Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study
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- 作者：Bernard De Saedeleer (1)
Michel Crucifix (1)
Sebastian Wieczorek (2) - 关键词：Climate models ; Milankovitch ; Oscillator ; Generalised synchronisation ; Lyapunov exponent ; Multistability
- 刊名：Climate Dynamics
- 出版年：2013
- 出版时间：2 - January 2013
- 年：2013
- 卷：40
- 期：1
- 页码：273-294
- 全文大小：2193KB
- 参考文献：1. Abarbanel HDI, Brown R, Kennel MB (1991) Variation of Lyapunov exponents on a strange attractor. J Nonlinear Sci 1(2):175鈥?99 CrossRef
2. Abarbanel HDI, Rulkov NF, Sushchik MM (1996) Generalized synchronization of chaos: the auxiliary system approach. Phys Rev E 53(5):4528鈥?535 CrossRef
3. Arnold V (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York (1988 second edition. English translation of the original russian publication: 鈥淒opolnitel鈥檔ye Glavy Teorii Obyknovennykh Differentsial鈥檔ykh Uravneni卯鈥?(Additional Chapters to the Theory of Ordinary Differential Equations, Moscow: Nauka, 1978))
4. Ashkenazy Y (2006) The role of phase locking in a simple model for glacial dynamics. Clim Dyn 27(4):421鈥?31 CrossRef
5. Balanov A, Janson N, Postnov D, Sosnovtseva O (2009) Synchronization: from simple to complex. Springer, Berlin
6. Barnes B, Grimshaw R (1997) Analytical and numerical studies of the bonhoeffer van der Pol system. ANZIAM J 38(04):427鈥?53 CrossRef
7. Belogortsev AB (1992) Quasiperiodic resonance and bifurcations of tori in the weakly nonlinear duffing oscillator. Physica D 59(4):417鈥?29 CrossRef
8. Benettin G, Galgani L, Giorgilli A, Strelcyn J-M (1980) Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. part 2: Numerical application. Meccanica 15(1):21鈥?0 CrossRef
9. Beno卯t E, Callot J, Diener F, Diener M (1981) Chasse au canard. Collectanea Mathematica 31鈥?2(1鈥?):37鈥?19
10. Benzi R, Parisi G, Sutera A, Vulpiani A (1982) Stochastic resonance in climatic change. Tellus 34(1):10鈥?6 CrossRef
11. Berger AL (1978) Long-term variations of daily insolation and quaternary climatic changes. J Atmos Sci 35:2362鈥?367 CrossRef
12. Braun H, Ditlevsen P, Kurths J (2009) New measures of multimodality for the detection of a ghost stochastic resonance. Chaos 19(4):043132 CrossRef
13. Broecker WS, van Donk J (1970) Insolation changes, ice volumes, and the O18 record in deep-sea cores. Rev Geophys 8(1):169鈥?98 CrossRef
14. Broer HW, Sim贸 C (1998) Hill鈥檚 equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bol Soc Brasil Mat (N.S.) 29:253鈥?93
15. Brown R, Kocarev L (2000) A unifying definition of synchronization for dynamical systems. Chaos 10(2):344鈥?49 CrossRef
16. Bryant P, Brown R, Abarbanel HDI (1990) Lyapunov exponents from observed time series. Phys Rev Lett 65(13):1523鈥?526 CrossRef
17. Chen J-H, Chen W-C (2008) Chaotic dynamics of the fractionally damped van der Pol equation. Chaos Solit Fract 35(1):188鈥?98 CrossRef
18. Crucifix M (2012) Oscillators and relaxation phenomena in Pleistocene climate theory. Philos Trans R Soc A 370:1140鈥?165
19. D鈥橝cunto M (2006) Determination of limit cycles for a modified van der pol oscillator. Mech Res Commun 33(1):93鈥?8 CrossRef
20. DegliEsposti Boschi C, Ortega GJ, Louis E (2002) Discriminating dynamical from additive noise in the van der pol oscillator. Physica D 171(1鈥?):8鈥?8 CrossRef
21. Dijkstra HA, Weijer W, Neelin JD (2003) Imperfections of the three-dimensional thermohaline circulation: hysteresis and unique-state regimes. J Phys Oceanogr 33:2796鈥?814 CrossRef
22. Doedel E, Champneys A, Dercole F, Fairgrieve T, Kuznetsov Y, Oldeman B, Paffenroth R, Sandstede B, Wang X, Zhang C (2009) Auto: software for continuation and bifurcation problems in ordinary differential equations. Technical report, Montreal
23. Donges JF, Zou Y, Marwan N, Kurths J (2009) The backbone of the climate network. EPL 87(4):48007 CrossRef
24. Feudel U, Kurths J, Pikovsky AS (1995) Strange non-chaotic attractor in a quasiperiodically forced circle map. Physica D 88(3鈥?):176鈥?86 CrossRef
