A mathematical framework for critical transitions: Bifurcations, fastslow systems and stochastic dynamics

Physica D: Nonlinear Phenomena

Volumn 240, Issue 12, 2011, Pages 1020-1035

  Kuehn, Christian ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Bifurcation delay; Critical transition; FokkerPlanck equation; Multiple time scales; Stochastic dynamics; Tipping point

Indexed keywords

BIFURCATION (MATHEMATICS); CLIMATE CHANGE; DIFFERENCE EQUATIONS; DYNAMICAL SYSTEMS; DYNAMICS; FOKKER PLANCK EQUATION;

BIFURCATION DELAY; CRITICAL TRANSITION; MULTIPLE TIME SCALE; STOCHASTIC DYNAMICS; TIPPING POINT;

STOCHASTIC SYSTEMS;

EID: 79958181387     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2011.02.012     Document Type: Article

References (91)

1. M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkhin, S.R. Carpenter, V. Dakos, H. Held, E.H. van Nes, M. Rietkerk, and G. Sugihara Early-warning signals for critical transitions Nature 461 2009 53 59
2. M. Scheffer, S. Carpenter, J.A. Foley, C. Folke, and B. Walker Catastrophic shifts in ecosystems Nature 413 2001 591 596 (Pubitemid 32964052)
3. M. Scheffer, and S.R. Carpenter Catastrophic regime shifts in ecosystems: linking theory to observation TRENDS Ecol. Evol. 18 12 2003 648 656 (Pubitemid 37399486)
4. J.G. Venegas, T. Winkler, G. Musch, M.F. Vidal Melo, D. Layfield, N. Tgavalekos, A.J. Fischman, R.J. Callahan, G. Bellani, and R.S. Harris Self-organized patchiness in asthma as a prelude to catastrophic shifts Nature 434 2005 777 782 (Pubitemid 44739522)
5. P.E. McSharry, L.A. Smith, and L. Tarassenko Prediction of epileptic seizures Nature Med. 9 2003 241 242
6. R. May, S.A. Levin, and G. Sugihara Ecology for bankers Nature 451 2008 893 895
7. J. Bakke, Ø. Lie, E. Heegaard, T. Dokken, G.H. Haug, H.H. Birks, P. Dulski, and T. Nilsen Rapid oceanic and atmospheric changes during the Younger Dryas cold period Nature Geosci. 2 2009 202 205
8. T.M. Lenton, H. Held, E. Kriegler, J.W. Hall, W. Lucht, S. Rahmstorf, and H.J. Schellnhuber Tipping elements in the earth's climate system Proc. Natl. Acad. Sci. USA 105 6 2008 1786 1793 (Pubitemid 351439417)
9. R.B. Alley, J. Marotzke, W.D. Nordhaus, J.T. Overpeck, D.M. Peteet, R.A. Pielke Jr., R.T. Pierrehumbert, P.B. Rhines, T.F. Stocker, L.D. Talley, and J.M. Wallace Abrupt climate change Science 299 2003 2005 2010 (Pubitemid 36383717)
10. M. Scheffer Critical Transitions in Nature and Society 2009 Princeton University Press
11. J. Guckenheimer, and P. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983 Springer
12. Yu.A. Kuznetsov Elements of Applied Bifurcation Theory 3rd edition 2004 Springer
13. J.M.T. Thompson, and J. Sieber Predicting climate tipping as a noisy bifurcation: a review Int. J. Bif. Chaos 2010 28pp.
14. J.M.T. Thompson, and J. Sieber Predicting climate tipping points B. Launder, M. Thompson, Geo-Engineering Climate Change: Environmental Necessity or Pandora's Box 2010 CUP 50 83
15. S.H. Strogatz Nonlinear Dynamics and Chaos 2000 Westview Press
16. N. Berglund, and B. Gentz Stochastic dynamic bifurcations and excitability C. Laing, G. Lord, Stochastic Methods in Neuroscience vol. 2 2009 OUP 65 93
17. N. Berglund, and B. Gentz Metastability in simple climate models: Pathwise analysis of slowly driven langevin equations Stoch. Dyn. 2 2002 327 356
18. V.N. Livina, and T.M. Lenton A modified method for detecing incipient bifurcations in a dynamical system Geophys. Res. Lett. 34 2007 L03712
19. T. Kleinen, H. Held, and G. Petschel-Held The potential role of spectral properties in detecting thresholds in the earth system: application to the thermohaline circulation Ocean Dyn. 53 2003 53 63 (Pubitemid 36770241)
20. C. Kuehn, Multiple Time Scale Dynamics. 2011 (book in preparation).
21. C.K.R.T. Jones Geometric singular perturbation theory Dynamical Systems (Montecatini Terme, 1994) Lecture Notes in Mathematics vol. 1609 1995 Springer 44 118
22. E.F. Mishchenko, and N.Kh. Rozov Differential Equations with Small Parameters and Relaxation Oscillations 1980 Plenum Press (translated from Russian)
