In the present chapter, we show that the use of the nondifferentiable mathematical procedures, developed in the Scale Relativity Theory with constant arbitrary fractal dimension, simplifies very much the dynamics analyses in the case of complex systems. By applying such a procedure to various complex systems dynamics (biological structures, ablation or discharge plasmas, etc.), we are able to observe that it starts from a steady (oscillating state) and as the external factor is varied the system undergoes significant changes. The systems evolve asymptotically through various transition, toward a chaotic regime (like bifurcations or intermittencies), but never reaching it. Another important reveal from the study of the system’s dynamics was the presence of various steady states depending on the resolution scale at which the theoretical investigations are performed.
Part of the book: Fractal Analysis
Various differentiable models are frequently used to describe the dynamics of complex systems (see the kinetic models, fluid models, etc.). Given the complexity of all the physical phenomena involved in the dynamics of such systems, it is required to introduce the dynamic variable dependencies both on the space-time coordinates and on the scale resolutions. Therefore, in this case an adequate theoretical approach may be the use of non-linear physical models either in the form of the Scale Relativity Theory or of the Extended Scale Relativity Theory, i.e., the Scale Relativity Theory with an arbitrary constant fractal dimension. In the framework of the Extended Scale Relativity Theory, fractal velocity field is described both by topological solitons of kink type and by non-topological soliton varieties of breather type. Applications for the blood flow are proposed. The results revealed the directional flow toward the walls, which can explain the thickening effect which is one of the source of arteriosclerosis.
Part of the book: Nonlinear Systems
A theory of space-time is built on a fractal/multifractal variety. Thus, considering that both the spatial coordinates and the time are fractal/multifractal, it is shown that both the energy and the non-differentiable mass of any biostructure depend on both the “state” of the biostructure and a speed limit of constant value. For the dynamics on Peano fractal/multifractal curves and Compton scale resolutions, it is shown that our results are reduced to those of Einstein relativity. In such a context, it has been shown that the “chameleon effect” of cholesterol corresponds to the HDL-LDL state transfer dictated by the spontaneous symmetry breaking through a fractal/multifractal tunnel effect. Then both HDL and LDL become distinct states of the same biostructure as in nuclear physics where proton and neutron are distinct states of the same nucleon.
Part of the book: Progress in Relativity