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 $K_{\mathrm{ab}}$ entering into the Jacobi equation. Attention focuses on ensembles of orbit segments evolved in representative two-dimensional 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, $〈\tilde{K}〉$ and ${\sigma }_{\tilde{K}},$ of the (suitably rescaled) trace of $K_{\mathrm{ab}}.$ 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 parametric 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 ${\sigma }_{\tilde{K}}$ 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 $〈\tilde{K}〉$ and ${\sigma }_{\tilde{K}},$ 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 $〈\tilde{K}〉$ and ${\sigma }_{\tilde{K}}$ (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.