23. J. Grasman Asymptotic Methods for Relaxation Oscillations and Applications 1987 Springer
24. N. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Differ. Equ. 31 1979 53 98
25. S. Wiggins Normally Hyperbolic Invariant Manifolds in Dynamical Systems 1994 Springer
26. A.N. Tikhonov Systems of differential equations containing small small parameters in the derivatives Mat. Sbornik N. S. 31 1952 575 586
27. C.K.R.T. Jones, T.J. Kaper, and N. Kopell Tracking invariant manifolds up tp exponentially small errors SIAM J. Math. Anal. 27 2 1996 558 577
28. M. Krupa, P. Szmolyan, Geometric analysis of the singularly perturbed fold, in: Multiple-Time-Scale Dynamical Systems, IMA, vol. 122, 2001, pp. 89116.
29. E. Benot Canards et enlacements Publ. Math. IHES 72 1990 63 91
30. R. Haiduc, Horseshoes in the forced van der Pol equation, Ph.D. Thesis - Cornell University, 2005.
31. C. Kuehn Characterizing slow exit points Electron. J. Differential Equations 2010 106 2010 1 20
32. N. Berglund, and B. Gentz Noise-Induced Phenomena in Slow-Fast Dynamical Systems 2006 Springer
33. E.H. van Nes, and M. Scheffer Slow recovery from perturbations as generic indicator of a nearby catastrophic shift Am. Nat. 169 6 2007 738 747 (Pubitemid 46847329)
34. J. Carr Applications of Centre Manifold Theory 1981 Springer
35. C. Kuehn, A mathematical framework for critical transitions: normal forms, variance and applications, 2011, pp. 155, arXiv: 1101.2908.
36. F. Dumortier, and R. Roussarie Canard cycles and center manifolds Mem. Amer. Math. Soc. 121 577 1996
37. M. Krupa, and P. Szmolyan Extending geometric singular perturbation theory to nonhyperbolic points fold and canard points in two dimensions SIAM J. Math. Anal. 33 2 2001 286 314 (Pubitemid 33787285)
38. A.I. Neishtadt Persistence of stability loss for dynamical bifurcations. I Differ. Equ. Trans. 23 1987 1385 1391
39. A.I. Neishtadt Persistence of stability loss for dynamical bifurcations. II Differ. Equ. Trans. 24 1988 171 176
40. A.I. Neishtadt On the stability loss delay for dynamical bifurcations Discr. Cont. Dyn. Sys. Series S 2 4 2009 897 909
41. S.M. Baer, T. Erneux, and J. Rinzel The slow passage through a hopf bifurcation: delay, memory effects, and resonance SIAM J. Appl. Math. 49 1 1989 55 71
42. M. Krupa, and P. Szmolyan Extending slow manifolds near transcritical and pitchfork singularities Nonlinearity 14 2001 1473 1491 (Pubitemid 33401561)
43. B. Øksendal Stochastic Differential Equations 2003 Springer
44. C. Gardiner Stochastic Methods 4th edition 2009 Springer
45. W. Horsthemke, and R. Lefever Noise-Induced Transitions 2006 Springer
46. H. Risken The FokkerPlanck Equation 1996 Springer
47. L. Arnold Random Dynamical Systems 2003 Springer
48. L. Arnold Random dynamical systems Dynamical systems (Montecatini Terme, 1994) 1995 Springer 1 43
49. L. Arnold, and H. Crauel Random dynamical systems Lyapunov Exponents (Oberwolfach, 1990) 1991 Springer 1 22
50. V. Arajo Random dynamical systems Encyclopedia of Mathematical Physics 2006 Elsevier 330 338
51. N. Sri Namachchivaya Stochastic bifurcation Appl. Math. Comput. 38 1990 101 159
52. N. Berglund, and B. Gentz Geometric singular perturbation theory for stochastic differential equations J. Diff. Equ. 191 2003 1 54 (Pubitemid 36593518)
53. R. Durrett Probability: Theory and Examples 2nd edition 1996 Duxbury
54. B. Schmalfuss, and K.R. Schneider Invariant manifolds for random dynamical systems with slow and fast variables J. Dyn. Diff. Equ. 20 1 2008 133 164
55. Y. Kabanov, and S. Pergamenshchikov Two-Scale Stochastic Systems 2003 Springer
56. M.I. Freidlin, and A.D. Wentzell Random Perturbations of Dynamical Systems 1998 Springer
57. G.R. Terrell Mathematical Statistics: A Unified Introduction 1999 Springer
58. Y.A. Kutoyants Statistical Inference for Ergodic Diffusion Processes 2004 Springer
59. L. Arnold, Recent progress in stochastic bifurcation theory, in: IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics, Springer, 2001, pp. 1527.
60. P. Baxendale Invariant measures for nonlinear stochastic differential equations Lyapunov Exponents. Proceedings, Oberwolfach 1990 1991 Springer 123 140
