Assessment of Complex System Dynamics via Harmonic Mapping in a Multifractal Paradigm

Abstract

In the present paper, nonlinear behaviors of complex system dynamics from a multifractal perspective of motion are analyzed. In the framework of scale relativity theory, by analyzing the dynamics of complex system entities based on continuous but non-differentiable curves (multifractal curves), both the Schrödinger and Madelung scenarios on the holographic implementations of dynamics are functional and complementary. In the Madelung scenario, the holographic implementation of dynamics (i.e., free of any external or internal constraints) has some important consequences explicated by means of various operational procedures. The selected procedures involve synchronous modes through SL (2R) transformation group based on a hidden symmetry, coherence domains through Riemann manifold embedded with a Poincaré metric based on a parallel transport of direction (in a Levi Civita sense). Other procedures used here relate to the stationary-non-stationary dynamics transition through harmonic mapping from the usual space to the hyperbolic one manifested as cellular and channel type self-structuring. Finally, the Madelung scenario on the holographic implementations of dynamics are discussed with respect to laser-produced plasma dynamics.

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

“…These dynamical systems have been selected from different sources. For example-Computer aided system, Basic experimental science, modeling issues of mechanical system, Mathematical formulation of any system [10][11][12][13]. Montassir, S. et.al.…”

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.…”