Geometric Interpretation of Chaos in Two-Dimensional Hamiltonian Systems

This paper exploits the fact that Hamiltonian flows associated with a time-independent H can be viewed as geodesic flows in a curved manifold, so that the problem of stability and the onset of chaos hinge on properties of the curvature Kab entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in representative twodimensional potentials, examining how such properties as orbit type, values of short time Lyapunov exponents χ, complexities of Fourier spectra, and locations of initial conditions on a surface of section correlate with the mean value and dispersion, 〈K̃〉 and σK̃ , of the (suitably rescaled) trace of Kab. Most analyses of chaos in this context have explored the effects of negative curvature, which implies a divergence of nearby trajectories. The aim here is to exploit instead a point stressed recently by Pettini (Phys. Rev. E 47, 828 [1993]), namely that geodesics can be chaotic even if K is everywhere positive, chaos in this case arising as a parameteric instability triggered by regular variations in K along the orbit. For ensembles of fixed energy, containing both regular and chaotic segments, simple patterns exist connecting 〈K̃〉 and σK̃ for different segments both with each other and with the short time χ: Often, but not always, there is a nearly one-to-one correlation between 〈K̃〉 and σK̃ , a plot of these two quantities approximating a simple curve. Overall χ varies smoothly along the curve, certain regions corresponding to regular and “confined” chaotic orbits where χ is especially small. Chaotic segments located furthest from the regular regions tend systematically to have the largest χ’s. The values of 〈K̃〉 and σK̃ (and in some cases χ) for regular orbits also vary smoothly as a function of the “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 one-dimensional Mathieu equation, in which parametric instability is introduced in the simplest possible way.