Non-Differentiable Dynamics in Complex Systems

Maricel Agcop
Gh. Asachi Technical University, Department of Physics, Iasi, Romania

Lacramioara Ochiuz
“Grigore.T. Popa” University of Medicine and Pharmacy from Iasi, Iasi, Romania

Dan Tesloianu
“Grigore.T. Popa” University of Medicine and Pharmacy from Iasi, Iasi, Romania

Călin Gheorghe Buzea
National Institute of Research and Development for Technical Physics, Iaşi, Romania

Ștefan Andrei Irimiciuc
Faculty of Physics, “Alexandru Ioan Cuza” University of Iasi, Iasi, Romania

Series: Systems Engineering Methods, Developments and Technology
BISAC: TEC009000

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The models commonly used to study complex systems are based on the assumption – otherwise unjustified – of the differentiability of the physical quantities that describe it. The success of differentiable models must be understood sequentially, i.e., on domains large enough that differentiability and integrability are valid. But a differential method fails when faced with the physical reality.

In order to describe the physical dynamics of complex systems, but still remain tributary to a differential hypothesis, it is necessary to introduce, in an explicit manner, the resolution scale in the expressions of the physical variables that describe these dynamics and, implicitly, in the fundamental equations of “evolution”. This means that any classically dependent dynamic variable on both spatial coordinates and time becomes, in this new context, dependent also on the resolution scale. In other words, instead of working with a dynamic variable, described through a strictly non-differentiable mathematical function, this book will just work with different approximations of that function, derived through its averaging at different resolution scales. Consequently, any dynamic variable acts as the limit of a function’s family, the function being non-differentiable for a null resolution scale and differentiable for a non-zero resolution scale.

This approach, which is well adapted for applications in the field of complex systems where any real determination is conducted at a finite resolution scale, clearly implies the development both of a new geometric structure and of a physical theory for which the motion laws, invariant to spatial and temporal coordinate transformations, are integrated with scale laws that are invariant at scale transformations. Such a theory that includes the geometric structure based on the above presented assumptions was developed via scale relativity theory, and more recently with an arbitrary constant fractal dimension. Both theories define the “fractal physics models” class (non-differentiable physical models).

The aim of this book is to showcase the far reach of non-differentiable procedures for the analysis of dynamics for a wide range of physical phenomena, starting from the deformable continuous media, fluids dynamics, and cancer growth to drug release mechanisms, approached from a complex system viewpoint.

Preface

Chapter 1. Dynamics in Complex Systems by Means of Fractal Information

Chapter 2. Non-differentiability in Complex Fluids Dynamics

Chapter 3. On a Holographic Type Model for the Evolution of Cancerous Diseases

Chapter 4. Phases in Temporal Multiscale Evolution of Drug Release Mechanism from IPN-Type Chitosan Based Hydrogels

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