Skip to main content
SpringerLink
Log in

Modeling the Dynamics of Glacial Cycles

  • Chapter
  • First Online:
Mathematics of Planet Earth

Part of the book series: Mathematics of Planet Earth ((MPE,volume 5))

  • 1051 Accesses

Abstract

This chapter is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman (J Geophys Res 95(D2):1955–1963, 1990), which is based on physical arguments and emphasizes the role of atmospheric CO2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO2 concentration, and the volume of the North Atlantic Deep Water. In this chapter, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov–Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov–Takens point into two, with different local dynamics in their neighborhoods.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., et al.: Theory of Bifurcations of Dynamic Systems on a Plane. Israel Program of Scientific Translations, Jerusalem (1971)

    Google Scholar 

  2. Arnold, V.: Instability of dynamical systems with several degrees of freedom. Soviet Math. Dokl. 5, 581–585 (1964)

    Google Scholar 

  3. Ashkenazy, Y., Tziperman, E.: Are the 41 kyr glacial oscillations a linear response to Milankovitch forcing? Quat. Sci. Rev. 23, 1879–1890 (2004)

    Article  Google Scholar 

  4. Ashwin, P., Ditlevsen, P.: The Mid-Pleistocene transition as a generic bifurcation on a slow manifold. Clim. Dyn. 45(9–10), 2683–2695 (2015)

    Article  Google Scholar 

  5. Bogdanov, R.I.: Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9(2), 144–145 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Broer, H.W., Krauskopf, B., Vegter, G.: Global Analysis of Dynamical Systems. Institute of Physics Publishing, London (2001)

    Book  MATH  Google Scholar 

  7. Carr, J.: Applications of Centre Manifold Theory. Springer, New York (1981)

    Book  MATH  Google Scholar 

  8. Chow, Y.K.: Melnikov’s method with applications. Tech. rep., MA thesis, The University of British Columbia (2001)

    Google Scholar 

  9. Clark, P., Alley, R., Pollard, D.: Northern hemisphere ice-sheet influences on global climate change. Science 5442, 1104–1111 (1999)

    Article  Google Scholar 

  10. Cook, K.H.: Climate Dynamics. Princeton University Press, Princeton (2013)

    Google Scholar 

  11. Crucifix, M.: Oscillators and relaxation phenomena in Pleistocene climate theory. Phil. Trans. R. Soc. A 370(1962), 1140–1165 (2012). https://doi.org/10.1098/rsta.2011.0315

    Article  Google Scholar 

  12. Cushman, R., Sanders, J.A.: A codimension two bifurcation with a third order Picard–Fuchs equation. J. Differ. Equ. 59, 243–256 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dangelmayr, G., Guckenheimer, J.: On a four-parameter family of planar vector fields. Arch. Ration. Mech. Anal. 97, 321–352 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dijkstra, H.A.: Nonlinear Climate Dynamics. Cambridge University Press, Berlin, Heidelberg (2013)

    Book  MATH  Google Scholar 

  15. Doedel, E.J.: AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr. Numer. 30, 265–284 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Doedel, E.J., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (i): Bifurcation in finite dimensions. Int. J. Bifurcat. Chaos pp. 493–520 (1991)

    Google Scholar 

  17. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., et al.: AUTO-07P: continuation and bifurcation software for ordinary differential equations. Tech. rep., Concordia University, Montreal, Canada (2007). http://cmvl.cs.concordia.ca/

    Google Scholar 

  18. Dumortier, F., Roussarie, R., Sotomayor, J.: Generic 3-Parameter Families of Planar Vector Fields, Unfoldings of Saddle, Focus and Elliptic Singularities With Nilpotent Linear Parts, vol. 1480, pp. 1–164. Springer Lecture Notes in Mathematics. Springer, Berlin (1991)

    Google Scholar 

  19. Engler, H., Kaper, H.G., Kaper, T.J., Vo, T.: Dynamical systems analysis of the Maasch–Saltzman model of glacial cycles. Phys. D 359, 1–20 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ghil, M.: Cryothermodynamics: the chaotic dynamics of paleoclimate. Phys. D 77, 130–159 (1994)

