Chaos. Classical and Quantum by P. Civitanovic

By P. Civitanovic

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We have omitted an infinity of unstable cycles; so why does approximating the dynamics by a finite number of the shortest cycle eigenvalues work so well? The convergence of cycle expansions of dynamical zeta functions is a consequence of the smoothness and analyticity of the underlying flow. 9; the key observation is that the long orbits are shadowed by sequences of shorter orbits. 11) where a and b are symbol sequences of the two shorter cycles. If all orbits are weighted equally (tp = z np ), such combinations cancel exactly; if orbits of similar symbolic dynamics have similar weights, the weights in such combinations almost cancel.

8. A GUIDE TO THE LITERATURE 25 Quantum chaos. Semiclassical propagators, density of states, trace formulas, semiclassical spectral determinants, billiards, semiclassical helium, diffraction, creeping, tunneling, higher-order corrections. This book concentrates on periodic orbit theory. 13], who noted that hidden in the apparent chaos is a rigid skeleton, a tree of cycles (periodic orbits) of increasing lengths and self-similar structure, and suggested that the cycles should be the key to chaotic dynamics.

The phase space can be enlarged, in the hope that in a sufficiently large phase space it is possible to determine an evolution rule, so we imagine that knowing the state of the atmosphere, measured over many points over the entire planet should be sufficient to determine the temperature tomorrow. Even that is not quite true, and we are less hopeful when it comes to stocks. ☞ chapter 12 For a deterministic system almost every point has a unique future, so trajectories cannot intersect. ) for which a trajectory is not defined.

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