Nonlinear models from time series: The earth's mean surface temperature

In the context of nonlinear dynamics and chaos theory, we study the apparent complexity of the earth's mean surface temperature. The apparent complexity refers to the irregular time dependence observed from the monthly measurements of the last 113 years (1356 data). Chaos theory is in some sense a formal method to extract simplicity from complexity, very complex behaviors can result from relatively simple nonlinear differential equations. Our main goal is to extract the possible simplicity underlying the earth's surface temperature time series. We briefly present some basic concepts of nonlinear dynamics and chaos theory including the invariant quantities of chaotic motion, i.e. those quantities not sensitive to changes in initial conditions. We then search for evidence of possible chaotic behavior in the Temperature time series.; The earth's surface mean temperature as a macroscopic natural phenomenon is continuous in time and so it can be naturally modeled by a system of autonomous ordinary differential equations (ODE's). When this system consists of three or more differential equations containing nonlinear terms, we have the minimum (but not always sufficient) conditions for chaotic behavior. The definition of an invariant measure on the attractor allows us to construct an orthonormal base of polynomials to expand the vector field of ODE's. This polynomial orthonormal base eliminates the problem of multiparameter optimization. Additionally, a joint probability criterion to obtain the actual data with a given set of parameters is used to truncate the expansion of the vector field and find the best model that fits the data.; The reconstructed fields are compared with models from first principles for the earth's surface temperature. In particular, for energy balance models that can be reasonably studied in temporal and spatial scales of monthly and global means. The sensitivity of the reconstructed vector field to variations on some reconstruction quantities is also investigated.