    Article  Google Scholar 

  21. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd printing, revised and corrected edn. Springer, New York (1985)

    Google Scholar 

  22. Hayes, J.D., Imbrie, J., Shackleton, N.J.: Variations in the Earth’s orbit: pacemaker of the ice ages. Science 194, 1121–1132 (1976)

    Article  Google Scholar 

  23. Holmes, P., Rand, D.: Phase portraits and bifurcations of the non-linear oscillator \(\ddot {x} + (\alpha + \gamma x) \dot {x} + \beta x + \delta x^3 = 0\). Int. J. Nonlinear Mech. 15, 449–458 (1980)

    Google Scholar 

  24. Huybers, P.: Pleistocene glacial variability and the integrated insolation forcing. Science 313, 508–511 (2006)

    Article  Google Scholar 

  25. Huybers, P.: Glacial variability over the last two million years: an extended depth-derived agemodel, continuous obliquity pacing, and the Pleistocene progression. Quat. Sci. Rev. 26, 37–55 (2007)

    Article  Google Scholar 

  26. Huybers, P., Wunsch, C.: Obliquity pacing of the late-Pleistocene glacial cycles. Nature 434, 491–494 (2005)

    Article  Google Scholar 

  27. Imbrie, J., Raymo, M.E., Shackleton, N.J., et al.: On the structure and origin of major glaciation cycles. 1. Linear responses to Milankovitch forcing. Paleoceanography 6, 205–226 (1992)

    Google Scholar 

  28. Kaper, H.G., Engler, H.: Mathematics & Climate. OT131. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)

    Google Scholar 

  29. Khibnik, A., Krauskopf, B., Rousseau, C.: Global study of a family of cubic Liénard equations. Nonlinearity 11, 1505–1519 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kuznetsov, Y.A.: Practical computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcations. Int. J. Bifurcation Chaos 15(11), 3535–3546 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (2013)

    Google Scholar 

  32. Laskar, J., Fienga, A., Gastineau, M., et al.: La2010: a new orbital solution for the long-term motion of the earth. Astron. Astrophys. 532, A89 (2011)

    Article  MATH  Google Scholar 

  33. Le Treut, H., Ghil, M.: Orbital forcing, climatic interactions, and glaciation cycles. J. Geophys. Res. Oceans 88(C9), 5167–5190 (1983). http://dx.doi.org/10.1029/JC088iC09p05167

    Article  Google Scholar 

  34. Lisiecki, L.E., Raymo, M.E.: A Pliocene-Pleistocene stack of 57 globally distributed benthic δ 18O records. Paleoceanography 20(1) (2005). http://dx.doi.org/10.1029/2004PA001071. PA1003

  35. Maasch, K.A., Saltzman, B.: A low-order dynamic model of global climate variability over the full Pleistocene. J. Geophys. Res. 95(D2), 1955–1963 (1990)

    Article  Google Scholar 

  36. Marshall, J., Plumb, R.A.: Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Academic Press, London (2007)

    Google Scholar 

  37. Melnikov, V.: On the stability of the center for time-periodic perturbations. Trans. Moscow Math. Soc. 12, 1–57 (1963)

    MathSciNet  Google Scholar 

  38. Milankovič, M.: Kanon der erdbestrahlung und seine anwendung auf das eiszeitenproblem. Tech. rep., University of Belgrade (1941)

    Google Scholar 

  39. Paillard, D.: Modèles simplifiés pour l’étude de la variabilité de la circulation thermohaline au cours des cycles glaciaire-interglaciaire. Ph.D. thesis, Univ. Paris-Sud (1995)

    Google Scholar 

  40. Paillard, D.: The timing of Pleistocene glaciations from a simple multiple-state climate model. Nature 391, 378–391 (1998)