25. Ganopolski A, Rahmstorf S (2002) Abrupt glacial climate changes due to stochastic resonance. Phys Rev Lett 88(3):038501 CrossRef
26. Gildor H, Tziperman E (2000) Sea ice as the glacial cycles climate switch: role of seasonal and orbital forcing. Paleoceanography 15:605鈥?15 CrossRef
27. Ginoux J-M, Rossetto B (2006) Differential geometry and mechanics: Applications to chaotic dynamical systems. Int J Bifurcat Chaos 16(4):887鈥?10 CrossRef
28. Glass L, Mackey M (1988) From clocks to chaos: the rhytms of life. Princeton University Press, Princeton
29. Glass L, Sun J (1994) Periodic forcing of a limit-cycle oscillator: fixed points, Arnold tongues, and the global organization of bifurcations. Phys Rev E 50:5077鈥?084 CrossRef
30. Glendinning P, Wiersig J (1999) Fine structure of mode-locked regions of the quasi-periodically forced circle map. Phys Lett A 257(1鈥?):65鈥?9 CrossRef
31. Grasman J, Verhulst F, Shih S (2005) The Lyapunov exponents of the Van der Pol oscillator. Math Methods Appl Sci 28:1131鈥?139 CrossRef
32. Grebogi C, Ott E, Pelikan S, Yorke JA (1984) Strange attractors that are not chaotic. Physica D 13(1鈥?):261鈥?68 CrossRef
33. Guckenheimer J, Haiduc R (2005) Canards at folded node. Mosc Math J 5:91鈥?03
34. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York
35. Guckenheimer J, Hoffman K, Weckesser W (2000) Numerical computation of canards. Int J Bifurcat Chaos 10(12):2269鈥?687 CrossRef
36. Hays JD, Imbrie J, Shackleton NJ (1976) Variations in the earth鈥檚 orbit: pacemaker of ice ages. Science 194:1121鈥?132 CrossRef
37. Hilborn R (2000) Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press, Oxford
38. Huybers P (2007) Glacial variability over the last two millions years: an extended depth-derived age model, continuous obliquity pacing, and the Pleistocene progression. Quat Sci Rev 26:37鈥?5 CrossRef
39. Hyde WT, Peltier WR (1985) Sensitivity experiments with a model of the ice age cycle: the response to harmonic forcing. J Atmos Sci 42(20):2170鈥?188 CrossRef
40. Hyde WT, Peltier WR (1987) Sensitivity experiments with a model of the ice age cycle: the response to Milankovitch forcing. J Atmos Sci 44(10):1351鈥?374 CrossRef
41. Imbrie J, Imbrie JZ (1980) Modelling the climatic response to orbital variations. Science 207:943鈥?53 CrossRef
42. Kantz H, Schreiber T (2004) Nonlinear time series analysis, 2nd edn. Cambridge University Press, Cambridge
43. Kloeden PE (2000) A Lyapunov function for pullback attractors of nonautonomous differential equations. Electronic J Diff Eqns Conf 05:91鈥?02
44. Kosmidis EK, Pakdaman K (2003) An analysis of the reliability phenomenon in the fitzhugh-nagumo model. J Comput Neurosci 14(1):5鈥?2 CrossRef
45. Langa JA, Robinson JC, Su谩rez A (2002) Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15(3):1鈥?7 CrossRef
46. Laskar J, Robutel P, Joutel F, Boudin F, Gastineau M, Correia ACM, Levrard B (2004) A long-term numerical solution for the insolation quantities of the earth. Astronom Astroph 428:261鈥?85 CrossRef
47. Le Treut H, Ghil M (1983) Orbital forcing, climatic interactions and glaciation cycles. J Geophys Res 88(C9):5167鈥?190 CrossRef
48. Lichtenberg AJ, Lieberman MA (1983) Regular and stochastic motion. Springer, New York
49. Lisiecki LE, Raymo ME (2005) A pliocene-pleistocene stack of 57 globally distributed benthic 未18 / O records. Paleoceanography 20:PA1003
50. Lisiecki LE, Raymo ME (2007) Plio-pleistocene climate evolution: trends and transitions in glacial cycle dynamics. Quat Sci Rev 26(1鈥?):56鈥?9 CrossRef
51. Liu H-F, Dai Z-H, Li W-F, Gong X, Yu Z-H (2005) Noise robust estimates of the largest Lyapunov exponent. Phys Lett A 341(1鈥?):119鈥?27 CrossRef
52. Luethi D, Le Floch M, Bereiter B, Blunier T, Barnola J-M, Siegenthaler U, Raynaud D, Jouzel J, Fischer H, Kawamura K, Stocker TF (2008) High-resolution carbon dioxide concentration record 650,000-800,000聽years before present. Nature 453(7193):379鈥?82 CrossRef
