Lack of sufficiently universal theory of SOC stimulates mathematical investigation of this originally physical field of study. Several mathematical models, like sandpiles or continuous energy model, were proposed to study the phenomenon of SOC. We mention important works by D. Dhar, H. Y. Zhang and a recent series of papers by Ph. Blanchard, B. Cessac and T. Krüger. Their formulations made possible to start a thorough mathematical investigation of SOC.
The Project is aiming to perform a further study in the direction of mathematical understanding of SOC. For this the methods of Dynamical Systems theory will be applied. It is especially important to understand how the crucial dynamical invariants, like entropy or Lyapunov spectrum, behave. We plan to stud y the proposed models from this point of view and put special efforts to describe low-dimensional systems.
The system was investigated in a series of papers in “The Journal of Statistical Physics”. However, the results obtained there are either incomplet e (full of hypotheses or lacking the proof) or even wrong. Recently new methods of studying dynamical systems with singularities were proposed. The most important are: Dimensional properties of the measure on attractor and the Multi-Fractal formalism (dev eloped by Y. Pesin, J. Schmeling, J. Buzzi and others). We plan to apply these to study the Zhang model of SOC by developing specifications of the general methods, which are known to fail with direct application.