Subtleties of Turbulence and Chaos and Mathematical and Physical Interpretation

This study elucidates the preliminary mathematical interpretation and differences between turbulence and chaos, proposing a novel mathematical framework to elaborate turbulence and local resonance. Traditionally, Rayleigh principle has been employed to explain physical resonance, positing that system integrity is compromised when forcing frequency equals the system’s natural frequency. However, with the accumulation of experimental data and advancement in observational techniques, a more profound un- derstanding of turbulence’s intrinsic nature has become increasingly imper- ative. Initially, the physical phenomena of turbulence and chaos have been considered the same or overlapping. Gradually, it becomes clear that the corresponding method to describe them differs, which leads to the different scientific technologies and engineering applications. Building upon this foundation, we propose a new conceptual framework for turbulence based on non-dissipative motion theory, which reveals that under specific conditions, local entropy may approach zero or exist in a non-dissipative state. Under these conditions, energy transfer within the system exhibits lossless characteristics, adhering to the physical laws of non-dissipative dynamics. Through the introduction of Hausdorff space concepts, we successfully elu- cidate the essential differences between turbulence and chaos mathemati- cally, proving that despite both exhibiting complexity, fundamental differ- ences exist in their topological structures and transport mechanisms. This research not only enriches non-equilibrium thermodynamics theory but also provides new perspectives for understanding the connection between local non-dissipative states and global dynamical systems, bearing significant im- plications for academic research, engineering applications, and technological development.