APPLICATION OF p-ADIC ANALYSIS TO TIME SERIES

Abstract

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree-like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells.

P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

“…Concluding this introductory section we remark that novel applications of ultrametric methods are not restricted to our recent studies in geophysics [4][5][6]. We can also point to a series of studies performed in [30][31][32][33].…”

“…[4][5][6]. On the other hand, by proceeding from the discrete tree-like dynamics, for which the physical meaning of dynamics' parameters are well defined, to "continuous" pseudo-differential equations, we also clarify the physical meaning of their parameters.…”

“…We point out that, although the ultrametric pseudo-differential equations studied in [4][5][6] describe some distinguishing features of propagation of fluids in capillary networks, they did not explicitly reflect coupling with the physical parameters of such networks. Roughly speaking, they were designed on the basis of abstract reasoning about the natural mathematical form of fluid dynamics on the tree-like configuration space.…”

“…The tree-like geometry of percolation networks was always of special interest of Mandelbrot, the father of fractal geometry. In the set of our previous publications [4][5][6] we have discussed the advantages of using the scaling, as well as hierarchical representations of several complex systems. Some simplified models of oil, oil-in-water, and water-in-oil emulsion propagation in capillary networks in random porous media based on their physically real tree-like geometry were proposed [4][5][6].…”

“…Mathematically, such tree-like fractals are represented as ultrametric spaces and there exist well developed theories of fractional (pseudo-)differential equations on such spaces [7][8][9][10][11][12][13][14][15], which were applied in [4][5][6] to represent the dynamics of fluids in these tree-like capillary networks. The class of homogeneous trees with the constant number of vertices, p > 1, leaving each vertex, the so called p-adic trees, is especially useful for theoretical modeling as representing a mathematically simple model which, at the same time, represents the basic features of fluids' propagation through capillary/fractures networks.…”