53. Marwan N, Donges JF, Zou Y, Donner RV, Kurths J (2009) Complex network approach for recurrence analysis of time series. Phys Lett A 373(46):4246鈥?254 CrossRef
54. McCaffrey DF, Ellner S, Gallant AR, Nychka DW (1992) Estimating the Lyapunov exponent of a chaotic system with nonparametric regression. J Am Stat Assoc 87(419):682鈥?95 CrossRef
55. Mettin R, Parlitz U, Lauterborn W (1993) Bifurcation structure of the driven van der Pol oscillator. Int J Bifurcat Chaos 3(6):1529鈥?555 CrossRef
56. Milankovitch M (1941) Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem. K枚niglich Serbische Akademie, Belgrade
57. Oseledec V (1968) A multiplicative ergodic theorem: Ljapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 19:197鈥?31
58. Osinga H, Wiersig J, Glendinning P, Feudel U (2000) Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map. ArXiv Nonlinear Sci e-prints: http://arxiv.org/abs/nlin/0005032v1
59. Ott E (2002) Chaos in dynamical systems. Cambridge University Press, Cambridge CrossRef
60. Paillard D (1998) The timing of pleistocene glaciations from a simple multiple-state climate model. Nature 391:378鈥?81 CrossRef
61. Paillard D (2001) Glacial cycles: toward a new paradigm. Rev Geophys 39(3):325鈥?46 CrossRef
62. Paillard D, Parrenin F (2004) The Antarctic ice sheet and the triggering of deglaciations. Earth Planet Sci Lett 227:263鈥?71 CrossRef
63. Parlitz U, Lauterborn W (1987) Period-doubling cascades and devil鈥檚 staircases of the driven van der pol oscillator. Phys Rev A 36(3):1428鈥?434 CrossRef
64. Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, New York CrossRef
65. Rahmstorf S, Crucifix M, Ganopolski A, Goosse H, Kamenkovich I, Knutti R, Lohmann G, Marsh R, Mysak LA, Wang Z, Weaver AJ (2005) Thermohaline circulation hysteresis: a model intercomparison. Geophys Res Lett 32:L23605 CrossRef
66. Ramasubramanian K, Sriram MS (2000) A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 139(1-2):72鈥?6 CrossRef
67. Rial JA, Saha R (2011) Modeling abrupt climate change as the interaction between sea ice extent and mean ocean temperature under orbital insolation forcing. In: Rashid H, Polyak L, Mosley-Thompson E (eds) AGU geophysics monograph 193, understanding the causes, mechanisms and extent of abrupt climate change, pp 57鈥?4
68. Rial JA, Yang M (2007) Is the frequency of abrupt climate change modulated by the orbital insolation? In: Hamming S (eds) AGU monograph 173, ocean circulation, mechanisms and impacts, pp 167鈥?74
69. Rosenstein MT, Collins JJ, Luca CJD (1993) A practical method for calculating largest Lyapunov exponents from small datasets. Physica D 65:117鈥?34 CrossRef
70. Ruelle D (1990) Deterministic chaos: the science and the fiction. Proc R Soc A Lond 427:241鈥?48 CrossRef
71. Ruihong L, Wei X, Shuang L (2008) Chaos control and synchronization of the 蠒6-van der pol system driven by external and parametric excitations. Nonlinear Dyn 53(3):261鈥?71 CrossRef
72. Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HDI (1995) Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 51(2):980鈥?94 CrossRef
73. Saltzman B (2002) Dynamical paleoclimatology: generalized theory of global climate change (international geophysics). Academic Press, London
74. Saltzman B, Maasch KA (1990) A first-order global model of late Cenozoic climate. Trans R Soc Edinburgh Earth Sci 81:315鈥?25 CrossRef
75. Saltzman B, Maasch KA (1991) A first-order global model of late Cenozoic climate. II further analysis based on a simplification of the CO2 dynamics. Clim Dyn 5:201鈥?10 CrossRef
76. Saltzman B, Hansen AR, Maasch KA (1984) The late Quaternary glaciations as the response of a 3-component feedback-system to earth-orbital forcing. J Atmos Sci 41(23):3380鈥?389 CrossRef
77. Savi MA (2005) Chaos and order in biomedical rhythms. J Braz Soc Mech Sci Eng 27(2):157鈥?69 CrossRef
