A concept of homeomorphic defect for defining mostly conjugate dynamical systems

Area of Science:

• Dynamical Systems Theory
• Mathematical Physics
• Chaos Theory

Background:

• Topological conjugacy is a key equivalence relation for comparing dynamical systems.
• Traditional methods often rely on normed linear spaces (e.g., L(2)) for comparison.
• A fundamental challenge lies in comparing 'toy models' to more complex systems.

Purpose of the Study:

• To generalize a fixed-point iteration scheme for comparing non-conjugate dynamical systems.
• To introduce the concept of a 'commuter' function for dissimilar systems.
• To develop methods for quantifying the degree of non-equivalence between dynamical systems.

Main Methods:

• Generalizing a functional fixed-point iteration scheme.
• Defining and utilizing 'commuter' functions as nonhomeomorphic coordinate changes.
• Quantifying the failure of commuters to be homeomorphisms to measure system dissimilarity.

Main Results:

• A fixed-point iteration scheme yields a limit point, termed a 'commuter,' for nonconjugate systems.
• Commuters translate between dissimilar systems by matching orbit structures.
• Quantifying commuter non-homeomorphism provides a dynamic-respecting comparison metric.

Conclusions:

• Commuters offer a novel framework for comparing non-equivalent dynamical systems.
• This approach respects the intrinsic orbit structures of the systems.
• Provides principled methods for assessing the representativeness of simplified models.