Physicists experimentalists use many observations of a phenomenon, which are the unknown equations that describe it, in order to understand the dynamics and obtain information on their future behavior. In this article we study the possibility of reproducing the dynamics of the phenomenon using only a measurement scale. The Whitney immersion theorem ideas are presented and generalization of Sauer for fractal sets to rebuild the asymptotic behavior of the phenomena and to investigate evidence of nonlinear dynamics in the reproduced dynamics using the Brock, Dechert, Scheinkman test (BDS). The applications are made in the financial market which are only known stock prices.

ResumenInformación sobre la complejidad de un fenómeno, puede ser obtenida midiendo sus grados de libertad. Asumiendo que las trayectorias del fenómeno convergen a un conjunto atractor, los grados de libertad del sistema, son los grados de libertad del atractor. En el presente articulo estudiamos la presencia de un conjunto atractor para la dinámica en el mercado financiero y estimamos su dimensión. Serán usadas las ideas de reconstrucción de atractores usando coordenadas de retraso de Takes y las aplicaciones son centradas en el mercado financiero peruano.

The transfer of energy in turbulent flows occurs as a product of breaking of smaller and smaller eddies, this implies that in a spectral formulation, the transfer occurs from small wavenumbers to large wavenumbers. In order to observe the energy cascading, dissipation scales must be reached, which depend on the Reynolds number, this makes direct simulations of the Navier-Stokes equation impractical. Reduced models were investigated in recent years, such as shell models. Shell models are built by mimicking the spectral model respecting the mechanisms that are preserved, such as energy conservation, scaling and symmetries. In this paper, we will use the Sabra shell model for the study of the energy cascading in turbulent flows and we will show numerically that the energy dissipation is approximately −1/3 which is in agreement with the K41 theory.

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