“Holographic Implementations” in the Complex Fluid Dynamics through a Fractal Paradigm

Abstract

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

Mathematical Modeling and Simulation in Mechanics and Dynamic Systems

“…The statistics of papers submitted to this Special Issue for both published and rejected items are as follows: 23 total submissions, of which 16 were published (69.6%) [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and 7 rejected (30.4%). The authors' geographical distribution is shown in Table 1, where it can be seen that the 67 authors are from 13 different countries.…”

Analytical Study of Series Solutions for the ‘Nonlinear Mechanical System’ with Stability Analysis

“…Not only that, how this kind of structure causes nonlinearity is also delineated with the help of simulation. In the reference [18], Saviuc, A. et.al. examined the 'non-differentiability' type behavior of the mechanical system which is very much beneficial for the future research work.…”

Implications and Consequences of SL(2R) as Invariance Group in the Description of Complex Systems Dynamics from a Multifractal Perspective of Motion

“…The simulations were performed in Maple software followed by data treatment with OriginPro. The meanings of the employed quantities are the usual ones from [8,9]. The results are presented in Figure 1, where we have represented Re (F), Im (F) and F for a fixed resolution scale dependent parameter (ω) and the fractal time.…”

“…From such a perspective, various operational procedures such as group invariants, joint invariant functions, Riemannian type differential geometries, parallel transport of directions in the Levi-Civita sense, harmonic mappings from the usual space to the hyperbolic one become functional in the description of the dynamics of various complex systems. For instance, in the recent papers [8][9][10], assimilating any complex system with a multifractal mathematical object, non-differentiable behaviors of complex systems are analyzed. Then in-phase coherence of the dynamics of any complex system structural units, which implies, for example, special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics assimilated with parallel transport of direction in Levi-Civita sense, special harmonic mapping imposed by means of parallel transport of direction, etc., make possible various scenarios to chaos (period doubling, intermittences, etc.…”