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By Cvitanovic et al.

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The motivation for recasting the eigenvalue problem in this form is sketched in fig. 10; exponentiation improves analyticity and trades in a divergence of the sect. 1 trace sum for a zero of the spectral determinant. The computation of the zeros of det (s − A) proceeds very much like the computations of sect. 3. 6 From chaos to statistical mechanics While the above replacement of dynamics of individual trajectories by evolution operators which propagate densities might feel like just another bit of mathematical voodoo, actually something very radical has taken place.

Each “comet” represents the same time interval of a trajectory, starting at the tail and ending at the head. The longer the comet, the faster the flow in that region. Newton’s laws, Lagrange’s method, or Hamilton’s method are all familiar procedures for obtaining a set of differential equations for the vector field v(x) that describes the evolution of a mechanical system. An equation that is second or higher order in time can always be rewritten as a set of first order equations. Here we are concerned with a much larger world of general flows, mechanical or not, all defined by a time independent vector field x(t) ˙ = v(x(t)) .

Spatiotemporal dynamical systems. Partial differential equations for dissipative systems, weak amplitude expansions, normal forms, symmetries and bifurcations, pseudospectral methods, spatiotemporal chaos. Quantum chaology. tex 15may2002 26 CHAPTER 1. OVERTURE mulas, semiclassical spectral determinants, billiards, semiclassical helium, diffraction, creeping, tunneling, higher corrections. This book does not discuss the random matrix theory approach to chaos in quantal spectra; no randomness assumptions are made here, rather the goal is to milk the deterministic chaotic dynamics for its full worth.

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Classical and quantum chaos book by Cvitanovic et al.

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