ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Parametrically forced Geophysical Model and Strange Non Chaotic Attractor

Journal of Environmental Accounting and Management 8(2) (2019) 305--325 | DOI:10.5890/JAND.2019.06.012

Rajarshi Middya$^{1}$, Asesh Roy Chowdhury$^{2}$

$^{1}$ Department of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata, 700032, India

$^{2}$ Department of Physics, Jadavpur University, Kolkata, 700032, India

Abstract

A classical glacial climate model previously developed by Saltzman et al. [1–4] and Nicollis et al. [5] is analysed when parametric forcing is present. This particular model was actually used in the study of glacier formation. Our analysis mainly focusses on the issues due to the periodic or quasiperiodic variation of these parameters with time. It is observed that this quasiperiodically driven Saltzman model leads to the generation of Strange Nonchaotic Attractor (SNA), which is very interesting because it is neither a periodic state nor a fully chaotic one. Due to this fact, it is usually very difficult to pinpoint its existence and generation. In particular, we focus on an intermittency transitions of type I and on subharmonic bifurcation leading to type III intermittency. The properties of the attractor are characterised by the finite time Lyapunov exponents, its variance and their distributions along with Poincar´e sections. The zone of existence of SNA for different parameter values have been found.

References

1.  [1] Saltzman, B. (2002), Dynamical Paleoclimatology: Generalized Theory of Global Climate Change, International geophysics series. Academic Press.
2.  [2] Saltzman, B. and Maasch, K.A. (1990), A first-order global model of late cenozoic climatic change, Transactions of the Royal Society of Edinburgh: Earth Sciences, 81(4), 315-325.
3.  [3] Saltzman, B. and Verbitsky, M.Y. (1992), Asthenospheric ice-load effects in a global dynamical-system model of the pleistocene climate, Climate Dynamics, 8(1), 1-11.
4.  [4] Saltzman, B. and Verbitsky, M.Y. (1993), Multiple instabilities and modes of glacial rhythmicity in the plio-pleistocene: a general theory of late cenozoic climatic change. Climate Dynamics, 9(1):1-15.
5.  [5] Nicolis, C. and Nicolis, G. (1995), From short-scale atmospheric variability to global climate dynamics:Toward a systematic theory of averaging, Journal of the Atmospheric Sciences, 52(11), 1903-1913.
6.  [6] Lorenz, E. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20(2), 130-141.
7.  [7] Benzi, R., Parisi, R., Sutera, A. and Vulpiani, A. (1982), Stochastic resonance in climatic change, Tellus, 34(1), 10-16.
8.  [8] Bountis, T. (2012), Chaotic Dynamics: Theory and Practice, Nato Science Series B: Springer US.
9.  [9] Crucifix, M. (2013), Why could ice ages be unpredictable? Climate of the Past, 9(5), 2253-2267.
10.  [10] Crucifix, M. (2011), How can a glacial inception be predicted? The Holocene, 21(5), 831-842.
11.  [11] Crucifix, M. (2012), Oscillators and relaxation phenomena in pleistocene climate theory. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 370, 1140-1165.
12.  [12] Ghil, M. (1994), Cryothermodynamics: the chaotic dynamics of paleoclimate. Physica D: Nonlinear Phenomena, 77(1), 130-159.
13.  [13] Hunt, B.R. and Ott, E. (2001), Fractal properties of robust strange nonchaotic attractors, Phys. Rev. Lett., 87, 254101.
14.  [14] Huybers, P. (2009), Pleistocene glacial variability as a chaotic response to obliquity forcing, Climate of the Past, 5(3), 481-488.
15.  [15] Oerlemans, J. (1982), Glacial cycles and ice-sheet modelling. Climatic Change, 4(4), 353-374.
16.  [16] Paillard, D. (1998), The timing of pleistocene glaciations from a simple multiple-state climate model, Nature, 391, 378-381.
17.  [17] Paillard, D (2001), Glacial cycles: Toward a new paradigm, Reviews of Geophysics, 39(3), 325-346.
18.  [18] Paillard, D. and Parrenin, F. (2004), The antarctic ice sheet and the triggering of deglaciations. Earth and Planetary Science Letters, 227(34), 263-271.
19.  [19] Andronov, A.A. (1971), Theory of bifurcations of dynamic systems on a plane, John Wiley.
20.  [20] Grebogi, C., Ott, E., Pelikan, S., and Yorke, J.A. (1984), Strange attractors that are not chaotic, Physica D:Nonlinear Phenomena, 13(1), 261-268.
21.  [21] Wojewoda, J. and Kapitaniak, T. Attractors of Quasiperiodically Forced Systems, World Scientific Series on Nonlinear Science Series A: Volume 12.
22.  [22] Kuznetsov, S., Feudel, U. and Pikovsky, A. (2006), Strange Nonchaotic Attractors Dynamics between Order and Chaos in Quasiperiodically Forced Systems, World Scientific, 2006.
23.  [23] Heagy, J.F. and Hammel, S.M. (1994), The birth of strange nonchaotic attractors. Physica D: Nonlinear Phenomena, 70(1), 140-153.
24.  [24] Kaneko, K. (1984), Fractalization of torus, Progress of Theoretical Physics, 71(5), 1112.