This book serves as a comprehensive and detailed collection of knowledge on two innovative aspects of our research: ultra Poincaré chaos and alpha labeling. The first concept represents a fundamental trait of dynamical complexity, while the second acts as an algorithmic tool to analyse and construct complexity within the dynamics, probabilities, and geometries of science and industry.

The manuscript aims to provide solid guidance for studying complexities through rigorous mathematical methods. The chaos is derived from the dynamical characteristic of alpha unpredictability, representing a modernized version of Poisson stability motion. It builds upon the foundational work of Poincaré and Birkhoff, incorporating key new insights that expand upon the French genius’s contributions to the recurrence theorem, applicable in contexts such as the three-body problem.

Key Features:

• Explores the complexities found in fractals, dynamical systems, and stochastic processes

• Introduces an innovative analytical method for assessing random processes

• Offers a fresh perspective on fractal geometries

• Highlights the advantages of ultra Poincaré chaos

• Includes historical and philosophical insights