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.

# Dynamical systems

Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Physica D. 1989;35 (3) :395–424.Abstract

.
Hopf Bifurcation in Quasi-geostrophic Channel Flow. SIAM J. Appl. Math. 2003;64 (1) :343–368.

.
Did celestial chaos kill the dinosaurs?. The Observatory. 2003;123 (1177) :328–333.

.
Natural disasters impacting a macroeconomic model with endogenous dynamics. Ecological Economics. 2008;68 (1-2) :582–592.Abstract

.
Averaging of time-periodic systems without a small parameter. Discrete and Continuous Dynamical Systems. 2010;14 (4) :753–782.

.
Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland Publ. Co., Amsterdam/New York; 1985 pp. 449.

.
Data assimilation with an extended Kalman filter for impact-produced shock-wave dynamics. Journal of Computational Physics. 2004;196 (2) :705–723.

.
Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. New York: Springer Briefs in Mathematics, Springer; 2015. Publisher's VersionAbstract

.
Successive bifurcations in a simple model of atmospheric zonal-flow vacillation. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2002;12 (2) :300–309.

.
Dynamical origin of low-frequency variability in a highly nonlinear midlatitude coupled model. Journal of Climate. 2006;19 (24).Abstract

.
Successive refinements in long-term integrations of planetary orbits. The Astrophysical Journal. 2003;592 (1) :620.

.
Understanding ENSO variability and its extrema: A delay differential equation approach. In: Extreme Events: Observations, Modeling and Economics. American Geophysical Union & Wiley ; 2015. pp. 63–78.Abstract

.
Dynamic Meteorology: Data Assimilation Methods. In: Applied Mathematical Sciences. Vol. 36. Dynamic Meteorology - Data Assimilation Methods. Springer-Verlag ; 1981. pp. 139–224.

.
Boolean delay equations on networks in economics and the geosciences. International Journal of Bifurcation and Chaos. 2011;21 (12) :3511–3548.

.
Asymptotics of the Coleman-Gurtin model. Discrete and Continuous Dynamical Systems - Series S. 2011;4 (2) :351–369.

.