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Geometric Interpretation of Chaos in TwoDimensional Hamiltonian Systems  Henry E. Kandrup
;  Date: 
9 Jul 1997  Subject:  Astrophysics; Chaotic Dynamics  astroph chaodyn nlin.CD  Affiliation:  University of Florida  Abstract:  Timeindependent Hamiltonian flows are viewed as geodesic flows in a curved manifold, so that the onset of chaos hinges on properties of the curvature twoform entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in 2D potentials, examining how various orbital properties correlate with the mean value and dispersion, and k, of the trace K of the curvature. Unlike most analyses, which have attributed chaos to negative curvature, this work exploits the fact that geodesics can be chaotic even if K is everywhere positive, chaos arising as a parameteric instability triggered by regular variations in K along the orbit. For ensembles of fixed energy, with both regular and chaotic segments, simple patterns connect the values of and k for different segments, both with each other and with the short time Lyapunov exponent X. Often, but not always, there is a near oneto one correlation between and k, a plot of these quantities approximating a simple curve. X varies smoothly along this curve, chaotic segments located furthest from the regular regions tending systematically to have the largest X’s. For regular orbits, and k also vary smoothly with ``distance’’ from the chaotic phase space regions, as probed, e.g., by the location of the initial condition on a surface of section. Many of these observed properties can be understood qualitatively in terms of a onedimensional Mathieu equation.  Source:  arXiv, astroph/9707114  Services:  Forum  Review  PDF  Favorites 


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