Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications

Area of Science:

• Nonlinear dynamics
• Complexity science
• Spacetime physics

Background:

• Fractal dimensionality quantifies complexity beyond integer dimensions.
• Fractal control mechanisms and implications in nonlinear systems remain unclear.
• Understanding fractal properties is crucial for modeling complex phenomena.

Purpose of the Study:

• To explore fractal control mechanisms and physical implications in spacetime.
• To investigate the relationship between fractal dimensionality and nonlinear dynamics.
• To clarify the measurability of fractals in self-regulating systems.

Main Methods:

• Development of a nonlinear integrated model using Newton's second law.
• Analysis of self-regulating systems within a spacetime context.
• Examination of fixed points, period cycles, and chaos in relation to fractal properties.

Main Results:

• Stochastic fixed points show self-similarity and long-term memory; deterministic fixed points primarily show self-similarity.
• Period cycles and chaos exhibit both long-term memory and self-similarity.
• Fractal levels are influenced by wave indicators (extrinsic vs. intrinsic forces) and system delays.

Conclusions:

• Fractal dimensionality in self-regulating systems is primarily governed by system dynamics and delays.
• Wave indicators significantly control the fractal level of stable fixed points.
• The study suggests that fractals in self-regulating systems can be quantified using integer dimensions.