The connection of Lyapunov spectra to fractal dimension is described by Farmer, Ott, and Yorke [Physica D 7 (1983)].  In two classic papers Bennettin [see my papers with Posch in Physical Review A 38, 473 (1988) and 39, 2175 (1989) for complete references] showed how to obtain Lyapunov spectra for many-body systems.  This work generalizes the numerical work of Spotswood Stoddard and Joseph Ford [Physical Review A 8, 1504 (1973)].  Harald Posch and I discovered an elegant Lagrange-Multiplier approach to this problem [Physics Letters A 113, 82 (1985) and 123, 227 (1987)] which has later been rediscovered by several others.  Our more recent work on the Lyapunov Instability of Classical Many-Body Systems was reviewed in the fall of 2005 by Harald at an Astrophysics Conference at Waseda University.  In 2006, as part of a project sponsored by the Academy of Applied Science (Concord, New Hampshire) at Great Basin College (Elko, Nevada), Carol and I compared the local values of the forward-in-time and reversed-in-time Lyapunov exponents with the local extremal phase-space growth rates as determined by a simple Monte Carlo method. The Lyapunov exponents fluctuate more rapidly than do the extremal rates. In correspondence with Florian Grond, stimulated by his 2003 and 2005 work in Chaos, Solitons, and Fractals, we developed a better (cheaper and more nearly accurate) approach based on singular value decomposition of the local dynamical matrix D [ D = df/dx where dx/dt = f(x) ]. The (real) extremal values turn out to depend only on the symmetrized part of the matrix. This approach avoids the typical complex eigenvalues described in our Polish paper . The work using singular values appears in our preprint (Fall 2006). The quantum analog of Lyapunov instability is hard to find beyond the simple correspondence of wave-packet dynamics with the classical trajectory (Ehrenfest's Theorem). For an attempt, using the Galton Board model see [Hoover, Moran, my wife Carol, and Will Evans, Physics Letters A 131, 211 (1988)].

Lyapunov Instability of Dense Lennard-Jones Fluids (with Harald Posch) [Physical Review A 38, 473 (1988)]

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