ABSTRACT

Since the publication of B. B. Mandelbrot’s book “The Fractal Geometry of Nature” (1977), much has been written about the use of fractal geometry and multifractal analysis as new mathematical tools to better understand natural shapes, processes and systems. A complementary approach to this question can be found in the book by I. Stewart, “Does God Play Dice?” (1989). Under this eye-catching title Stewart introduces the state of the art of the mathematics that links fractals and multifractals with two major theories of mathematical physics, the theory of dynamical systems and the theory of chaos and nonlinear dynamics. The origins of

Departamento de Matemática Aplicada a la Ingeniería Agronómica, Escuela Técnica Superior de Ingenieros Agrónomos, Avd Complutense s/n, Universidad Politécnica de Madrid, 28040 Madrid, Spain. aE-mail: fernando.sanjose@upm.es bE-mail: j.caniego@upm.es *Corresponding author

the mathematical discipline of dynamical systems can be traced back to Isaac Newton in the second half of the 17th century, while the theory of chaos and nonlinear dynamics was initiated by Henri Poincaré at the end of the 19th century. It is perhaps a coincidence that Newton developed his theory to explain planetary motions, while Poincaré extended it by introducing new geometrical methods to determine the stability of the solar system. Poincaré investigated this question in the reduced setting of the asymptotic dynamics of three bodies moving under the mutual gravitational attraction of Newton’s second law. Nowadays the theory of chaos and nonlinear dynamics is a fundamental in understanding phenomena in physics, chemistry, biology and geosciences. In this context, fractals and multifractals provide mathematical tools to characterize the dynamics of chaotic nonlinear dynamical systems. This framework was introduced by soil scientists (Smeck et al. 1983, Culling 1987, 1988, Ibáñez et al. 1990, Ibáñez and García-Álvarez 1991, Phillips 1993, 1998, 1999, 2001) for Earth surface systems from Jenny’s version of the state-factor model for soil formation (Jenny 1941). This view was initiated in the late 19th century. At that time the Russian geologist, geographer and soil scientist, V.V. Dokuchaev, argued that soils are a unique object produced by the combined action of animals, plants, climate, relief, geology and time.