the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Review Article: Dynamical Systems, Algebraic Topology, and the Climate Sciences
Denisse Sciamarella
Abstract. The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports; and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefitted from remarkable technological advances in computing resources, in throughput as well as storage, and in observational capabilities, regarding both platforms and instruments.
Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of nonautonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change; and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing.
Michael Ghil and Denisse Sciamarella
Status: open (until 12 Apr 2023)
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CC1: 'Complimentary comment on egusphere-2023-216', Paul PUKITE, 24 Feb 2023
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Thank you for this review article for it's mix of perspectives on autonomous versus non-autonomous systems, and of natural versus forced (pullback attractors) responses. It's known that non-linear otherwise chaotic systems can also deterministically follow the forcing applied. This is where the forced response overrides the natural response. Doesn’t mean that it’s easy to figure out what the response is (based partly on “hawkmoth” structural uncertainty ), but like other forced responses, the dependence on initial conditions becomes irrelevant once it synchronizes with the forcing applied. This means that there may be hope in predicting dynamical climate once the patterns of forcing and responses are better understood.
The excerpt attached from “Synchronization in Oscillatory Networks”, Osipov et al (Springer, 2007)
Citation: https://doi.org/10.5194/egusphere-2023-216-CC1 -
CC2: 'Reply on CC1', Paul PUKITE, 24 Feb 2023
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Upload of image excerpt didn't work in the last reply comment so here is a link to the image
https://imagizer.imageshack.com/img923/3113/FbOxei.png
Citation: https://doi.org/10.5194/egusphere-2023-216-CC2 -
AC1: 'Reply to “Complimentary comment on egusphere-2023-216” CC1 & CC2', Denisse Sciamarella, 26 Feb 2023
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Thank you, Dr. Pukite, for your interesting comment on our review paper. Your comment raises two questions: (i) that of structural uncertainty or instability, sometimes labeled the “hawkmoth effect”; and (ii) that of the connection between the theory of nonautonomous and random dynamical systems (NDSs and RDSs) and synchronization theory (e.g., Osipov et al., 2007; Duane et al., 2017).
(i) Structural stability refers to the stability of a system’s behavior under perturbations of its parameters or, more generally, of its governing equations. It is distinct from and complementary to the usual stability of steady states, in particular, or, more generally, to that of other invariant solutions, such as periodic (limit cycles) or quasi-periodic (tori) ones to perturbations in initial conditions. Hence the term hawkmoth effect for the former, which parallels the term “butterfly effect” for the latter. Ghil (1976) referred to these two types of stability as external vs. internal.
Structural stability was introduced into dynamical systems theory by Andronov and Pontryagin (1937) and it is by now well understood for autonomous dynamical systems (Arnold, 1983; Guckenheimer and Holmes, 1983). Similar results are available for NDSs and RDSs (e.g., Caraballo and Han, 2017; Kloeden and Rasmussen, 2011). Our review article is already quite long and we do not plan, at this stage, to add material on this important but somewhat technical topic. The concept of pullback attraction — as opposed to the forward attraction of autonomous systems, which we do present and discuss in our paper — plays a key role in the NDS and RDS case of structural stability, too.
(ii) Synchronization, in its simplest form, is a particular manifestation of an oscillator’s frequency becoming entrained by that of a forcing. A well-known example is that of circadian rhythms in humans and other animals, as well as in plants (Winfree, 1980). Ghil and Childress (1987/2012, Ch. 12) have discussed the more general case of quasi-periodic forcing of a climatic oscillator during the Quaternary glaciation cycles. Riechers et al. (2022) have considered this case from the point of view of NDS theory.
Mutual synchronization between two or more oscillators is another form of this widespread phenomenon in the physical and life sciences. More recently, the limit of this case to large networks and continuous media has been actively considered (Duane et al., 2017, and references therein). It would be of substantial interest to study the connection between synchronization in this limit and NDS theory. Again, such considerations go well beyond what’s feasible in the present paper.
References
1. Andronov, A. A., and L. Pontryagin. Systèmes grossiers. Dokl. Akad. Nauk. SSSR, 14, 247–251, 1937.
2. Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York, 1983.
3. Caraballo, T. and Han, X., Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems. Springer, 2017.
4. Duane, G., C. Grabow, F. Selten, and M. Ghil, Introduction to focus issue: Synchronization in large networks and continuous media—data, models, and supermodels’, Chaos, 27, 126601 (9 pp.), doi: 10.1063/1.5018728 , 2017.
5. Ghil, M.: Climate stability for a Sellers-type model, J. Atmos. Sci., 33, 3–20, 1976.
6. Ghil, M., and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Springer Science & Business Media, doi:10.1007/978-1-4612-1052-8, ISBN 978-F0-387-96475-l, pp. xv + 485. 1987; reissued as an eBook by Springer, ISBN 978-1-4612-1052-8, 2012.
7. Kloeden, P. E., and M. Rasmussen. Nonautonomous Dynamical Systems. American Mathematical Society, 2011.
8. Osipov, G.V., Kurths, J. and Zhou, C., Synchronization in Oscillatory Networks. Springer Science & Business Media, 2007.
9. Riechers, K., T. Mitsui, N. Boers, and M. Ghil, Orbital insolation variations, intrinsic climate variability, and Quaternary glaciations, Clim. Past, 18(1), 863–893, doi:10.5194/cp-18-863-2022, 2022.
10. Winfree, A. T., 1980: The Geometry of Biological Time, Springer, New York, 530 pp.Citation: https://doi.org/10.5194/egusphere-2023-216-AC1
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AC1: 'Reply to “Complimentary comment on egusphere-2023-216” CC1 & CC2', Denisse Sciamarella, 26 Feb 2023
reply
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CC2: 'Reply on CC1', Paul PUKITE, 24 Feb 2023
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Michael Ghil and Denisse Sciamarella
Michael Ghil and Denisse Sciamarella
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