78. Shimada I, Nagashima T (1979) A numerical approach to ergodic problem of dissipative dynamical systems. Prog Theor Phys 61(6):1605鈥?616 CrossRef
79. Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity). Studies in nonlinearity, 1st edn. Perseus Books Group, Cambridge
80. Svensson C-M, Coombes S (2009) Mode locking in a spatially extended neuron model: active soma and compartmental tree. Int J Bifurcat Chaos 19(8):2597鈥?607 CrossRef
81. Tziperman E, Gildor H (2003) On the mid-Pleistocene transtion to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times. Paleoceanography 18(1):1001 CrossRef
82. Tziperman E, Raymo ME, Huybers P, Wunsch C (2006) Consequences of pacing the Pleistocene 100 kyr ice ages by nonlinear phase locking to Milankovitch forcing. Paleoceanography 21:PA4206 CrossRef
83. van der Pol B (1926) On relaxation oscillations. Phil Mag 2(11):978鈥?92
84. Wieczorek S (2009) Stochastic bifurcation in noise-driven lasers and Hopf oscillators. Phys Rev E 79(3):036209 CrossRef
85. Wieczorek SM (2011) Noise synchronisation and stochastic bifurcations in lasers. http://arxiv.org/abs/1104.4052
86. Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Texts in applied mathematics, 2nd edn. Springer, Berlin
87. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Physica D 16(3):285鈥?17 CrossRef
88. Wu L, Zhu S, Li J (2006) Synchronization on fast and slow dynamics in drive-response systems. Physica D 223(2):208鈥?13 CrossRef - 作者单位：Bernard De Saedeleer (1)
Michel Crucifix (1)
Sebastian Wieczorek (2)
1. Earth and Life Institute, Georges Lema卯tre Centre for Earth and Climate Research, Universit茅 catholique de Louvain, Louvain-la-Neuve, Belgium
2. Mathematics Research Institute, University of Exeter, Exeter, UK - ISSN：1432-0894
文摘
There is evidence that ice age cycles are paced by astronomical forcing, suggesting some kind of synchronisation phenomenon. Here, we identify the type of such synchronisation and explore systematically its uniqueness and robustness using a simple paleoclimate model akin to the van der Pol relaxation oscillator and dynamical system theory. As the insolation is quite a complex quasiperiodic signal involving different frequencies, the traditional concepts used to define synchronisation to periodic forcing are no longer applicable. Instead, we explore a different concept of generalised synchronisation in terms of (coexisting) synchronised solutions for the forced system, their basins of attraction and instabilities. We propose a clustering technique to compute the number of synchronised solutions, each of which corresponds to a different paleoclimate history. In this way, we uncover multistable synchronisation (reminiscent of phase- or frequency-locking to individual periodic components of astronomical forcing) at low forcing strength, and monostable or unique synchronisation at stronger forcing. In the multistable regime, different initial conditions may lead to different paleoclimate histories. To study their robustness, we analyse Lyapunov exponents that quantify the rate of convergence towards each synchronised solution (local stability), and basins of attraction that indicate critical levels of external perturbations (global stability). We find that even though synchronised solutions are stable on a long term, there exist short episodes of desynchronisation where nearby climate trajectories diverge temporarily (for about 50 kyr). As the attracting trajectory can sometimes lie close to the boundary of its basin of attraction, a small perturbation could quite easily make climate to jump between different histories, reducing the predictability. Our study brings new insight into paleoclimate dynamics and reveals a possibility for the climate system to wander throughout different climatic histories related to preferential synchronisation regimes on obliquity, precession or combinations of both, all over the history of the Pleistocene.