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.
Highlights
Published: 18 December 2021Nonlinearity is accepted as one of the most fundamental properties of any complex system dynamics
By considering that any complex system dynamics can be assimilated with a mathematical object of multifractal type, various non-linear behaviors in the framework of the scale relativity theory of motion are developed
Exploring at various scale resolutions a hidden symmetry of stationary dynamics in the Madelung description, synchronization modes are seen forming through the SL (2R) group between the complex system entities
Summary
Nonlinearity is accepted as one of the most fundamental properties of any complex system dynamics. In this class of models (non-differentiable), the complex system’s structural unit’s dynamics can be described by continuous but non-differentiable movement curves (multifractal motion curves) These curves exhibit self-similarity as their main property at any of the points forming the curve, which translates into behaviors of holographic type (every part reflects the global system). SL(2R) groups through dynamic synchronization among the complex system structural units, fractal Riemann manifolds induced by fractal cubics and embedded with a Poincaré metric through apolar transport of cubes, and harmonic mapping from the usual space to the hyperbolic one These procedures become operational so that several possible scenarios towards chaos (fractal periodic doubling scenario), but without fully transitioning into chaos, (non-manifest chaos) can be obtained. Accessing complex systems’ nonstationary dynamics is performed thorough harmonic mapping from the usual space to the hyperbolic one
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