Retrospective-Prospective Differential Inclusions and Their Control by the Differential Connection Tensors of Their Evolutions: The Trendometer

There are two different motivations of this study: retrospectiveprospective differential inclusions and differential connection tensors in networks of both continuous-time nonsmooth functions and time series, and several consequences: control of dynamical systems by differential connectionist tensors, detection by a “trendometer” of all local extrema of differentiable functions, but “wild” as the sum of three sines, as well as of economic and financial time series. It provides us a “trend reversal” of the Fermat rule, using the zeros of the derivative for finding all the local extrema of any numerical function of one variable. Instead, the extrema of the primitive of a function allow us to find zeros of the function. Once detected, the trendometer measures the “jerkiness” of their trend reversals. It provides an efficient econometric tool for detecting crisis (e.g., the dot.com and subprime crises), the dates of the trend reversals, and their jerkiness, helping qualitative analysts to focus their attention on the dates when quantitative jerkiness of the extrema is high. The differential connection matrix plays for evolutions (and discrete time series) a dynamic role analogous to the static role played by the covariance matrix of a family of random variables measuring the covariance entries between two random coefficients. Covariance matrices deal with random variables. Differential connection tensors deal with temporal series or continuous-time evolutions. They are therefore different and cannot be compared, since they deal with different mathematical universes.