar
Reed, M. & Simon, B. (1980) Methods of Modern Mathematical Physics I. Functional Analysis, Academic Press Inc, London Ltd.Google Scholar
Sankaran, S., Majumder, S., Kéfi, S. & Guttal, V. (2018) Implications of being discrete and spatial for detecting early warning signals of regime shifts. Ecol. Indic. 94(1), 503–511.CrossRefGoogle Scholar
Scheffer, M., Bascompte, J., Brock, W. A., Brovkhin, V., Carpenter, S. R., Dakos, V., Held, H., van Nes, E. H., Rietkerk, M. & Sugihara, G. (2009) Early-warning signals for critical transitions. Nature 461, 53–59.CrossRefGoogle ScholarPubMed
Schmüdgen, K. (2012) Unbounded Self-adjoint Operators on Hilbert Space, Springer Netherlands.CrossRefGoogle Scholar
van Belzen, J., van de Koppel, J., Kirwan, M. L., van der Wal, D., Herman, P. M., Dakos, V., Kéfi, S., Scheffer, M., Guntenspergen, G. R. & Bouma, T. J. (2017) Vegetation recovery in tidal marshes reveals critical slowing down under increased inundation. Nat. Comm. 8, 15811.CrossRefGoogle ScholarPubMed
Venegas, J. G., Winkler, T., Musch, G., Vidal Melo, M. F., Layfield, D., Tgavalekos, N., Fischman, A. J., Callahan, R. J., Bellani, G. & Harris, R. S. (2005) Self-organized patchiness in asthma as a prelude to catastrophic shifts. Nature 434, 777–782.CrossRefGoogle ScholarPubMed
Wiesenfeld, K. (1985) Noisy precursors of nonlinear instabilities. J. Stat. Phys. 38(5), 1071–1097.CrossRefGoogle Scholar
Zhang, X., Hallerberg, S. & Kuehn, C. (2015) Predictability of critical transitions. Phys. Rev. E 92, 052905.CrossRefGoogle ScholarPubMed