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**1 - 2**of**2**### Project Summary

"... A dynamical theory of turbulence is one of the grand outstanding challenges of classical physics: how are we to describe unstable dynamics of very many degrees of freedom? The recurrent patterns program proposed here is an undertaking very different from most current turbulence research. Here everyt ..."

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A dynamical theory of turbulence is one of the grand outstanding challenges of classical physics: how are we to describe unstable dynamics of very many degrees of freedom? The recurrent patterns program proposed here is an undertaking very different from most current turbulence research. Here everything follows from deterministic dynamics, with no probabilistic velocity distribution, external stochastic forcing, or homogeneity of turbulence assumptions. The theory emphatically does not seek to be universal; its goal is to accurately predict measurable effects of a turbulent flow, such as the frictional drag, for a given system with given physical parameters and boundary conditions. Inspired by the unstable coherent structures observed in turbulence, the proposed theory follows a path also radically different from the long-time PDE simulation approaches. It postulates, in a nutshell, that (mesh size× time) # fields ≈ 10 10 is a vastly too large a number, and that a finite set of building blocks, of order of 10 2 − 10 4 recurrent patterns, is all that is required for the eventual assembly of a predictive capability for non-equilibrium turbulent flows. The key issues are (1) how to identify, and (2) what to do with these patterns? In the initial phase, a new method for determining recurrent patterns will be applied to a turbulent 1-d

### Appendix A A brief history of chaos

"... 1. Arnol’d’s Law: everything that is discovered is named after someone else (including Arnol’d’s law) 2. Berry’s Law: sometimes, the sequence of an-tecedents seems endless. So, nothing is discovered for the first time. 3. Whiteheads’s Law: Everything of importance has been said before by someone who ..."

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1. Arnol’d’s Law: everything that is discovered is named after someone else (including Arnol’d’s law) 2. Berry’s Law: sometimes, the sequence of an-tecedents seems endless. So, nothing is discovered for the first time. 3. Whiteheads’s Law: Everything of importance has been said before by someone who did not discover it. —Sir Michael V. Berry Writing a history of anything is a reckless undertaking, especially a history of something that has preoccupied at one time or other any serious thinker from ancient Sumer to today’s Hong Kong. A mathematician, to take an example, might see it this way: “History of dynamical systems. ” Nevertheless, here comes yet another very imperfect attempt. A.1 Chaos is born I’ll maybe discuss more about its history when I learn more about it.