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TL;DR

This paper explores the mathematical modeling of dynamical systems, illustrating their behaviors through examples, and demonstrates that certain fundamental problems in understanding these systems are inherently unsolvable.

Contribution

It introduces the technical framework for analyzing dynamical systems and proves the unsolvability of key problems within this mathematical context.

Findings

Illustrates qualitative and quantitative behaviors of dynamical systems

Introduces the mathematical framework for these systems

Shows certain problems are unsolvable in a rigorous sense

Abstract

This note discusses dynamical systems-systems that evolve through time. We start with two contemporary examples illustrating the qualitative and the quantitative behavior of dynamical systems. These are two broad categories, usually called the study of the smooth behavior and ergodic theory. We then introduce the technical framework necessary to state the problems mathematically. Finally we show that the problems are unsolvable in a rigorous sense.