# Modeling and Scale dynamics

by Frédéric Pierret (IMCCE, Observatoire de Paris)

Modeling phenomena, for example in Astronomy, Physics or Biology, always begin with a choice of hypothesis on the dynamics such as determinism, smooth or non-smooth motion etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following:

"With a finite set of data concerning a phenomenon, can we recover its underlying nature?"

From this problem, we introduce in this talk the defnition of multi- scale functions, scale calculus and scale dynamics based on the time-scale calculus. These definitions will be illustrated on the multi-scale Okamoto's functions. The introduced formalism explains why there exists different continuous models associated to an equation with different scale regimes whereas the equation is scale invariant. A typical example of such an equation, is the Euler-Lagrange equation and particularly the Newton's equation which will be discussed. Notably, we obtain a non-linear diffusion equation via the scale Newton's equation and also the non-linear Schroedinger equation via the scale Newton's equation. Under special assumptions, we recover the classical diffusion equation and the Schroedinger equation.

This talk is based on the joint work with Jacky Cresson (LMA/UPPA, France):

J. Cresson and F. Pierret, Multiscale functions, Scale dynamics and Applications to partial differential equations. arXiv, 1509.01048, 2015.