61. P. Baxendale A stochastic Hopf bifurcation Probab. Theory Related Fields 99 4 1994 581 616
62. H. Crauel, and F. Flandoli Attractors for random dynamical systems Probab. Theory Related Fields 100 3 1994 365 393
63. K.R. Schenk-Hopp Random attractors general properties, existence and applications to stochastic bifurcation theory DCDS-A 4 1 1998 99 130
64. L. Arnold, and P. Boxler Stochastic bifurcation: instructive examples in dimension one Diffusion Processes and Related Problems in Analysis, Volume II: Stochastic Flows 1992 Birkhuser 241 255
65. E.C. Zeeman Stability of dynamical systems Nonlinearity 1 1988 115 155
66. E.C. Zeeman On the classification of dynamical systems Bull. London Math. Soc. 20 1988 545 557
67. G.N. Milstein, and M.V. Tretyakov Stochastic Numerics for Mathematical Physics 2004 Springer
68. P.E. Kloeden, and E. Platen Numerical Solution of Stochastic Differential Equations 2010 Springer
69. B. Lindner, J. Garcia-Ojalvo, A. Neiman, and L. Schimansky-Geier Effects of noise in excitable systems Phys. Rep. 392 2004 321 424 (Pubitemid 38231071)
70. D.J. Highham An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Rev. 43 3 2001 525 546
71. C.B. Muratov, and E. Vanden-Eijnden Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle Chaos 18 2008 015111
72. C.B. Muratov, E. Vanden-Eijnden, and W.E. Self-induced stochastic resonance in excitable systems Physica D 210 2005 227 240 (Pubitemid 41348619)
73. N. Berglund, B. Gentz, C. Kuehn, Hunting French ducks in a noisy environment, 2010, pp. 160, arXiv: 1011.3193.
74. H. Held, and T. Kleinen Detection of climate system bifurcations by degenerate fingerprinting Geophys. Res. Lett. 31 2004 L23207
75. V. Dakos, M. Scheffer, E.H. van Nes, V. Brovkin, V. Petoukhov, and H. Held Slowing down as an early warning signal for abrupt climate change Proc. Natl. Acad. Sci. USA 105 38 2008 14308 14312
76. R. Donangelo, H. Fort, V. Dakos, M. Scheffer, and E.H. Van Nes Early warnings for catastrophic shifts in ecosystems: comparison between spatial and temporal indicators Int. J. Bif. Chaos 20 2 2010 315 321
77. M. Rietkerk, S.C. Dekker, P. de Ruiter, and J. van de Koppel Self-organized patchiness and catastrophic shifts in ecosystems Science 305 2 2004 1926 1929 (Pubitemid 39287984)
78. B. Sandstede, A. Scheel, and C. Wulff Bifurcations and dynamics of spiral waves J. Nonlin. Sci. 9 1999 439 478 (Pubitemid 129559095)
79. R. Hoyle Pattern Formation: An Introduction to Methods 2006 Cambridge University Press
80. J. Garcia-Ojalvo, and J. Sancho Noise in Spatially Extended Systems 1999 Springer
81. H. Keller, and G. Ochs Numerical approximation of random attractors Stochastic Dynamics 1999 Springer 93 116
82. N. Berglund, and B. Gentz On the noise-induced passage through an unstable periodic orbit I: two-level model J. Statist. Phys. 114 5 2004 1577 1618 (Pubitemid 39015525)
83. A.J. Homburg, and B. Sandstede Homoclinic and heteroclinic bifurcations in vector fields H. Broer, F. Takens, B. Hasselblatt, Handbook of Dynamical Systems III 2010 Elsevier 379 524
84. S. Wiggins Global Bifurcations and Chaos 1998 Springer
85. B. Hao Critical slowing down in one-dimensional maps and beyond J. Stat. Phys. 121 5 2005 749 757 (Pubitemid 41788300)
86. K.T. Alligood, T.D. Sauer, and J.A. Yorke Chaos: An Introduction to Dynamical Systems 1996 Springer
87. J. Guckenheimer Singular Hopf bifurcation in systems with two slow variables SIAM J. Appl. Dyn. Syst. 7 4 2008 1355 1377
88. 3 J. Differ. Equ. 177 2001 419 453
89. M. Desroches, J. Guckenheimer, C. Kuehn, B. Krauskopf, H. Osinga, M. Wechselberger, Mixed-mode oscillations with multiple time scales. SIAM Rev., (2011), http://www.mpipks-dresden.mpg.de/~ckuehn/PDFfiles/mmosurveypreprint.pdf (in press).
90. R. Kuske Probability densities for noisy delay bifurcation J. Stat. Phys. 96 3 1999 797 816
91. N. Berglund, and B. Gentz Pathwise description of dynamic pitchfork bifurcations with additive noise Probab. Theory Related Fields 3 2002 341 388 (Pubitemid 36185842)