System Dynamics Control through the Fractal Potential

Abstract

Implications of the fractal potential in the system dynamics using an extended scale relativity model assuming the fractal character of the particle movements, are established. So, in the dissipative approximation of the model it is shown that the fractal potential comes from the non-differentiability of the space-time, i.e. by means of imaginary part of a complex speed field. In the dispersive approximation of the same model, the fractalization of the differential part of the complex speed field induces a normalized fractal potential which controls through coherence the system dynamics. In such context the type I superconductivity results: the temperature dependences of the superconducting parameter, the accumulator effect etc.

Brain Dynamics Explained by Means of Spectral-Structural Neuronal Networks

“…Moreover, its own space (the one generated by the brain) is structurally, a fractal in the most general sense given by Mandelbrot [24]. In such space, the only possible functionalities (which are compatible with the brain structure) are achieved on continuous but non-differentiable curves [21][22][23][24][25][26][27][28].…”

“…If we divide by dt and neglect the terms that contain differential factors (for details see the method from [21][22][23][24][25][26][27][28]) we obtain:…”

Implications of Non-Differentiable Entropy on a Space-Time Manifold

“…If we divide by dτ and neglect the terms that contain differential factors, using the method from [5][6][7][8][9][10][11], we obtain:…”

“…In order to eliminate this contradiction, we will assume that the temporal coordinate of the fractal curve is also a fractal one. Thus, most elements of the non-relativistic approach of scale relativity theory with arbitrary constant fractal dimension, as described in [5][6][7][8][9][10][11], remain valid, but the time differential element dt is now replaced by the proper time differential element dτ . In this way, not only the space, but the entire space-time continuum is considered to be non-differentiable and, therefore, fractal.…”

Dimensionality in Nanostructures by Means of Non-Differentiability

“…One is using fractional integro-differential calculus, [2][3][4][5][6][7] which was applied in different domain of physics. [8][9][10][11][12][13][14][15] The second is the theory of scale relativity, introduced by Lorand Nottale, [16][17][18] as an attempt to tackling the various problems raised by giving up of the hypothesis of differentiability. The latter is the subject of this paper.…”