    Article  Google Scholar 

  41. Paillard, D.: Glacial cycles: toward a new paradigm. Rev. Geophys. 39, 325–346 (2001)

    Article  Google Scholar 

  42. Paillard, D., Parrenin, F.: The Antarctic ice sheet and the triggering of deglaciation. Earth Planet. Sci. Lett. 227, 263–271 (2004)

    Article  Google Scholar 

  43. Petit, J.R., Davis, M., Delaygue, G., et al.: Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature 399, 429–436 (1999)

    Article  Google Scholar 

  44. Poincaré, H.: Sur les équations de la dynamique et le problème des trois corps. Acta Math. 13, 1–270 (1890)

    MathSciNet  MATH  Google Scholar 

  45. Raymo, M.E., Nisancioglu, K.H.: The 41 kyr world: Milankovitch’s other unsolved mystery. Paleoceanography 18(1) (2003). http://dx.doi.org/10.1029/2002PA000791. PA1011

  46. Raymo, M., Oppo, D., Curry, W.: The mid-Pleistocene climate transition: a deep sea carbon isotopic perspective. Paleoceanography 12, 546–559 (1997)

    Article  Google Scholar 

  47. Rutherford, S., D’hondt, S.: Early onset and tropical forcing of 100,000-year Pleistocene glacial cycles. Nature 408(6808), 72–75 (2000)

    Article  Google Scholar 

  48. Saltzman, B.: Carbon dioxide and the δ 18O record of late Quaternary climate change: a global model. Clim. Dyn. 1, 77–85 (1987)

    Article  Google Scholar 

  49. Saltzman, B., Maasch, K.A.: Carbon cycle instability as a cause of the late Pleistocene ice age oscillations: modeling the asymmetric response. Global Biogeochem. Cycles 2, 177–185 (1988)

    Article  Google Scholar 

  50. Saltzman, B., Maasch, K.A.: A first-order global model of late Cenozoic climatic change II. Further analysis based on a simplification of CO2 dynamics. Clim. Dyn. 5, 201–210 (1991)

    Google Scholar 

  51. Shackleton, N.J.: The 100,000-year ice-age cycle identified and found to lag temperature, carbon dioxide and orbital eccentricity. Science 289, 1897–1902 (2000)

    Article  Google Scholar 

  52. Sun, D.Z., Bryan, F.: Climate Dynamics: Why Does Climate Vary? Wiley, Wiley (2013)

    Google Scholar 

  53. Takens, F.: Forced oscillations and bifurcations. Tech. Rep. 3, Mathematics Institute, Rijksuniversiteit Utrecht, the Netherlands (1974). Reprinted in Chapter 1: Broer, H.W., Krauskopf, B., Vegter, G.: Global Analysis of Dynamical Systems. Institute of Physics Publishing, London (2001)

    Google Scholar 

  54. Tziperman, E., Gildor, H.: On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times. Paleoceanography 18(1) (2003). http://dx.doi.org/10.1029/2001pa000627. PA1001

Download references

Acknowledgement

The work of T.J. Kaper and Th. Vo was supported in part by NSF grant DMS-1616064.

Author information

Authors and Affiliations

  1. Mathematics and Statistics, Georgetown University, Washington, DC, USA

    Hans Engler & Hans G. Kaper

  2. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Tasso J. Kaper

  3. School of Mathematics, Monash University, Clayton, Victoria, Australia

    Theodore Vo

Authors
  1. Hans Engler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans G. Kaper
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tasso J. Kaper
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Theodore Vo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans G. Kaper .

Editor information

Editors and Affiliations

  1. Mathematics and Statistics, Georgetown University, Washington, DC, USA

    Hans G. Kaper

  2. DIMACS Center, Rutgers University, Piscataway, NJ, USA

    Fred S. Roberts

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Engler, H., Kaper, H.G., Kaper, T.J., Vo, T. (2019). Modeling the Dynamics of Glacial Cycles. In: Kaper, H., Roberts, F. (eds) Mathematics of Planet Earth. Mathematics of Planet Earth, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-22044-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation