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

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.

Towards Possible Laminar Channels through Turbulent Atmospheres in a Multifractal Paradigm

In this paper, developments are made towards simulating complex atmospheric behavior using turbulent energy cascade staging models developed through scale relativity theories. Such theoretical considerations imply gauges that describe atmospheric parameters as multifractal functions undertaking scale symmetry breaking at each stage of the turbulent energy cascade. It is found that gauges of higher complexity (in this case, a Riccati-type gauge) can exhibit more complex behavior accordingly, such as both dilation and contraction, but properly parameterizing the solutions formed by these gauges in terms of turbulent staging can be challenging given the multiple constants and parameters. However, it is found that a logistic-type approximation of the multifractal equations of motion that describe turbulent atmospheric entities can be coupled with a model produced by a simpler gauge, and this combination can reveal instances of laminar, or otherwise non-chaotic, behavior in a given turbulent flow at certain scales. Employing the theory with elastic lidar data, quasi-laminar behavior is found in the vicinity of the planetary boundary layer height, and laminar channels are revealed throughout an atmospheric column—these might be used to reveal complex vertical transport behavior in the atmospheric column.

Novel Approach for EEG Signal Analysis in a Multifractal Paradigm of Motions. Epileptic and Eclamptic Seizures as Scale Transitions

Two different operational procedures are proposed for evaluating and predicting the onset of epileptic and eclamptic seizures. The first procedure analyzes the electrical activity of the brain (EEG signals) using nonlinear dynamic methods (the time variations of the standard deviation, the variance, the skewness and the kurtosis; the evolution in time of the spatial–temporal entropy; the variations of the Lyapunov coefficients, etc.). The second operational procedure reconstructs any type of EEG signal through harmonic mappings from the usual space to the hyperbolic one using the time homographic invariance of a multifractal-type Schrödinger equation in the framework of the scale relativity theory (i.e., in a multifractal paradigm of motions). More precisely, the explicit differential descriptions of the brain activity in the form of 2 × 2 matrices with real elements disclose, through the in-phase coherences at various scale resolutions (i.e., as scale transitions), the multitude of brain neuronal dynamics, especially sequences of epileptic and eclamptic seizures. These two operational procedures are not mutually exclusive, but rather become complementary, offering valuable information concerning epileptic and eclamptic seizures. In such context, the prediction of epileptic and eclamptic seizures becomes fundamental for patients not responding to medical treatment and also presenting an increased rate of seizure recurrence.

Multifractality through Non-Markovian Stochastic Processes in the Scale Relativity Theory. Acute Arterial Occlusions as Scale Transitions

By assimilating biological systems, both structural and functional, into multifractal objects, their behavior can be described in the framework of the scale relativity theory, in any of its forms (standard form in Nottale’s sense and/or the form of the multifractal theory of motion). By operating in the context of the multifractal theory of motion, based on multifractalization through non-Markovian stochastic processes, the main results of Nottale’s theory can be generalized (specific momentum conservation laws, both at differentiable and non-differentiable resolution scales, specific momentum conservation law associated with the differentiable–non-differentiable scale transition, etc.). In such a context, all results are explicated through analyzing biological processes, such as acute arterial occlusions as scale transitions. Thus, we show through a biophysical multifractal model that the blocking of the lumen of a healthy artery can happen as a result of the “stopping effect” associated with the differentiable-non-differentiable scale transition. We consider that blood entities move on continuous but non-differentiable (multifractal) curves. We determine the biophysical parameters that characterize the blood flow as a Bingham-type rheological fluid through a normal arterial structure assimilated with a horizontal “pipe” with circular symmetry. Our model has been validated based on experimental clinical data.

Novel Approach for EKG Signals Analysis Based on Markovian and Non-Markovian Fractalization Type in Scale Relativity Theory

Two distinct operational procedures are proposed for diagnosis and tracking of heart disease evolution (in particular atrial fibrillations). The first procedure, based on the application of non-linear dynamic methods (strange attractors, skewness, kurtosis, histograms, Lyapunov exponent, etc.) analyzes the electrical activity of the heart (electrocardiogram signals). The second procedure, based on multifractalization through Markovian and non-Markovian-type stochasticizations in the framework of the scale relativity theory, reconstructs any type of EKG signal by means of harmonic mappings from the usual space to the hyperbolic one. These mappings mime various scale transitions by differential geometries, in Riemann spaces with symmetries of SL(2R)-type. Then, the two operational procedures are not mutually exclusive, but rather become complementary, through their finality, which is gaining valuable information concerning fibrillation crises. As such, the author’s proposed method could be used for developing new models for medical diagnosis and evolution tracking of heart diseases (patterns dynamics, signal reconstruction, etc.).

Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of SL(2R) Type

In the motion fractal theory, the scale relativity dynamics of any complex system are described through various Schrödinger or hydrodynamic type fractal “regimes”. In the one dimensional stationary case of Schrödinger type fractal “regimes”, synchronizations of complex system entities implies a joint invariant function with the simultaneous action of two isomorphic groups of the S L ( 2 R ) type as solutions of Stoka type equations. Among these joint invariant functions, Gaussians become in the Jeans’s sense, probability density (i.e., stochasticity) whenever the information on the complex system analyzed is fragmentary. In the two-dimensional case of hydrodynamic type fractal “regimes” at a non-differentiable scale, the soliton and soliton-kink of fractal type of the velocity field generate the minimal vortex of fractal type that becomes the source of all turbulences in the complex systems dynamics. Some correlations of our model to experimental data were also achieved.