Dynamical systems

Ghil, Michael, Tian Ma, and Shouhong Wang. 2001. “Structural bifurcation of 2-D incompressible flows.” Indiana University Mathematics Journal 50: 159–180.
Speich, S, H Dijkstra, and M Ghil. 1995. “Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation.” Nonlinear Processes in Geophysics 2: 241–268. Abstract
Climate - the "coarse-gridded" state of the coupled ocean - atmosphere system - varies on many time and space scales. The challenge is to relate such variation to specific mechanisms and to produce verifiable quantitative explanations. In this paper, we study the oceanic component of the climate system and, in particular, the different circulation regimes of the mid-latitude win driven ocean on the interannual time scale. These circulations are dominated by two counterrotating, basis scale gyres: subtropical and subpolar. Numerical techniques of bifurcation theory are used to stud the multiplicity and stability of the steady-state solution of a wind-driven, double-gyre, reduced-gravity, shallow water model. Branches of stationary solutions and their linear stability are calculated systematically as parameter are varied. This is one of the first geophysical studies i which such techniques are applied to a dynamical system with tens of thousands of degrees of freedom. Multiple stationary solutions obtain as a result of nonlinear interactions between the two main recirculating cell (cyclonic and anticyclonic) of the large- scale double-gyre flow. These equilibria appear for realistic values of the forcing and dissipation parameters. They undergo Hop bifurcation and transition to aperiodic solutions eventually occurs. The periodic and chaotic behaviour is probably related to an increased number of vorticity cells interaction with each other. A preliminary comparison with observations of the Gulf Stream and Kuroshio Extensions suggests that the intern variability of our simulated mid-latitude ocean is a important factor in the observed interannual variability o these two current systems.
Miller, Robert N., Michael Ghil, and François Gauthiez. 1994. “Advanced Data Assimilation in Strongly Nonlinear Dynamical Systems.” Journal of Atmospheric Sciences 51: 1037–1056.
Ghil, Michael. 1994. “Cryothermodynamics: the chaotic dynamics of paleoclimate.” Physica D 77 (1-3). Amsterdam, The Netherlands, The Netherlands: Elsevier Science Publishers B. V.: 130–159.
Jin, F.-F., J. David Neelin, and Michael Ghil. 1994. “El Niño on the Devil's Staircase: Annual subharmonic steps to chaos.” Science 264: 70–72.
Vautard, Robert, and Michael Ghil. 1989. “Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series.” Physica D 35 (3): 395–424. Abstract

We distinguish between two dimensions of a dynamical system given by experimental time series. Statistical dimension gives a theoretical upper bound for the minimal number of degrees of freedom required to describe tje attractor up to the accuracy of the data, taking into account sampling and noise problems. The dynamical dimension is the intrinsic dimension of the attractor and does not depend on the quality of the data. Singular Spectrum Analysis (SSA) provides estimates of the statistical dimension. SSA also describes the main physical phenomena reflected by the data. It gives adaptive spectral filters associated with the dominant oscillations of the system and clarifies the noise characteristics of the data. We apply SSA to four paleoclimatic records. The principal climatic oscillations, and the regime changes in their amplitude are detected. About 10 degrees of freedom are statistically significant in the data. Large noise and insufficient sample length do not allow reliable estimates of the dynamical dimension.

Ghil, Michael, and S. Childress. 1987. Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics. Springer-Verlag, New York/Berlin, 485.
Ghil, Michael, R. Benzi, and G. Parisi, ed. 1985. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland Publ. Co., Amsterdam/New York, 449.
Ghil, Michael, and John Tavantzis. 1983. “Global Hopf Bifurcation in a Simple Climate Model.” Siam Journal on Applied Mathematics 43 (5). Society for Industrial and Applied Mathematics: 1019–1041. Publisher's Version Abstract
The mathematical structure of a simple climate model is investigated. The model is governed by a system of two nonlinear, autonomous differential equations for the evolution in time of global temperature $T$ and meridional ice-sheet extent $L$. The system's solutions are studied by a combination of qualitative reasoning with explicit calculations, both analytical and numerical. For plausible values of the physical parameters, a branch of periodic solutions obtains, which is both orbitally and structurally stable. The amplitude of the stable periodic solutions in $T$ and $L$ correspond roughly to that obtained from proxy records of Quaternary glaciation cycles. The period of these solutions increases along the branch, until it becomes infinite, while the amplitude of the limiting solution is finite. The limiting solution is a homoclinic orbit formed by the reconnecting separatrix of a saddle. The exchange of stability between the branch of periodic solutions and the steady solution from which it arises is studied by a slight simplification of known methods [20], [21].
Ghil, Michael, and H. Le Treut. 1981. “A climate model with cryodynamics and geodynamics.” Journal of Geophysical Research 86: 5262–5270. Publisher's Version
Ghil, Michael, J. Tavantzis S. Coho, K. Bube, and E. Isaacson. 1981. “Dynamic Meteorology: Data Assimilation Methods.” Applied Mathematical Sciences, edited by L. Bengtsson, Michael Ghil, and E. Källén, Dynamic Meteorology - Data Assimilation Methods, 36: 139–224. Springer-Verlag.